Patent Publication Number: US-2023137884-A1

Title: Transform-based image coding method and apparatus therefor

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of earlier filing date and right of priority to U.S. Provisional Application No. 63/006,698, filed on Apr. 7, 2020, the contents of which are all hereby incorporated by reference herein in their entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates generally to an image coding technology and, more particularly, to an image coding method based on a transform in an image coding system and an apparatus therefor. 
     BACKGROUND ART 
     Nowadays, the demand for high-resolution and high-quality images/videos such as 4K, 8K or more ultra high definition (UHD) images/videos has been increasing in various fields. As the image/video data becomes higher resolution and higher quality, the transmitted information amount or bit amount increases as compared to the conventional image data. Therefore, when image data is transmitted using a medium such as a conventional wired/wireless broadband line or image/video data is stored using an existing storage medium, the transmission cost and the storage cost thereof are increased. 
     Further, nowadays, the interest and demand for immersive media such as virtual reality (VR), artificial reality (AR) content or hologram, or the like is increasing, and broadcasting for images/videos having image features different from those of real images, such as a game image is increasing. 
     Accordingly, there is a need for a highly efficient image/video compression technique for effectively compressing and transmitting or storing, and reproducing information of high resolution and high quality images/videos having various features as described above. 
     DISCLOSURE 
     Technical Problem 
     A technical problem of the present disclosure is to provide a method and an apparatus for increasing image coding efficiency. 
     Another technical problem of the present disclosure is to provide a method and an apparatus for increasing efficiency in coding an LFNST index. 
     Another technical problem of the present disclosure is to provide a method and apparatus for increasing a coding efficiency of an LFNST index when a current block is of a single type. 
     Technical Solution 
     According to an embodiment of the present disclosure, an image decoding method performed by a decoding apparatus is provided. The method may comprise deriving a transform coefficient for a current block based on the residual information; receiving an LFNST index related to an LFNST matrix in response to a case where an LFNST is applied to the current block; and deriving a modified transform coefficient for the current block based on the LFNST matrix, and the LFNST index may be signaled for a luma component of the current block. 
     The LFNST index may be signaled in a transform unit level for the current block after the transform coefficient for the current block is derived. 
     In response to a case where a tree type of the current block is a dual tree chroma, the LFNST index may not be signaled. 
     In response to a case where an ISP mode is applied to the current block, the LFNST index may be signaled after a transform coefficient for a last partition block of the current block is derived. 
     The method may further comprise receiving an MTS index related to an MTS matrix in response to a case where an MTS is applied to the current block, 
     The MTS index may be signaled immediately after signaling of the LFNST index in the transform unit level. 
     The LFNST index may be signaled in a coding unit level for the current block. 
     According to an embodiment of the present disclosure, an image encoding method performed by an encoding apparatus is provided. The method may comprise deriving residual samples for a current block based on prediction samples; deriving a transform coefficient for the current block from the residual samples based on an LFNST or an MTS; and outputting by encoding an LFNST index related to an LFNST matrix in response to a case where the LFNST is applied to the current block, and the LFNST index may be applied on a luma component of the current block. 
     According to another embodiment of the present disclosure, there may be provided a digital storage medium that stores image data including encoded image information and a bitstream generated according to an image encoding method performed by an encoding apparatus. 
     According to yet another embodiment of the present disclosure, there may be provided a digital storage medium that stores image data including encoded image information and a bitstream to cause a decoding apparatus to perform the image decoding method. 
     Technical Effects 
     According to the present disclosure, it is possible to increase overall image/video compression efficiency. 
     According to the present disclosure, it is possible to increase efficiency in coding an LFNST index. 
     According to the present disclosure, it is possible to increase coding efficiency of an LFNST index according to a tree type of a current block. 
     The effects that can be obtained through specific examples of the present disclosure are not limited to the effects listed above. For example, there may be various technical effects that a person having ordinary skill in the related art can understand or derive from the present disclosure. Accordingly, specific effects of the present disclosure are not limited to those explicitly described in the present disclosure and may include various effects that can be understood or derived from the technical features of the present disclosure. 
    
    
     
       DESCRIPTION OF DIAGRAMS 
         FIG.  1    schematically illustrates an example of a video/image coding system to which the present disclosure is applicable. 
         FIG.  2    is a diagram schematically illustrating a configuration of a video/image encoding apparatus to which the present disclosure is applicable. 
         FIG.  3    is a diagram schematically illustrating a configuration of a video/image decoding apparatus to which the present disclosure is applicable. 
         FIG.  4    illustrates the structure of a content streaming system to which the present disclosure is applied. 
         FIG.  5    schematically illustrates a multiple transform technique according to an embodiment of the present disclosure. 
         FIG.  6    exemplarily shows intra directional modes of 65 prediction directions. 
         FIG.  7    is a diagram for explaining RST according to an embodiment of the present. 
         FIG.  8    is a diagram illustrating a sequence of arranging output data of a forward primary transformation into a one-dimensional vector according to an example. 
         FIG.  9    is a diagram illustrating a sequence of arranging output data of a forward secondary transform into a two-dimensional block according to an example. 
         FIG.  10    is a diagram illustrating wide-angle intra prediction modes according to an embodiment of the present document. 
         FIG.  11    is a diagram illustrating a block shape to which the LFNST is applied. 
         FIG.  12    is a diagram illustrating an arrangement of output data of a forward LFNST according to an example. 
         FIG.  13    illustrates zeroing-out in a block to which a 4×4 LFNST is applied according to an example. 
         FIG.  14    illustrates zeroing-out in a block to which an 8×8 LFNST is applied according to an example. 
         FIG.  15    is a flowchart illustrating an operation of a video decoding apparatus according to an embodiment of the present disclosure. 
         FIG.  16    is a flowchart illustrating an operation of a video encoding apparatus according to an embodiment of the present disclosure. 
     
    
    
     MODE FOR INVENTION 
     While the present disclosure may be susceptible to various modifications and include various embodiments, specific embodiments thereof have been shown in the drawings by way of example and will now be described in detail. However, this is not intended to limit the present disclosure to the specific embodiments disclosed herein. The terminology used herein is for the purpose of describing specific embodiments only, and is not intended to limit technical idea of the present disclosure. The singular forms may include the plural forms unless the context clearly indicates otherwise. The terms such as “include” and “have” are intended to indicate that features, numbers, steps, operations, elements, components, or combinations thereof used in the following description exist, and thus should not be understood as that the possibility of existence or addition of one or more different features, numbers, steps, operations, elements, components, or combinations thereof is excluded in advance. 
     Meanwhile, each component on the drawings described herein is illustrated independently for convenience of description as to characteristic functions different from each other, and however, it is not meant that each component is realized by a separate hardware or software. For example, any two or more of these components may be combined to form a single component, and any single component may be divided into plural components. The embodiments in which components are combined and/or divided will belong to the scope of the patent right of the present disclosure as long as they do not depart from the essence of the present disclosure. 
     Hereinafter, preferred embodiments of the present disclosure will be explained in more detail while referring to the attached drawings. In addition, the same reference signs are used for the same components on the drawings, and repeated descriptions for the same components will be omitted. 
     This document relates to video/image coding. For example, the method/example disclosed in this document may relate to a VVC (Versatile Video Coding) standard (ITU-T Rec. H.266), a next-generation video/image coding standard after VVC, or other video coding related standards (e.g., HEVC (High Efficiency Video Coding) standard (ITU-T Rec. H.265), EVC (essential video coding) standard, AVS2 standard, etc.). 
     In this document, a variety of embodiments relating to video/image coding may be provided, and, unless specified to the contrary, the embodiments may be combined to each other and be performed. 
     In this document, a video may mean a set of a series of images over time. Generally a picture means a unit representing an image at a specific time zone, and a slice/tile is a unit constituting a part of the picture. The slice/tile may include one or more coding tree units (CTUs). One picture may be constituted by one or more slices/tiles. One picture may be constituted by one or more tile groups. One tile group may include one or more tiles. 
     A pixel or a pel may mean a smallest unit constituting one picture (or image). Also, ‘sample’ may be used as a term corresponding to a pixel. A sample may generally represent a pixel or a value of a pixel, and may represent only a pixel/pixel value of a luma component or only a pixel/pixel value of a chroma component. Alternatively, the sample may refer to a pixel value in the spatial domain, or when this pixel value is converted to the frequency domain, it may refer to a transform coefficient in the frequency domain. 
     A unit may represent the basic unit of image processing. The unit may include at least one of a specific region and information related to the region. One unit may include one luma block and two chroma (e.g., cb, cr) blocks. The unit and a term such as a block, an area, or the like may be used in place of each other according to circumstances. In a general case, an M×N block may include a set (or an array) of samples (or sample arrays) or transform coefficients consisting of M columns and N rows. 
     In this document, the term “I” and “,” should be interpreted to indicate “and/or.” For instance, the expression “A/B” may mean “A and/or B.” Further, “A, B” may mean “A and/or B.” Further, “A/B/C” may mean “at least one of A, B, and/or C.” Also, “A/B/C” may mean “at least one of A, B, and/or C.” 
     Further, in the document, the term “or” should be interpreted to indicate “and/or.” For instance, the expression “A or B” may include 1) only A, 2) only B, and/or 3) both A and B. In other words, the term “or” in this document should be interpreted to indicate “additionally or alternatively.” 
     In the present disclosure, “at least one of A and B” may mean “only A”, “only B”, or “both A and B”. In addition, in the present disclosure, the expression “at least one of A or B” or “at least one of A and/or B” may be interpreted as “at least one of A and B”. 
     In addition, in the present disclosure, “at least one of A, B, and C” may mean “only A”, “only B”, “only C”, or “any combination of A, B, and C”. In addition, “at least one of A, B, or C” or “at least one of A, B, and/or C” may mean “at least one of A, B, and C”. 
     In addition, a parenthesis used in the present disclosure may mean “for example”. Specifically, when indicated as “prediction (intra prediction)”, it may mean that “intra prediction” is proposed as an example of “prediction”. That is, “prediction” in the present disclosure is not limited to “intra prediction”, and “intra prediction” may be proposed as an example of “prediction”. In addition, when indicated as “prediction (i.e., intra prediction)”, it may also mean that “intra prediction” is proposed as an example of “prediction”. 
     Technical features individually described in one figure in the present disclosure may be individually implemented or may be simultaneously implemented. 
       FIG.  1    schematically illustrates an example of a video/image coding system to which the present disclosure is applicable. 
     Referring to  FIG.  1   , the video/image coding system may include a first device (source device) and a second device (receive device). The source device may deliver encoded video/image information or data in the form of a file or streaming to the receive device via a digital storage medium or network. 
     The source device may include a video source, an encoding apparatus, and a transmitter. The receive device may include a receiver, a decoding apparatus, and a renderer. The encoding apparatus may be called a video/image encoding apparatus, and the decoding apparatus may be called a video/image decoding apparatus. The transmitter may be included in the encoding apparatus. The receiver may be included in the decoding apparatus. The renderer may include a display, and the display may be configured as a separate device or an external component. 
     The video source may obtain a video/image through a process of capturing, synthesizing, or generating a video/image. The video source may include a video/image capture device and/or a video/image generating device. The video/image capture device may include, for example, one or more cameras, video/image archives including previously captured video/images, or the like. The video/image generating device may include, for example, a computer, a tablet and a smartphone, and may (electronically) generate a video/image. For example, a virtual video/image may be generated through a computer or the like. In this case, the video/image capturing process may be replaced by a process of generating related data. 
     The encoding apparatus may encode an input video/image. The encoding apparatus may perform a series of procedures such as prediction, transform, and quantization for compression and coding efficiency. The encoded data (encoded video/image information) may be output in the form of a bitstream. 
     The transmitter may transmit the encoded video/image information or data output in the form of a bitstream to the receiver of the receive device through a digital storage medium or a network in the form of a file or streaming. The digital storage medium may include various storage mediums such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, and the like. The transmitter may include an element for generating a media file through a predetermined file format, and may include an element for transmission through a broadcast/communication network. The receiver may receive/extract the bitstream and transmit the received/extracted bitstream to the decoding apparatus. 
     The decoding apparatus may decode a video/image by performing a series of procedures such as dequantization, inverse transform, prediction, and the like corresponding to the operation of the encoding apparatus. 
     The renderer may render the decoded video/image. The rendered video/image may be displayed through the display. 
       FIG.  2    is a diagram schematically illustrating a configuration of a video/image encoding apparatus to which the present disclosure is applicable. Hereinafter, what is referred to as the video encoding apparatus may include an image encoding apparatus. 
     Referring to  FIG.  2   , the encoding apparatus  200  may include an image partitioner  210 , a predictor  220 , a residual processor  230 , an entropy encoder  240 , an adder  250 , a filter  260 , and a memory  270 . The predictor  220  may include an inter predictor  221  and an intra predictor  222 . The residual processor  230  may include a transformer  232 , a quantizer  233 , a dequantizer  234 , an inverse transformer  235 . The residual processor  230  may further include a subtractor  231 . The adder  250  may be called a reconstructor or reconstructed block generator. The image partitioner  210 , the predictor  220 , the residual processor  230 , the entropy encoder  240 , the adder  250 , and the filter  260 , which have been described above, may be constituted by one or more hardware components (e.g., encoder chipsets or processors) according to an embodiment. Further, the memory  270  may include a decoded picture buffer (DPB), and may be constituted by a digital storage medium. The hardware component may further include the memory  270  as an internal/external component. 
     The image partitioner  210  may partition an input image (or a picture or a frame) input to the encoding apparatus  200  into one or more processing units. As one example, the processing unit may be called a coding unit (CU). In this case, starting with a coding tree unit (CTU) or the largest coding unit (LCU), the coding unit may be recursively partitioned according to the Quad-tree binary-tree ternary-tree (QTBTTT) structure. For example, one coding unit may be divided into a plurality of coding units of a deeper depth based on the quad-tree structure, the binary-tree structure, and/or the ternary structure. In this case, for example, the quad-tree structure may be applied first and the binary-tree structure and/or the ternary structure may be applied later. Alternatively, the binary-tree structure may be applied first. The coding procedure according to the present disclosure may be performed based on the final coding unit which is not further partitioned. In this case, the maximum coding unit may be used directly as a final coding unit based on coding efficiency according to the image characteristic. Alternatively, the coding unit may be recursively partitioned into coding units of a further deeper depth as needed, so that the coding unit of an optimal size may be used as a final coding unit. Here, the coding procedure may include procedures such as prediction, transform, and reconstruction, which will be described later. As another example, the processing unit may further include a prediction unit (PU) or a transform unit (TU). In this case, the prediction unit and the transform unit may be split or partitioned from the above-described final coding unit. The prediction unit may be a unit of sample prediction, and the transform unit may be a unit for deriving a transform coefficient and/or a unit for deriving a residual signal from a transform coefficient. 
     The unit and a term such as a block, an area, or the like may be used in place of each other according to circumstances. In a general case, an M×N block may represent a set of samples or transform coefficients consisting of M columns and N rows. The sample may generally represent a pixel or a value of a pixel, and may represent only a pixel/pixel value of a luma component, or only a pixel/pixel value of a chroma component. The sample may be used as a term corresponding to a pixel or a pel of one picture (or image). 
     The subtractor  231  subtracts a prediction signal (predicted block, prediction sample array) output from the inter predictor  221  or the intra predictor  222  from an input image signal (original block, original sample array) to generate a residual signal (residual block, residual sample array), and the generated residual signal is transmitted to the transformer  232 . In this case, as shown, a unit which subtracts the prediction signal (predicted block, prediction sample array) from the input image signal (original block, original sample array) in the encoder  200  may be called the subtractor  231 . The predictor may perform prediction on a processing target block (hereinafter, referred to as ‘current block’), and may generate a predicted block including prediction samples for the current block. The predictor may determine whether intra prediction or inter prediction is applied on a current block or CU basis. As discussed later in the description of each prediction mode, the predictor may generate various information relating to prediction, such as prediction mode information, and transmit the generated information to the entropy encoder  240 . The information on the prediction may be encoded in the entropy encoder  240  and output in the form of a bitstream. 
     The intra predictor  222  may predict the current block by referring to samples in the current picture. The referred samples may be located in the neighbor of or apart from the current block according to the prediction mode. In the intra prediction, prediction modes may include a plurality of non-directional modes and a plurality of directional modes. The non-directional modes may include, for example, a DC mode and a planar mode. The directional mode may include, for example, 33 directional prediction modes or 65 directional prediction modes according to the degree of detail of the prediction direction. However, this is merely an example, and more or less directional prediction modes may be used depending on a setting. The intra predictor  222  may determine the prediction mode applied to the current block by using the prediction mode applied to the neighboring block. 
     The inter predictor  221  may derive a predicted block for the current block based on a reference block (reference sample array) specified by a motion vector on a reference picture. At this time, in order to reduce the amount of motion information transmitted in the inter prediction mode, the motion information may be predicted on a block, subblock, or sample basis based on correlation of motion information between the neighboring block and the current block. The motion information may include a motion vector and a reference picture index. The motion information may further include inter prediction direction (L0 prediction, L1 prediction, Bi prediction, etc.) information. In the case of inter prediction, the neighboring block may include a spatial neighboring block existing in the current picture and a temporal neighboring block existing in the reference picture. The reference picture including the reference block and the reference picture including the temporal neighboring block may be same to each other or different from each other. The temporal neighboring block may be called a collocated reference block, a collocated CU (colCU), and the like, and the reference picture including the temporal neighboring block may be called a collocated picture (colPic). For example, the inter predictor  221  may configure a motion information candidate list based on neighboring blocks and generate information indicating which candidate is used to derive a motion vector and/or a reference picture index of the current block. Inter prediction may be performed based on various prediction modes. For example, in the case of a skip mode and a merge mode, the inter predictor  221  may use motion information of the neighboring block as motion information of the current block. In the skip mode, unlike the merge mode, the residual signal may not be transmitted. In the case of the motion information prediction (motion vector prediction, MVP) mode, the motion vector of the neighboring block may be used as a motion vector predictor and the motion vector of the current block may be indicated by signaling a motion vector difference. 
     The predictor  220  may generate a prediction signal based on various prediction methods. For example, the predictor may apply intra prediction or inter prediction for prediction on one block, and, as well, may apply intra prediction and inter prediction at the same time. This may be called combined inter and intra prediction (CIIP). Further, the predictor may be based on an intra block copy (IBC) prediction mode, or a palette mode in order to perform prediction on a block. The IBC prediction mode or palette mode may be used for content image/video coding of a game or the like, such as screen content coding (SCC). Although the IBC basically performs prediction in a current block, it can be performed similarly to inter prediction in that it derives a reference block in a current block. That is, the IBC may use at least one of inter prediction techniques described in the present disclosure. 
     The prediction signal generated through the inter predictor  221  and/or the intra predictor  222  may be used to generate a reconstructed signal or to generate a residual signal. The transformer  232  may generate transform coefficients by applying a transform technique to the residual signal. For example, the transform technique may include at least one of a discrete cosine transform (DCT), a discrete sine transform (DST), a Karhunen-Loéve transform (KLT), a graph-based transform (GBT), or a conditionally non-linear transform (CNT). Here, the GBT means transform obtained from a graph when relationship information between pixels is represented by the graph. The CNT refers to transform obtained based on a prediction signal generated using all previously reconstructed pixels. In addition, the transform process may be applied to square pixel blocks having the same size or may be applied to blocks having a variable size rather than the square one. 
     The quantizer  233  may quantize the transform coefficients and transmit them to the entropy encoder  240 , and the entropy encoder  240  may encode the quantized signal (information on the quantized transform coefficients) and output the encoded signal in a bitstream. The information on the quantized transform coefficients may be referred to as residual information. The quantizer  233  may rearrange block type quantized transform coefficients into a one-dimensional vector form based on a coefficient scan order, and generate information on the quantized transform coefficients based on the quantized transform coefficients of the one-dimensional vector form. The entropy encoder  240  may perform various encoding methods such as, for example, exponential Golomb, context-adaptive variable length coding (CAVLC), context-adaptive binary arithmetic coding (CABAC), and the like. The entropy encoder  240  may encode information necessary for video/image reconstruction other than quantized transform coefficients (e.g. values of syntax elements, etc.) together or separately. Encoded information (e.g., encoded video/image information) may be transmitted or stored on a unit basis of a network abstraction layer (NAL) in the form of a bitstream. The video/image information may further include information on various parameter sets such as an adaptation parameter set (APS), a picture parameter set (PPS), a sequence parameter set (SPS), a video parameter set (VPS) or the like. Further, the video/image information may further include general constraint information. In the present disclosure, information and/or syntax elements which are transmitted/signaled to the decoding apparatus from the encoding apparatus may be included in video/image information. The video/image information may be encoded through the above-described encoding procedure and included in the bitstream. The bitstream may be transmitted through a network, or stored in a digital storage medium. Here, the network may include a broadcast network, a communication network and/or the like, and the digital storage medium may include various storage media such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, and the like. A transmitter (not shown) which transmits a signal output from the entropy encoder  240  and/or a storage (not shown) which stores it may be configured as an internal/external element of the encoding apparatus  200 , or the transmitter may be included in the entropy encoder  240 . 
     Quantized transform coefficients output from the quantizer  233  may be used to generate a prediction signal. For example, by applying dequantization and inverse transform to quantized transform coefficients through the dequantizer  234  and the inverse transformer  235 , the residual signal (residual block or residual samples) may be reconstructed. The adder  155  adds the reconstructed residual signal to a prediction signal output from the inter predictor  221  or the intra predictor  222 , so that a reconstructed signal (reconstructed picture, reconstructed block, reconstructed sample array) may be generated. When there is no residual for a processing target block as in a case where the skip mode is applied, the predicted block may be used as a reconstructed block. The adder  250  may be called a reconstructor or a reconstructed block generator. The generated reconstructed signal may be used for intra prediction of a next processing target block in the current block, and as described later, may be used for inter prediction of a next picture through filtering. 
     Meanwhile, in the picture encoding and/or reconstructing process, luma mapping with chroma scaling (LMCS) may be applied. 
     The filter  260  may improve subjective/objective video quality by applying the filtering to the reconstructed signal. For example, the filter  260  may generate a modified reconstructed picture by applying various filtering methods to the reconstructed picture, and may store the modified reconstructed picture in the memory  270 , specifically in the DPB of the memory  270 . The various filtering methods may include, for example, deblocking filtering, sample adaptive offset, an adaptive loop filter, a bilateral filter or the like. As discussed later in the description of each filtering method, the filter  260  may generate various information relating to filtering, and transmit the generated information to the entropy encoder  240 . The information on the filtering may be encoded in the entropy encoder  240  and output in the form of a bitstream. 
     The modified reconstructed picture which has been transmitted to the memory  270  may be used as a reference picture in the inter predictor  221 . Through this, the encoding apparatus can avoid prediction mismatch in the encoding apparatus  100  and a decoding apparatus when the inter prediction is applied, and can also improve coding efficiency. 
     The memory  270  DPB may store the modified reconstructed picture in order to use it as a reference picture in the inter predictor  221 . The memory  270  may store motion information of a block in the current picture, from which motion information has been derived (or encoded) and/or motion information of blocks in an already reconstructed picture. The stored motion information may be transmitted to the inter predictor  221  to be utilized as motion information of a neighboring block or motion information of a temporal neighboring block. The memory  270  may store reconstructed samples of reconstructed blocks in the current picture, and transmit them to the intra predictor  222 . 
       FIG.  3    is a diagram schematically illustrating a configuration of a video/image decoding apparatus to which the present disclosure is applicable. 
     Referring to  FIG.  3   , the video decoding apparatus  300  may include an entropy decoder  310 , a residual processor  320 , a predictor  330 , an adder  340 , a filter  350  and a memory  360 . The predictor  330  may include an inter predictor  331  and an intra predictor  332 . The residual processor  320  may include a dequantizer  321  and an inverse transformer  321 . The entropy decoder  310 , the residual processor  320 , the predictor  330 , the adder  340 , and the filter  350 , which have been described above, may be constituted by one or more hardware components (e.g., decoder chipsets or processors) according to an embodiment. Further, the memory  360  may include a decoded picture buffer (DPB), and may be constituted by a digital storage medium. The hardware component may further include the memory  360  as an internal/external component. 
     When a bitstream including video/image information is input, the decoding apparatus  300  may reconstruct an image correspondingly to a process by which video/image information has been processed in the encoding apparatus of  FIG.  2   . For example, the decoding apparatus  300  may derive units/blocks based on information relating to block partition obtained from the bitstream. The decoding apparatus  300  may perform decoding by using a processing unit applied in the encoding apparatus. Therefore, the processing unit of decoding may be, for example, a coding unit, which may be partitioned along the quad-tree structure, the binary-tree structure, and/or the ternary-tree structure from a coding tree unit or a largest coding unit. One or more transform units may be derived from the coding unit. And, the reconstructed image signal decoded and output through the decoding apparatus  300  may be reproduced through a reproducer. 
     The decoding apparatus  300  may receive a signal output from the encoding apparatus of  FIG.  2    in the form of a bitstream, and the received signal may be decoded through the entropy decoder  310 . For example, the entropy decoder  310  may parse the bitstream to derive information (e.g., video/image information) required for image reconstruction (or picture reconstruction). The video/image information may further include information on various parameter sets such as an adaptation parameter set (APS), a picture parameter set (PPS), a sequence parameter set (SPS), a video parameter set (VPS) or the like. Further, the video/image information may further include general constraint information. The decoding apparatus may decode a picture further based on information on the parameter set and/or the general constraint information. In the present disclosure, signaled/received information and/or syntax elements, which will be described later, may be decoded through the decoding procedure and be obtained from the bitstream. For example, the entropy decoder  310  may decode information in the bitstream based on a coding method such as exponential Golomb encoding, CAVLC, CABAC, or the like, and may output a value of a syntax element necessary for image reconstruction and quantized values of a transform coefficient regarding a residual. More specifically, a CABAC entropy decoding method may receive a bin corresponding to each syntax element in a bitstream, determine a context model using decoding target syntax element information and decoding information of neighboring and decoding target blocks, or information of symbol/bin decoded in a previous step, predict bin generation probability according to the determined context model and perform arithmetic decoding of the bin to generate a symbol corresponding to each syntax element value. Here, the CABAC entropy decoding method may update the context model using information of a symbol/bin decoded for a context model of the next symbol/bin after determination of the context model. Information on prediction among information decoded in the entropy decoder  310  may be provided to the predictor (inter predictor  332  and intra predictor  331 ), and residual values, that is, quantized transform coefficients, on which entropy decoding has been performed in the entropy decoder  310 , and associated parameter information may be input to the residual processor  320 . The residual processor  320  may derive a residual signal (residual block, residual samples, residual sample array). Further, information on filtering among information decoded in the entropy decoder  310  may be provided to the filter  350 . Meanwhile, a receiver (not shown) which receives a signal output from the encoding apparatus may further constitute the decoding apparatus  300  as an internal/external element, and the receiver may be a component of the entropy decoder  310 . Meanwhile, the decoding apparatus according to the present disclosure may be called a video/image/picture coding apparatus, and the decoding apparatus may be classified into an information decoder (video/image/picture information decoder) and a sample decoder (video/image/picture sample decoder). The information decoder may include the entropy decoder  310 , and the sample decoder may include at least one of the dequantizer  321 , the inverse transformer  322 , the adder  340 , the filter  350 , the memory  360 , the inter predictor  332 , and the intra predictor  331 . 
     The dequantizer  321  may output transform coefficients by dequantizing the quantized transform coefficients. The dequantizer  321  may rearrange the quantized transform coefficients in the form of a two-dimensional block. In this case, the rearrangement may perform rearrangement based on an order of coefficient scanning which has been performed in the encoding apparatus. The dequantizer  321  may perform dequantization on the quantized transform coefficients using quantization parameter (e.g., quantization step size information), and obtain transform coefficients. 
     The deqauntizer  322  obtains a residual signal (residual block, residual sample array) by inverse transforming transform coefficients. 
     The predictor may perform prediction on the current block, and generate a predicted block including prediction samples for the current block. The predictor may determine whether intra prediction or inter prediction is applied to the current block based on the information on prediction output from the entropy decoder  310 , and specifically may determine an intra/inter prediction mode. 
     The predictor may generate a prediction signal based on various prediction methods. For example, the predictor may apply intra prediction or inter prediction for prediction on one block, and, as well, may apply intra prediction and inter prediction at the same time. This may be called combined inter and intra prediction (CIIP). In addition, the predictor may perform intra block copy (IBC) for prediction on a block. The intra block copy may be used for content image/video coding of a game or the like, such as screen content coding (SCC). Although the IBC basically performs prediction in a current block, it can be performed similarly to inter prediction in that it derives a reference block in a current block. That is, the IBC may use at least one of inter prediction techniques described in the present disclosure. 
     The intra predictor  331  may predict the current block by referring to the samples in the current picture. The referred samples may be located in the neighbor of or apart from the current block according to the prediction mode. In the intra prediction, prediction modes may include a plurality of non-directional modes and a plurality of directional modes. The intra predictor  331  may determine the prediction mode applied to the current block by using the prediction mode applied to the neighboring block. 
     The inter predictor  332  may derive a predicted block for the current block based on a reference block (reference sample array) specified by a motion vector on a reference picture. At this time, in order to reduce the amount of motion information transmitted in the inter prediction mode, the motion information may be predicted on a block, subblock, or sample basis based on correlation of motion information between the neighboring block and the current block. The motion information may include a motion vector and a reference picture index. The motion information may further include inter prediction direction (L0 prediction, L1 prediction, Bi prediction, etc.) information. In the case of inter prediction, the neighboring block may include a spatial neighboring block existing in the current picture and a temporal neighboring block existing in the reference picture. For example, the inter predictor  332  may configure a motion information candidate list based on neighboring blocks, and derive a motion vector and/or a reference picture index of the current block based on received candidate selection information. Inter prediction may be performed based on various prediction modes, and the information on prediction may include information indicating a mode of inter prediction for the current block. 
     The adder  340  may generate a reconstructed signal (reconstructed picture, reconstructed block, reconstructed sample array) by adding the obtained residual signal to the prediction signal (predicted block, prediction sample array) output from the predictor  330 . When there is no residual for a processing target block as in a case where the skip mode is applied, the predicted block may be used as a reconstructed block. 
     The adder  340  may be called a reconstructor or a reconstructed block generator. The generated reconstructed signal may be used for intra prediction of a next processing target block in the current block, and as described later, may be output through filtering or be used for inter prediction of a next picture. 
     Meanwhile, in the picture decoding process, luma mapping with chroma scaling (LMCS) may be applied. 
     The filter  350  may improve subjective/objective video quality by applying the filtering to the reconstructed signal. For example, the filter  350  may generate a modified reconstructed picture by applying various filtering methods to the reconstructed picture, and may transmit the modified reconstructed picture in the memory  360 , specifically in the DPB of the memory  360 . The various filtering methods may include, for example, deblocking filtering, sample adaptive offset, an adaptive loop filter, a bilateral filter or the like. 
     The (modified) reconstructed picture which has been stored in the DPB of the memory  360  may be used as a reference picture in the inter predictor  332 . The memory  360  may store motion information of a block in the current picture, from which motion information has been derived (or decoded) and/or motion information of blocks in an already reconstructed picture. The stored motion information may be transmitted to the inter predictor  260  to be utilized as motion information of a neighboring block or motion information of a temporal neighboring block. The memory  360  may store reconstructed samples of reconstructed blocks in the current picture, and transmit them to the intra predictor  331 . 
     In this specification, the examples described in the predictor  330 , the dequantizer  321 , the inverse transformer  322 , and the filter  350  of the decoding apparatus  300  may be similarly or correspondingly applied to the predictor  220 , the dequantizer  234 , the inverse transformer  235 , and the filter  260  of the encoding apparatus  200 , respectively. 
     As described above, prediction is performed in order to increase compression efficiency in performing video coding. Through this, a predicted block including prediction samples for a current block, which is a coding target block, may be generated. Here, the predicted block includes prediction samples in a space domain (or pixel domain). The predicted block may be identically derived in the encoding apparatus and the decoding apparatus, and the encoding apparatus may increase image coding efficiency by signaling to the decoding apparatus not original sample value of an original block itself but information on residual (residual information) between the original block and the predicted block. The decoding apparatus may derive a residual block including residual samples based on the residual information, generate a reconstructed block including reconstructed samples by adding the residual block to the predicted block, and generate a reconstructed picture including reconstructed blocks. 
     The residual information may be generated through transform and quantization procedures. For example, the encoding apparatus may derive a residual block between the original block and the predicted block, derive transform coefficients by performing a transform procedure on residual samples (residual sample array) included in the residual block, and derive quantized transform coefficients by performing a quantization procedure on the transform coefficients, so that it may signal associated residual information to the decoding apparatus (through a bitstream). Here, the residual information may include value information, position information, a transform technique, transform kernel, a quantization parameter or the like of the quantized transform coefficients. The decoding apparatus may perform a quantization/dequantization procedure and derive the residual samples (or residual sample block), based on residual information. The decoding apparatus may generate a reconstructed block based on a predicted block and the residual block. The encoding apparatus may derive a residual block by dequantizing/inverse transforming quantized transform coefficients for reference for inter prediction of a next picture, and may generate a reconstructed picture based on this. 
       FIG.  4    illustrates the structure of a content streaming system to which the present disclosure is applied. 
     Further, the contents streaming system to which the present disclosure is applied may largely include an encoding server, a streaming server, a web server, a media storage, a user equipment, and a multimedia input device. 
     The encoding server functions to compress to digital data the contents input from the multimedia input devices, such as the smart phone, the camera, the camcoder and the like, to generate a bitstream, and to transmit it to the streaming server. As another example, in a case where the multimedia input device, such as, the smart phone, the camera, the camcoder or the like, directly generates a bitstream, the encoding server may be omitted. The bitstream may be generated by an encoding method or a bitstream generation method to which the present disclosure is applied. And the streaming server may store the bitstream temporarily during a process to transmit or receive the bitstream. 
     The streaming server transmits multimedia data to the user equipment on the basis of a user&#39;s request through the web server, which functions as an instrument that informs a user of what service there is. When the user requests a service which the user wants, the web server transfers the request to the streaming server, and the streaming server transmits multimedia data to the user. In this regard, the contents streaming system may include a separate control server, and in this case, the control server functions to control commands/responses between respective equipments in the content streaming system. 
     The streaming server may receive contents from the media storage and/or the encoding server. For example, in a case the contents are received from the encoding server, the contents may be received in real time. In this case, the streaming server may store the bitstream for a predetermined period of time to provide the streaming service smoothly. 
     For example, the user equipment may include a mobile phone, a smart phone, a laptop computer, a digital broadcasting terminal, a personal digital assistant (PDA), a portable multimedia player (PMP), a navigation, a slate PC, a tablet PC, an ultrabook, a wearable device (e.g., a watch-type terminal (smart watch), a glass-type terminal (smart glass), a head mounted display (HMD)), a digital TV, a desktop computer, a digital signage or the like. Each of servers in the contents streaming system may be operated as a distributed server, and in this case, data received by each server may be processed in distributed manner. 
       FIG.  5    schematically illustrates a multiple transform technique according to an embodiment of the present disclosure. 
     Referring to  FIG.  5   , a transformer may correspond to the transformer in the foregoing encoding apparatus of  FIG.  2   , and an inverse transformer may correspond to the inverse transformer in the foregoing encoding apparatus of  FIG.  2   , or to the inverse transformer in the decoding apparatus of  FIG.  3   . 
     The transformer may derive (primary) transform coefficients by performing a primary transform based on residual samples (residual sample array) in a residual block (S 510 ). This primary transform may be referred to as a core transform. Herein, the primary transform may be based on multiple transform selection (MTS), and when a multiple transform is applied as the primary transform, it may be referred to as a multiple core transform. 
     The multiple core transform may represent a method of transforming additionally using discrete cosine transform (DCT) type 2 and discrete sine transform (DST) type 7, DCT type 8, and/or DST type 1. That is, the multiple core transform may represent a transform method of transforming a residual signal (or residual block) of a space domain into transform coefficients (or primary transform coefficients) of a frequency domain based on a plurality of transform kernels selected from among the DCT type 2, the DST type 7, the DCT type 8 and the DST type 1. Herein, the primary transform coefficients may be called temporary transform coefficients from the viewpoint of the transformer. 
     That is, when the conventional transform method is applied, transform coefficients may be generated by applying transform from a space domain to a frequency domain to a residual signal (or residual block) based on DCT type 2. However, when the multiple core transform is applied, transform coefficients (or primary transform coefficients) may be generated by applying transform from a space domain to a frequency domain to a residual signal (or residual block) based on DCT type 2, DST type 7, DCT type 8, and/or DST type 1. Here, DCT type 2, DST type 7, DCT type 8, and DST type 1 may be referred to as transform types, transform kernels, or transform cores. These DCT/DST types may be defined based on basis functions. 
     If the multiple core transform is performed, then a vertical transform kernel and a horizontal transform kernel for a target block may be selected from among the transform kernels, a vertical transform for the target block may be performed based on the vertical transform kernel, and a horizontal transform for the target block may be performed based on the horizontal transform kernel. Here, the horizontal transform may represent a transform for horizontal components of the target block, and the vertical transform may represent a transform for vertical components of the target block. The vertical transform kernel/horizontal transform kernel may be adaptively determined based on a prediction mode and/or a transform index of a target block (CU or sub-block) including a residual block. 
     Further, according to an example, if the primary transform is performed by applying the MTS, a mapping relationship for transform kernels may be set by setting specific basis functions to predetermined values and combining basis functions to be applied in the vertical transform or the horizontal transform. For example, when the horizontal transform kernel is expressed as trTypeHor and the vertical direction transform kernel is expressed as trTypeVer, a trTypeHor or trTypeVer value of 0 may be set to DCT2, a trTypeHor or trTypeVer value of 1 may be set to DST-7, and a trTypeHor or trTypeVer value of 2 may be set to DCT-8. 
     In this case, MTS index information may be encoded and signaled to the decoding apparatus to indicate any one of a plurality of transform kernel sets. For example, an MTS index of 0 may indicate that both trTypeHor and trTypeVer values are 0, an MTS index of 1 may indicate that both trTypeHor and trTypeVer values are 1, an MTS index of 2 may indicate that the trTypeHor value is 2 and the trTypeVer value. Is 1, an MTS index of 3 may indicate that the trTypeHor value is 1 and the trTypeVer value is 2, and an MTS index of 4 may indicate that both both trTypeHor and trTypeVer values are 2. 
     In one example, transform kernel sets according to MTS index information are illustrated in the following table. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 tu_mts_idx[ x0 ][ y0 ] 
                 0 
                 1 
                 2 
                 3 
                 4 
               
               
                   
                   
               
             
            
               
                   
                 trTypeHor 
                 0 
                 1 
                 2 
                 1 
                 2 
               
               
                   
                 trTypeVer 
                 0 
                 1 
                 1 
                 2 
                 2 
               
               
                   
                   
               
            
           
         
       
     
     The transformer may derive modified (secondary) transform coefficients by performing the secondary transform based on the (primary) transform coefficients (S 520 ). The primary transform is a transform from a spatial domain to a frequency domain, and the secondary transform refers to transforming into a more compressive expression by using a correlation existing between (primary) transform coefficients. The secondary transform may include a non-separable transform. In this case, the secondary transform may be called a non-separable secondary transform (NSST), or a mode-dependent non-separable secondary transform (MDNSST). The non-separable secondary transform may represent a transform which generates modified transform coefficients (or secondary transform coefficients) for a residual signal by secondary-transforming, based on a non-separable transform matrix, (primary) transform coefficients derived through the primary transform. At this time, the vertical transform and the horizontal transform may not be applied separately (or horizontal and vertical transforms may not be applied independently) to the (primary) transform coefficients, but the transforms may be applied at once based on the non-separable transform matrix. In other words, the non-separable secondary transform may represent a transform method in which is not separately applied in the vertical direction and the horizontal direction for the (primary) transform coefficients, and for example, two-dimensional signals (transform coefficients) are re-arranged to a one-dimensional signal through a certain determined direction (e.g., row-first direction or column-first direction), and then modified transform coefficients (or secondary transform coefficients) are generated based on the non-separable transform matrix. For example, according to a row-first order, M×N blocks are disposed in a line in an order of a first row, a second row, . . . , and an Nth row. According to a column-first order, M×N blocks are disposed in a line in an order of a first column, a second column, . . . , and an Nth column. The non-separable secondary transform may be applied to a top-left region of a block configured with (primary) transform coefficients (hereinafter, may be referred to as a transform coefficient block). For example, if the width (W) and the height (H) of the transform coefficient block are all equal to or greater than 8, an 8×8 non-separable secondary transform may be applied to a top-left 8×8 region of the transform coefficient block. Further, if the width (W) and the height (H) of the transform coefficient block are all equal to or greater than 4, and the width (W) or the height (H) of the transform coefficient block is less than 8, then a 4×4 non-separable secondary transform may be applied to a top-left min(8,W)×min(8,H) region of the transform coefficient block. However, the embodiment is not limited to this, and for example, even if only the condition that the width (W) or height (H) of the transform coefficient block is equal to or greater than 4 is satisfied, the 4×4 non-separable secondary transform may be applied to the top-left min(8,W)×min(8,H) region of the transform coefficient block. 
     Specifically, for example, if a 4×4 input block is used, the non-separable secondary transform may be performed as follows. 
     The 4×4 input block X may be represented as follows. 
     
       
         
           
             
               
                 
                   X 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             X 
                             00 
                           
                         
                         
                           
                             X 
                             01 
                           
                         
                         
                           
                             X 
                             02 
                           
                         
                         
                           
                             X 
                             03 
                           
                         
                       
                       
                         
                           
                             X 
                             10 
                           
                         
                         
                           
                             X 
                             11 
                           
                         
                         
                           
                             X 
                             12 
                           
                         
                         
                           
                             X 
                             13 
                           
                         
                       
                       
                         
                           
                             X 
                             20 
                           
                         
                         
                           
                             X 
                             21 
                           
                         
                         
                           
                             X 
                             22 
                           
                         
                         
                           
                             X 
                             23 
                           
                         
                       
                       
                         
                           
                             X 
                             30 
                           
                         
                         
                           
                             X 
                             31 
                           
                         
                         
                           
                             X 
                             32 
                           
                         
                         
                           
                             X 
                             33 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     1 
                   
                   ] 
                 
               
             
           
         
       
     
     If the X is represented in the form of a vector, the vector   may be represented as below. 
         =[ X   00    X   01    X   02    X   03    X   10    X   11    X   12    X   13    X   20    X   21    X   22    X   23    X   30    X   31    X   32    X   33 ] T   [Equation 2]
 
     In Equation 2, the vector   is a one-dimensional vector obtained by rearranging the two-dimensional block X of Equation 1 according to the row-first order. 
     In this case, the secondary non-separable transform may be calculated as below. 
         = T·         [Equation 3]
 
     In this equation,   represents a transform coefficient vector, and T represents a 16×16 (non-separable) transform matrix. 
     Through foregoing Equation 3, a 16×1 transform coefficient vector   may be derived, and the   may be re-organized into a 4×4 block through a scan order (horizontal, vertical, diagonal and the like). However, the above-described calculation is an example, and hypercube-Givens transform (HyGT) or the like may be used for the calculation of the non-separable secondary transform in order to reduce the computational complexity of the non-separable secondary transform. 
     Meanwhile, in the non-separable secondary transform, a transform kernel (or transform core, transform type) may be selected to be mode dependent. In this case, the mode may include the intra prediction mode and/or the inter prediction mode. 
     As described above, the non-separable secondary transform may be performed based on an 8×8 transform or a 4×4 transform determined based on the width (W) and the height (H) of the transform coefficient block. The 8×8 transform refers to a transform that is applicable to an 8×8 region included in the transform coefficient block when both W and H are equal to or greater than 8, and the 8×8 region may be a top-left 8×8 region in the transform coefficient block. Similarly, the 4×4 transform refers to a transform that is applicable to a 4×4 region included in the transform coefficient block when both W and H are equal to or greater than 4, and the 4×4 region may be a top-left 4×4 region in the transform coefficient block. For example, an 8×8 transform kernel matrix may be a 64×64/16×64 matrix, and a 4×4 transform kernel matrix may be a 16×16/8×16 matrix. 
     Here, to select a mode-dependent transform kernel, two non-separable secondary transform kernels per transform set for a non-separable secondary transform may be configured for both the 8×8 transform and the 4×4 transform, and there may be four transform sets. That is, four transform sets may be configured for the 8×8 transform, and four transform sets may be configured for the 4×4 transform. In this case, each of the four transform sets for the 8×8 transform may include two 8×8 transform kernels, and each of the four transform sets for the 4×4 transform may include two 4×4 transform kernels. 
     However, as the size of the transform, that is, the size of a region to which the transform is applied, may be, for example, a size other than 8×8 or 4×4, the number of sets may be n, and the number of transform kernels in each set may be k. 
     The transform set may be referred to as an NSST set or an LFNST set. A specific set among the transform sets may be selected, for example, based on the intra prediction mode of the current block (CU or subblock). A low-frequency non-separable transform (LFNST) may be an example of a reduced non-separable transform, which will be described later, and represents a non-separable transform for a low frequency component. 
     For reference, for example, the intra prediction mode may include two non-directinoal (or non-angular) intra prediction modes and 65 directional (or angular) intra prediction modes. The non-directional intra prediction modes may include a planar intra prediction mode of No. 0 and a DC intra prediction mode of No. 1, and the directional intra prediction modes may include 65 intra prediction modes of Nos. 2 to 66. However, this is an example, and this document may be applied even when the number of intra prediction modes is different. Meanwhile, in some cases, intra prediction mode No. 67 may be further used, and the intra prediction mode No. 67 may represent a linear model (LM) mode. 
       FIG.  6    exemplarily shows intra directional modes of 65 prediction directions. 
     Referring to  FIG.  6   , on the basis of intra prediction mode 34 having a left upward diagonal prediction direction, the intra prediction modes may be divided into intra prediction modes having horizontal directionality and intra prediction modes having vertical directionality. In  FIG.  6   , H and V denote horizontal directionality and vertical directionality, respectively, and numerals −32 to 32 indicate displacements in 1/32 units on a sample grid position. These numerals may represent an offset for a mode index value. Intra prediction modes 2 to 33 have the horizontal directionality, and intra prediction modes 34 to 66 have the vertical directionality. Strictly speaking, intra prediction mode 34 may be considered as being neither horizontal nor vertical, but may be classified as belonging to the horizontal directionality in determining a transform set of a secondary transform. This is because input data is transposed to be used for a vertical direction mode symmetrical on the basis of intra prediction mode 34, and an input data alignment method for a horizontal mode is used for intra prediction mode 34. Transposing input data means that rows and columns of two-dimensional M×N block data are switched into N×M data. Intra prediction mode 18 and intra prediction mode 50 may represent a horizontal intra prediction mode and a vertical intra prediction mode, respectively, and intra prediction mode 2 may be referred to as a right upward diagonal intra prediction mode because intra prediction mode 2 has a left reference pixel and performs prediction in a right upward direction. Likewise, intra prediction mode 34 may be referred to as a right downward diagonal intra prediction mode, and intra prediction mode 66 may be referred to as a left downward diagonal intra prediction mode. 
     According to an example, the four transform sets according to the intra prediction mode may be mapped, for example, as shown in the following table. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 predModeIntra 
                 lfnstTrSetIdx 
               
               
                   
                   
               
             
            
               
                   
                 predModeIntra &lt; 0 
                 1 
               
               
                   
                 0 &lt;= predModeIntra &lt;= 1 
                 0 
               
               
                   
                 2 &lt;= predModeIntra &lt;= 12 
                 1 
               
               
                   
                 13 &lt;= predModeIntra &lt;= 23 
                 2 
               
               
                   
                 24 &lt;= predModeIntra &lt;= 44 
                 3 
               
               
                   
                 45 &lt;= predModeIntra &lt;= 55 
                 2 
               
               
                   
                 56 &lt;= predModeIntra &lt;= 80 
                 1 
               
               
                   
                   
               
            
           
         
       
     
     As shown in Table 2, any one of the four transform sets, that is, lfnstTrSetIdx, may be mapped to any one of four indexes, that is, 0 to 3, according to the intra prediction mode. 
     When it is determined that a specific set is used for the non-separable transform, one of k transform kernels in the specific set may be selected through a non-separable secondary transform index. An encoding apparatus may derive a non-separable secondary transform index indicating a specific transform kernel based on a rate-distortion (RD) check and may signal the non-separable secondary transform index to a decoding apparatus. The decoding apparatus may select one of the k transform kernels in the specific set based on the non-separable secondary transform index. For example, lfnst index value 0 may refer to a first non-separable secondary transform kernel, lfnst index value 1 may refer to a second non-separable secondary transform kernel, and lfnst index value 2 may refer to a third non-separable secondary transform kernel. Alternatively, lfnst index value 0 may indicate that the first non-separable secondary transform is not applied to the target block, and lfnst index values 1 to 3 may indicate the three transform kernels. 
     The transformer may perform the non-separable secondary transform based on the selected transform kernels, and may obtain modified (secondary) transform coefficients. As described above, the modified transform coefficients may be derived as transform coefficients quantized through the quantizer, and may be encoded and signaled to the decoding apparatus and transferred to the dequantizer/inverse transformer in the encoding apparatus. 
     Meanwhile, as described above, if the secondary transform is omitted, (primary) transform coefficients, which are an output of the primary (separable) transform, may be derived as transform coefficients quantized through the quantizer as described above, and may be encoded and signaled to the decoding apparatus and transferred to the dequantizer/inverse transformer in the encoding apparatus. 
     The inverse transformer may perform a series of procedures in the inverse order to that in which they have been performed in the above-described transformer. The inverse transformer may receive (dequantized) transformer coefficients, and derive (primary) transform coefficients by performing a secondary (inverse) transform (S 550 ), and may obtain a residual block (residual samples) by performing a primary (inverse) transform on the (primary) transform coefficients (S 560 ). In this connection, the primary transform coefficients may be called modified transform coefficients from the viewpoint of the inverse transformer. As described above, the encoding apparatus and the decoding apparatus may generate the reconstructed block based on the residual block and the predicted block, and may generate the reconstructed picture based on the reconstructed block. 
     The decoding apparatus may further include a secondary inverse transform application determinator (or an element to determine whether to apply a secondary inverse transform) and a secondary inverse transform determinator (or an element to determine a secondary inverse transform). The secondary inverse transform application determinator may determine whether to apply a secondary inverse transform. For example, the secondary inverse transform may be an NSST, an RST, or an LFNST and the secondary inverse transform application determinator may determine whether to apply the secondary inverse transform based on a secondary transform flag obtained by parsing the bitstream. In another example, the secondary inverse transform application determinator may determine whether to apply the secondary inverse transform based on a transform coefficient of a residual block. 
     The secondary inverse transform determinator may determine a secondary inverse transform. In this case, the secondary inverse transform determinator may determine the secondary inverse transform applied to the current block based on an LFNST (NSST or RST) transform set specified according to an intra prediction mode. In an embodiment, a secondary transform determination method may be determined depending on a primary transform determination method. Various combinations of primary transforms and secondary transforms may be determined according to the intra prediction mode. Further, in an example, the secondary inverse transform determinator may determine a region to which a secondary inverse transform is applied based on the size of the current block. 
     Meanwhile, as described above, if the secondary (inverse) transform is omitted, (dequantized) transform coefficients may be received, the primary (separable) inverse transform may be performed, and the residual block (residual samples) may be obtained. As described above, the encoding apparatus and the decoding apparatus may generate the reconstructed block based on the residual block and the predicted block, and may generate the reconstructed picture based on the reconstructed block. 
     Meanwhile, in the present disclosure, a reduced secondary transform (RST) in which the size of a transform matrix (kernel) is reduced may be applied in the concept of NSST in order to reduce the amount of computation and memory required for the non-separable secondary transform. 
     Meanwhile, the transform kernel, the transform matrix, and the coefficient constituting the transform kernel matrix, that is, the kernel coefficient or the matrix coefficient, described in the present disclosure may be expressed in 8 bits. This may be a condition for implementation in the decoding apparatus and the encoding apparatus, and may reduce the amount of memory required to store the transform kernel with a performance degradation that can be reasonably accommodated compared to the existing 9 bits or 10 bits. In addition, the expressing of the kernel matrix in 8 bits may allow a small multiplier to be used, and may be more suitable for single instruction multiple data (SIMD) instructions used for optimal software implementation. 
     In the present specification, the term “RST” may mean a transform which is performed on residual samples for a target block based on a transform matrix whose size is reduced according to a reduced factor. In the case of performing the reduced transform, the amount of computation required for transform may be reduced due to a reduction in the size of the transform matrix. That is, the RST may be used to address the computational complexity issue occurring at the non-separable transform or the transform of a block of a great size. 
     RST may be referred to as various terms, such as reduced transform, reduced secondary transform, reduction transform, simplified transform, simple transform, and the like, and the name which RST may be referred to as is not limited to the listed examples. Alternatively, ince the RST is mainly performed in a low frequency region including a non-zero coefficient in a transform block, it may be referred to as a Low-Frequency Non-Separable Transform (LFNST). The transform index may be referred to as an LFNST index. 
     Meanwhile, when the secondary inverse transform is performed based on RST, the inverse transformer  235  of the encoding apparatus  200  and the inverse transformer  322  of the decoding apparatus  300  may include an inverse reduced secondary transformer which derives modified transform coefficients based on the inverse RST of the transform coefficients, and an inverse primary transformer which derives residual samples for the target block based on the inverse primary transform for the modified transform coefficients. The inverse primary transform refers to the inverse transform of the primary transform applied to the residual. In the present disclosure, deriving a transform coefficient based on a transform may refer to deriving a transform coefficient by applying the transform. 
       FIG.  7    is a diagram illustrating an RST according to an embodiment of the present disclosure. 
     In the present disclosure, a “target block” may refer to a current block to be coded, a residual block, or a transform block. 
     In the RST according to an example, an N-dimensional vector may be mapped to an R-dimensional vector located in another space, so that the reduced transform matrix may be determined, where R is less than N. N may mean the square of the length of a side of a block to which the transform is applied, or the total number of transform coefficients corresponding to a block to which the transform is applied, and the reduced factor may mean an R/N value. The reduced factor may be referred to as a reduced factor, reduction factor, simplified factor, simple factor or other various terms. Meanwhile, R may be referred to as a reduced coefficient, but according to circumstances, the reduced factor may mean R. Further, according to circumstances, the reduced factor may mean the N/R value. 
     In an example, the reduced factor or the reduced coefficient may be signaled through a bitstream, but the example is not limited to this. For example, a predefined value for the reduced factor or the reduced coefficient may be stored in each of the encoding apparatus  200  and the decoding apparatus  300 , and in this case, the reduced factor or the reduced coefficient may not be signaled separately. 
     The size of the reduced transform matrix according to an example may be R×N less than N×N, the size of a conventional transform matrix, and may be defined as in Equation 4 below. 
     
       
         
           
             
               
                 
                   
                     T 
                     RxN 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             t 
                             11 
                           
                         
                         
                           
                             t 
                             12 
                           
                         
                         
                           
                             t 
                             13 
                           
                         
                         
                           
                             … 
                           
                         
                         
                           
                             t 
                             
                               1 
                               ⁢ 
                               N 
                             
                           
                         
                       
                       
                         
                           
                             t 
                             21 
                           
                         
                         
                           
                             t 
                             22 
                           
                         
                         
                           
                             t 
                             23 
                           
                         
                         
                             
                         
                         
                           
                             t 
                             
                               2 
                               ⁢ 
                               N 
                             
                           
                         
                       
                       
                         
                             
                         
                         
                           ⋮ 
                         
                         
                             
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             t 
                             
                               R 
                               ⁢ 
                               1 
                             
                           
                         
                         
                           
                             t 
                             
                               R 
                               ⁢ 
                               2 
                             
                           
                         
                         
                           
                             t 
                             
                               R 
                               ⁢ 
                               3 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             t 
                             RN 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     4 
                   
                   ] 
                 
               
             
           
         
       
     
     The matrix T in the Reduced Transform block shown in  FIG.  7 ( a )  may mean the matrix T R×N  of Equation 4. As shown in  FIG.  7 ( a ) , when the reduced transform matrix T R×N  is multiplied to residual samples for the target block, transform coefficients for the target block may be derived. 
     In an example, if the size of the block to which the transform is applied is 8×8 and R=16 (i.e., R/N=16/64=1/4), then the RST according to  FIG.  7 ( a )  may be expressed as a matrix operation as shown in Equation 5 below. In this case, memory and multiplication calculation can be reduced to approximately 1/4 by the reduced factor. 
     In the present disclosure, a matrix operation may be understood as an operation of multiplying a column vector by a matrix, disposed on the left of the column vector, to obtain a column vector. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             t 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                         
                           
                             t 
                             
                               1 
                               , 
                               2 
                             
                           
                         
                         
                           
                             t 
                             
                               1 
                               , 
                               3 
                             
                           
                         
                         
                           
                             … 
                           
                         
                         
                           
                             t 
                             
                               1 
                               , 
                               64 
                             
                           
                         
                       
                       
                         
                           
                             t 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               2 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               3 
                             
                           
                         
                         
                             
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               64 
                             
                           
                         
                       
                       
                         
                             
                         
                         
                           ⋮ 
                         
                         
                             
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             t 
                             
                               16 
                               , 
                               1 
                             
                           
                         
                         
                           
                             t 
                             
                               16 
                               , 
                               2 
                             
                           
                         
                         
                           
                             t 
                             
                               R 
                               ⁢ 
                               3 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             t 
                             RN 
                           
                         
                       
                     
                     ] 
                   
                   × 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     r 
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             ⋮ 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       ⋮ 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                           
                         
                       
                       
                         
                           
                             r 
                             64 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 5, r 1  to r 64  may represent residual samples for the target block and may be specifically transform coefficients generated by applying a primary transform. As a result of the calculation of Equation 5 transform coefficients c i  for the target block may be derived, and a process of deriving c i  may be as in Equation 6. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 for i from 1 to R: 
                 [Equation 6] 
               
            
           
           
               
               
            
               
                   
                  c i  = 0 
               
               
                   
                  for j from 1 to N: 
               
               
                   
                   c i  + = t i,j  * r j   
               
               
                   
               
            
           
         
       
     
     As a result of the calculation of Equation 6, transform coefficients c 1  to c R  for the target block may be derived. That is, when R=16, transform coefficients c 1  to c 16  for the target block may be derived. If, instead of RST, a regular transform is applied and a transform matrix of 64×64 (N×N) size is multiplied to residual samples of 64×1 (N×1) size, then only 16 (R) transform coefficients are derived for the target block because RST was applied, although 64 (N) transform coefficients are derived for the target block. Since the total number of transform coefficients for the target block is reduced from N to R, the amount of data transmitted by the encoding apparatus  200  to the decoding apparatus  300  decreases, so efficiency of transmission between the encoding apparatus  200  and the decoding apparatus  300  can be improved. 
     When considered from the viewpoint of the size of the transform matrix, the size of the regular transform matrix is 64×64 (N×N), but the size of the reduced transform matrix is reduced to 16×64 (R×N), so memory usage in a case of performing the RST can be reduced by an R/N ratio when compared with a case of performing the regular transform. In addition, when compared to the number of multiplication calculations N×N in a case of using the regular transform matrix, the use of the reduced transform matrix can reduce the number of multiplication calculations by the R/N ratio (R×N). 
     In an example, the transformer  232  of the encoding apparatus  200  may derive transform coefficients for the target block by performing the primary transform and the RST-based secondary transform on residual samples for the target block. These transform coefficients may be transferred to the inverse transformer of the decoding apparatus  300 , and the inverse transformer  322  of the decoding apparatus  300  may derive the modified transform coefficients based on the inverse reduced secondary transform (RST) for the transform coefficients, and may derive residual samples for the target block based on the inverse primary transform for the modified transform coefficients. 
     The size of the inverse RST matrix T N×R  according to an example is N×R less than the size N×N of the regular inverse transform matrix, and is in a transpose relationship with the reduced transform matrix T R×N  shown in Equation 4. 
     The matrix T t  in the Reduced Inv. Transform block shown in  FIG.  7 ( b )  may mean the inverse RST matrix T R×N   T  (the superscript T means transpose). When the inverse RST matrix T R×N   T  is multiplied to the transform coefficients for the target block as shown in  FIG.  7 ( b ) , the modified transform coefficients for the target block or the residual samples for the current block may be derived. The inverse RST matrix T R×N   T  may be expressed as (T R×N   T )N×R. 
     More specifically, when the inverse RST is applied as the secondary inverse transform, the modified transform coefficients for the target block may be derived when the inverse RST matrix T R×N   T  is multiplied to the transform coefficients for the target block. Meanwhile, the inverse RST may be applied as the inverse primary transform, and in this case, the residual samples for the target block may be derived when the inverse RST matrix T R×N   T  is multiplied to the transform coefficients for the target block. 
     In an example, if the size of the block to which the inverse transform is applied is 8×8 and R=16 (i.e., R/N=16/64=1/4), then the RST according to  FIG.  7 ( b )  may be expressed as a matrix operation as shown in Equation 7 below. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             t 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         
                             
                         
                         
                           
                             t 
                             
                               16 
                               , 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             t 
                             
                               1 
                               , 
                               2 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               2 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             t 
                             
                               16 
                               , 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             t 
                             
                               1 
                               , 
                               3 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               3 
                             
                           
                         
                         
                             
                         
                         
                           
                             t 
                             
                               16 
                               , 
                               3 
                             
                           
                         
                       
                       
                         
                             
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             t 
                             
                               1 
                               , 
                               64 
                             
                           
                         
                         
                           
                             t 
                             
                               2 
                               , 
                               64 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             t 
                             
                               16 
                               , 
                               64 
                             
                           
                         
                       
                     
                     ] 
                   
                   × 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         c 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       
                                         c 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                           
                         
                       
                       
                         
                           
                             c 
                             16 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                         
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 7, c 1  to c 16  may represent the transform coefficients for the target block. As a result of the calculation of Equation 7, r i  representing the modified transform coefficients for the target block or the residual samples for the target block may be derived, and the process of deriving r i  may be as in Equation 8. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 For i from 1 to N 
                 [Equation 8] 
               
            
           
           
               
               
            
               
                   
                  r j =0 
               
               
                   
                   for j from 1 to R 
               
               
                   
                    r i  + =t ji  * c j   
               
               
                   
               
            
           
         
       
     
     As a result of the calculation of Equation 8, r 1  to r N  representing the modified transform coefficients for the target block or the residual samples for the target block may be derived. When considered from the viewpoint of the size of the inverse transform matrix, the size of the regular inverse transform matrix is 64×64 (N×N), but the size of the reduced inverse transform matrix is reduced to 64×16 (R×N), so memory usage in a case of performing the inverse RST can be reduced by an R/N ratio when compared with a case of performing the regular inverse transform. In addition, when compared to the number of multiplication calculations N×N in a case of using the regular inverse transform matrix, the use of the reduced inverse transform matrix can reduce the number of multiplication calculations by the R/N ratio (N×R). 
     A transform set configuration shown in Table 2 may also be applied to an 8×8 RST. That is, the 8×8 RST may be applied according to a transform set in Table 2. Since one transform set includes two or three transforms (kernels) according to an intra prediction mode, it may be configured to select one of up to four transforms including that in a case where no secondary transform is applied. In a transform where no secondary transform is applied, it may be considered to apply an identity matrix. Assuming that indexes 0, 1, 2, and 3 are respectively assigned to the four transforms (e.g., index 0 may be allocated to a case where an identity matrix is applied, that is, a case where no secondary transform is applied), a transform index or an lfnst index as a syntax element may be signaled for each transform coefficient block, thereby designating a transform to be applied. That is, for a top-left 8×8 block, through the transform index, it is possible to designate an 8×8 RST in an RST configuration, or to designate an 8×8 lfnst when the LFNST is applied. The 8×8 lfnst and the 8×8 RST refer to transforms applicable to an 8×8 region included in the transform coefficient block when both W and H of the target block to be transformed are equal to or greater than 8, and the 8×8 region may be a top-left 8×8 region in the transform coefficient block. Similarly, a 4×4 lfnst and a 4×4 RST refer to transforms applicable to a 4×4 region included in the transform coefficient block when both W and H of the target block to are equal to or greater than 4, and the 4×4 region may be a top-left 4×4 region in the transform coefficient block. 
     According to an embodiment of the present disclosure, for a transform in an encoding process, only 48 pieces of data may be selected and a maximum 16×48 transform kernel matrix may be applied thereto, rather than applying a 16×64 transform kernel matrix to 64 pieces of data forming an 8×8 region. Here, “maximum” means that m has a maximum value of 16 in an m×48 transform kernel matrix for generating m coefficients. That is, when an RST is performed by applying an m×48 transform kernel matrix (m≤16) to an 8×8 region, 48 pieces of data are input and m coefficients are generated. When m is 16, 48 pieces of data are input and 16 coefficients are generated. That is, assuming that 48 pieces of data form a 48×1 vector, a 16×48 matrix and a 48×1 vector are sequentially multiplied, thereby generating a 16×1 vector. Here, the 48 pieces of data forming the 8×8 region may be properly arranged, thereby forming the 48×1 vector. For example, a 48×1 vector may be constructed based on 48 pieces of data constituting a region excluding the bottom right 4×4 region among the 8×8 regions. Here, when a matrix operation is performed by applying a maximum 16×48 transform kernel matrix, 16 modified transform coefficients are generated, and the 16 modified transform coefficients may be arranged in a top-left 4×4 region according to a scanning order, and a top-right 4×4 region and a bottom-left 4×4 region may be filled with zeros. 
     For an inverse transform in a decoding process, the transposed matrix of the foregoing transform kernel matrix may be used. That is, when an inverse RST or LFNST is performed in the inverse transform process performed by the decoding apparatus, input coefficient data to which the inverse RST is applied is configured in a one-dimensional vector according to a predetermined arrangement order, and a modified coefficient vector obtained by multiplying the one-dimensional vector and a corresponding inverse RST matrix on the left of the one-dimensional vector may be arranged in a two-dimensional block according to a predetermined arrangement order. 
     In summary, in the transform process, when an RST or LFNST is applied to an 8×8 region, a matrix operation of 48 transform coefficients in top-left, top-right, and bottom-left regions of the 8×8 region excluding the bottom-right region among transform coefficients in the 8×8 region and a 16×48 transform kernel matrix. For the matrix operation, the 48 transform coefficients are input in a one-dimensional array. When the matrix operation is performed, 16 modified transform coefficients are derived, and the modified transform coefficients may be arranged in the top-left region of the 8×8 region. 
     On the contrary, in the inverse transform process, when an inverse RST or LFNST is applied to an 8×8 region, 16 transform coefficients corresponding to a top-left region of the 8×8 region among transform coefficients in the 8×8 region may be input in a one-dimensional array according to a scanning order and may be subjected to a matrix operation with a 48×16 transform kernel matrix. That is, the matrix operation may be expressed as (48×16 matrix)*(16×1 transform coefficient vector)=(48×1 modified transform coefficient vector). Here, an n×1 vector may be interpreted to have the same meaning as an n×1 matrix and may thus be expressed as an n×1 column vector. Further, * denotes matrix multiplication. When the matrix operation is performed, 48 modified transform coefficients may be derived, and the 48 modified transform coefficients may be arranged in top-left, top-right, and bottom-left regions of the 8×8 region excluding a bottom-right region. 
     When a secondary inverse transform is based on an RST, the inverse transformer  235  of the encoding apparatus  200  and the inverse transformer  322  of the decoding apparatus  300  may include an inverse reduced secondary transformer to derive modified transform coefficients based on an inverse RST on transform coefficients and an inverse primary transformer to derive residual samples for the target block based on an inverse primary transform on the modified transform coefficients. The inverse primary transform refers to the inverse transform of a primary transform applied to a residual. In the present disclosure, deriving a transform coefficient based on a transform may refer to deriving the transform coefficient by applying the transform. 
     The above-described non-separated transform, the LFNST, will be described in detail as follows. The LFNST may include a forward transform by the encoding apparatus and an inverse transform by the decoding apparatus. 
     The encoding apparatus receives a result (or a part of a result) derived after applying a primary (core) transform as an input, and applies a forward secondary transform (secondary transform). 
         y=G   T   x   [Equation 9]
 
     In Equation 9, x and y are inputs and outputs of the secondary transform, respectively, and G is a matrix representing the secondary transform, and transform basis vectors are composed of column vectors. In the case of an inverse LFNST, when the dimension of the transformation matrix G is expressed as [number of rows×number of columns], in the case of an forward LFNST, the transposition of matrix G becomes the dimension of G T . 
     For the inverse LFNST, the dimensions of matrix G are [48×16], [48×8], [16×16], [16×8], and the [48×8] matrix and the [16×8] matrix are partial matrices that sampled 8 transform basis vectors from the left of the [48×16] matrix and the [16×16] matrix, respectively. 
     On the other hand, for the forward LFNST, the dimensions of matrix G T  are [16×48], [ 8×48], [16×16], [8×16], and the [8×48] matrix and the [8×16] matrix are partial matrices obtained by sampling 8 transform basis vectors from the top of the [16×48] matrix and the [16×16] matrix, respectively. 
     Therefore, in the case of the forward LFNST, a [48×1] vector or [16×1] vector is possible as an input x, and a [16×1] vector or a [8×1] vector is possible as an output y. In video coding and decoding, the output of the forward primary transform is two-dimensional (2D) data, so to construct the [48×1] vector or the [16×1] vector as the input x, a one-dimensional vector must be constructed by properly arranging the 2D data that is the output of the forward transformation. 
       FIG.  8    is a diagram illustrating a sequence of arranging output data of a forward primary transformation into a one-dimensional vector according to an example. The left diagrams of (a) and (b) of  FIG.  8    show the sequence for constructing a [48×1] vector, and the right diagrams of (a) and (b) of  FIG.  8    shows the sequence for constructing a [16×1] vector. In the case of the LFNST, a one-dimensional vector x can be obtained by sequentially arranging 2D data in the same order as in (a) and (b) of  FIG.  8   . 
     The arrangement direction of the output data of the forward primary transform may be determined according to an intra prediction mode of the current block. For example, when the intra prediction mode of the current block is in the horizontal direction with respect to the diagonal direction, the output data of the forward primary transform may be arranged in the order of (a) of  FIG.  8   , and when the intra prediction mode of the current block is in the vertical direction with respect to the diagonal direction, the output data of the forward primary transform may be arranged in the order of (b) of  FIG.  8   . 
     According to an example, an arrangement order different from the arrangement orders of (a) and (b)  FIG.  8    may be applied, and in order to derive the same result (y vector) as when the arrangement orders of (a) and (b)  FIG.  8    is applied, the column vectors of the matrix G may be rearranged according to the arrangement order. That is, it is possible to rearrange the column vectors of G so that each element constituting the x vector is always multiplied by the same transform basis vector. 
     Since the output y derived through Equation 9 is a one-dimensional vector, when two-dimensional data is required as input data in the process of using the result of the forward secondary transformation as an input, for example, in the process of performing quantization or residual coding, the output y vector of Equation 9 must be properly arranged as 2D data again. 
       FIG.  9    is a diagram illustrating a sequence of arranging output data of a forward secondary transform into a two-dimensional block according to an example. 
     In the case of the LFNST, output values may be arranged in a 2D block according to a predetermined scan order. (a) of  FIG.  9    shows that when the output y is a [16×1] vector, the output values are arranged at 16 positions of the 2D block according to a diagonal scan order. (b) of  FIG.  9    shows that when the output y is a [8×1] vector, the output values are arranged at 8 positions of the 2D block according to the diagonal scan order, and the remaining 8 positions are filled with zeros. X in (b) of  FIG.  9    indicates that it is filled with zero. 
     According to another example, since the order in which the output vector y is processed in performing quantization or residual coding may be preset, the output vector y may not be arranged in the 2D block as shown in  FIG.  9   . However, in the case of the residual coding, data coding may be performed in 2D block (e.g., 4×4) units such as CG (Coefficient Group), and in this case, the data are arranged according to a specific order as in the diagonal scan order of  FIG.  9   . 
     Meanwhile, the decoding apparatus may configure the one-dimensional input vector y by arranging two-dimensional data output through a dequantization process or the like according to a preset scan order for the inverse transformation. The input vector y may be output as the output vector x by the following equation. 
         x=Gy   [Equation 10]
 
     In the case of the inverse LFNST, an output vector x can be derived by multiplying an input vector y, which is a [16×1] vector or a [8×1] vector, by a G matrix. For the inverse LFNST, the output vector x can be either a [48×1] vector or a [16×1] vector. 
     The output vector x is arranged in a two-dimensional block according to the order shown in  FIG.  8    and is arranged as two-dimensional data, and this two-dimensional data becomes input data (or a part of input data) of the inverse primary transformation. 
     Accordingly, the inverse secondary transformation is the opposite of the forward secondary transformation process as a whole, and in the case of the inverse transformation, unlike in the forward direction, the inverse secondary transformation is first applied, and then the inverse primary transformation is applied. 
     In the inverse LFNST, one of 8 [48×16] matrices and 8 [16×16] matrices may be selected as the transformation matrix G. Whether to apply the [48×16] matrix or the [16×16] matrix depends on the size and shape of the block. 
     In addition, 8 matrices may be derived from four transform sets as shown in Table 2 above, and each transform set may consist of two matrices. Which transform set to use among the 4 transform sets is determined according to the intra prediction mode, and more specifically, the transform set is determined based on the value of the intra prediction mode extended by considering the Wide Angle Intra Prediction (WAIP). Which matrix to select from among the two matrices constituting the selected transform set is derived through index signaling. More specifically, 0, 1, and 2 are possible as the transmitted index value, 0 may indicate that the LFNST is not applied, and 1 and 2 may indicate any one of two transform matrices constituting a transform set selected based on the intra prediction mode value. 
       FIG.  10    is a diagram illustrating wide-angle intra prediction modes according to an embodiment of the present document. 
     The general intra prediction mode value may have values from 0 to 66 and 81 to 83, and the intra prediction mode value extended due to WAIP may have a value from −14 to 83 as shown. Values from 81 to 83 indicate the CCLM (Cross Component Linear Model) mode, and values from −14 to −1 and values from 67 to 80 indicate the intra prediction mode extended due to the WAIP application. 
     When the width of the prediction current block is greater than the height, the upper reference pixels are generally closer to positions inside the block to be predicted. Therefore, it may be more accurate to predict in the bottom-left direction than in the top-right direction. Conversely, when the height of the block is greater than the width, the left reference pixels are generally close to positions inside the block to be predicted. Therefore, it may be more accurate to predict in the top-right direction than in the bottom-left direction. Therefore, it may be advantageous to apply remapping, ie, mode index modification, to the index of the wide-angle intra prediction mode. 
     When the wide-angle intra prediction is applied, information on the existing intra prediction may be signaled, and after the information is parsed, the information may be remapped to the index of the wide-angle intra prediction mode. Therefore, the total number of the intra prediction modes for a specific block (eg, a non-square block of a specific size) may not change, and that is, the total number of the intra prediction modes is 67, and intra prediction mode coding for the specific block may not be changed. 
     Table 3 below shows a process of deriving a modified intra mode by remapping an intra prediction mode to an wide-angle intra prediction mode. 
     
       
         
           
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                 Inputs to this process are: 
               
            
           
           
               
               
            
               
                 - 
                 a variable predModeIntra specifying the intra prediction mode, 
               
               
                 - 
                 a variable nTbW specifying the transform block width, 
               
               
                 - 
                 a variable nTbH specifying the transform block height, 
               
               
                 - 
                 a variable cIdx specifying the colour component of the current block. 
               
            
           
           
               
            
               
                 Output of this process is file modified intra prediction mode predModeIntra. 
               
               
                 The variables nW and nH are derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If IntraSubPartitionsSplitType is equal to ISP_NO_SPLIT or Idx is not equal to 0, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                  nW = nTbW 
                  (8-97) 
               
               
                   
                  nH = nTbH 
                  (8-98) 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise ( IntraSubPartitionsSplitType is not equal to ISP_NO_SPLIT and cIdx is equal to 0 ), the 
               
               
                   
                 following applies: 
               
            
           
           
               
               
               
            
               
                   
                  nW = nCbW 
                  (8-99) 
               
               
                   
                  nH = nCbH 
                 (8-100) 
               
            
           
           
               
            
               
                 The variable whRatio is set equal to Abs( Log2( nW / nH ) ). 
               
               
                 For non-square blocks (nW is not equal to nH), the intra prediction mode predModeIntra is modified as 
               
               
                 follows: 
               
            
           
           
               
               
            
               
                 - 
                 If all of the following conditions are true, predModeIntra is set equal to ( predModeIntra + 65 ). 
               
            
           
           
               
               
               
            
               
                   
                 - 
                 nW is greater than nH 
               
               
                   
                 - 
                 predModeIntra is greater than or equal to 2 
               
               
                   
                 - 
                 predModeIntra is less than ( whRatio &gt; 1 ) ? ( 8 + 2 * whRatio ) : 8 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, if all of the following conditions are true, predModeIntra is set equal to 
               
               
                   
                 ( predModeIntra − 67 ). 
               
            
           
           
               
               
               
            
               
                   
                 - 
                 nH is greater than nW 
               
               
                   
                 - 
                 predModeIntra is less than or equal to 66 
               
               
                   
                 - 
                 predModeIntra is greater than ( whRatio &gt; 1 ) ? ( 60 − 2 * whRatio ) : 60 
               
               
                   
                   
               
            
           
         
       
     
     In Table 3, the extended intra prediction mode value is finally stored in the variable predModeIntra, and ISP_NO_SPLIT indicates that a CU block is not divided into sub-partitions by the intra sub partitions (ISP) technique currently adopted in the VVC standard, and the variable cIdx values of 0, 1, and 2 indicate the case of luma, Cb, and Cr components, respectively. The function Log 2 shown in Table 3 returns a log value with a base of 2, and the function Abs returns an absolute value. 
     A variable predModeIntra indicating an intra prediction mode, a height and a width of a transform block, etc. are used as input values of an wide angle intra prediction mode mapping process, and an output value is a modified intra prediction mode (the modified intra prediction mode predModeIntra). The height and width of the transform block or the coding block may be the height and width of the current block for remapping of the intra prediction mode. In this case, the variable whRatio reflecting a ratio of the height and width may be set as Abs(Log 2(nW/nH)). 
     For a non-square block, the intra prediction mode may be classified into two cases and modified. 
     First, if all of conditions (1) the width of the current block is greater than the height of the current block, (2) the intra prediction mode before the modification is equal to or greater than 2, and (3) the intra prediction mode is less than a value derived as (8+2*whRatio) when the variable whRatio is greater than 1, or 8 when the variable whRatio is less than or equal to 1[predModeIntra is less than (whRatio&gt;1)?(8+2*whRatio): 8], are satisfied, the intra prediction mode is set as a value greater by 65 than the intra prediction mode [predModeIntra is set equal to (predModeIntra+65)]. 
     In a case different from the above, if all of conditions (1) the height of the current block is greater than the width, (2) the intra prediction mode before the modification is less than or equal to 66, and (3) the intra prediction mode is greater than a value as (60−2*whRatio) when the variable whRatio is greater than 1, or 60 when the variable whRatio is less than or equal to 1 [predModeIntra is greater than (whRatio&gt;1)?(60−2*whRatio): 60], are satisfied, the intra prediction mode is set as a value smaller by 67 than the intra prediction mode [predModeIntra is set equal to (predModeIntra−67)]. 
     Table 2 above shows how a transform set is selected based on the intra prediction mode value extended by WAIP in LFNST. As shown in  FIG.  10   , the modes 14 to 33 and 35 to 80 are symmetric with respect to a prediction direction based on mode 34. For example, mode 14 and mode 54 are symmetric with respect to the direction corresponding to mode 34. Therefore, the same transform set is applied to modes located in mutually symmetrical directions, and this symmetry is also reflected in Table 2. 
     However, it is assumed that a forward LFNST input data for mode 54 is symmetric with a forward LFNST input data for mode 14. For example, for mode 14 and mode 54, 2D data is rearranged into 1D data according to an arrangement order shown in  FIGS.  8 ( a ) and  8 ( b ) , respectively, and the order of the pattern shown in  FIGS.  8 ( a ) and  8 ( b )  are symmetric with respect to a direction indicated by mode 34 (diagonal direction). 
     Meanwhile, as described above, which transform matrix of the [48×16] matrix and the [16×16] matrix is applied to the LFNST is determined by the size and shape of the transform target block. 
       FIG.  11    is a diagram illustrating a block shape to which the LFNST is applied. (a) of  FIG.  11    shows 4×4 blocks, (b) shows 4×8 and 8×4 blocks, (c) shows 4×N or N×4 blocks in which N is 16 or more, (d) shows 8×8 blocks, (e) shows M×N blocks where M≥8, N≥8, and N&gt;8 or M&gt;8. 
     In  FIG.  11   , blocks with thick borders indicate regions to which the LFNST is applied. For the blocks of  FIGS.  13  ( a ) and ( b ) , the LFNST is applied to the top-left 4×4 region, and for the block of  FIG.  11  ( c ) , the LFNST is applied individually the two top-left 4×4 regions are continuously arranged. In (a), (b), and (c) of  FIG.  11   , since the LFNST is applied in units of 4×4 regions, this LFNST will be hereinafter referred to as “4×4 LFNST”. Based on the matrix dimension for G, a [16×16] or [16×8] matrix may be applied. 
     More specifically, the [16×8] matrix is applied to the 4×4 block (4×4 TU or 4×4 CU) of  FIG.  11  ( a )  and the [16×16] matrix is applied to the blocks in (b) and (c) of  FIG.  11   . This is to adjust the computational complexity for the worst case to 8 multiplications per sample. 
     With respect to (d) and (e) of  FIG.  11   , the LFNST is applied to the top-left 8×8 region, and this LFNST is hereinafter referred to as “8×8 LFNST”. As a corresponding transformation matrix, a [48×16] matrix or [48×8] matrix may be applied. In the case of the forward LFNST, since the [48×1] vector (x vector in Equation 9) is input as input data, all sample values of the top-left 8×8 region are not used as input values of the forward LFNST. That is, as can be seen in the left order of  FIG.  8  ( a )  or the left order of  FIG.  8  ( b ) , the [48×1] vector may be constructed based on samples belonging to the remaining 3 4×4 blocks while leaving the bottom-right 4×4 block as it is. 
     The [48×8] matrix may be applied to an 8×8 block (8×8 TU or 8×8 CU) in  FIG.  11    ( d ), and the [48×16] matrix may be applied to the 8×8 block in  FIG.  11 ( e ) . This is also to adjust the computational complexity for the worst case to 8 multiplications per sample. 
     Depending on the block shape, when the corresponding forward LFNST (4×4 LFNST or 8×8 LFNST) is applied, 8 or 16 output data (y vector in Equation 9, [8×1] or [16×1] vector) is generated. In the forward LFNST, the number of output data is equal to or less than the number of input data due to the characteristics of the matrix GT. 
       FIG.  12    is a diagram illustrating an arrangement of output data of a forward LFNST according to an example, and shows a block in which output data of the forward LFNST is arranged according to a block shape. 
     The shaded area at the top-left of the block shown in  FIG.  12    corresponds to the area where the output data of the forward LFNST is located, the positions marked with 0 indicate samples filled with 0 values, and the remaining area represents regions that are not changed by the forward LFNST. In the area not changed by the LFNST, the output data of the forward primary transform remains unchanged. 
     As described above, since the dimension of the transform matrix applied varies according to the shape of the block, the number of output data also varies. As  FIG.  12   , the output data of the forward LFNST may not completely fill the top-left 4×4 block. In the case of (a) and (d) of  FIG.  12   , a [16×8] matrix and a [48×8] matrix are applied to the block indicated by a thick line or a partial region inside the block, respectively, and a [8×1] vector as the output of the forward LFNST is generated. That is, according to the scan order shown in (b) of  FIG.  9   , only 8 output data may be filled as shown in (a) and (d) of  FIG.  12   , and 0 may be filled in the remaining 8 positions. In the case of the LFNST applied block of  FIG.  11  ( d ) , as shown in  FIG.  12 ( d ) , two 4×4 blocks in the top-right and bottom-left adjacent to the top-left 4×4 block are also filled with 0 values. 
     As described above, basically, by signaling the LFNST index, whether to apply the LFNST and the transform matrix to be applied are specified. As shown  FIG.  12   , when the LFNST is applied, since the number of output data of the forward LFNST may be equal to or less than the number of input data, a region filled with a zero value occurs as follows. 
     1) As shown in (a) of  FIG.  12   , samples from the 8th and later positions in the scan order in the top-left 4×4 block, that is, samples from the 9th to the 16th. 
     2) As shown in (d) and (e) of  FIG.  12   , when the [48×16] matrix or the [48×8] matrix is applied, two 4×4 blocks adjacent to the top-left 4×4 block or the second and third 4×4 blocks in the scan order. 
     Therefore, if non-zero data exists by checking the areas 1) and 2), it is certain that the LFNST is not applied, so that the signaling of the corresponding LFNST index can be omitted. 
     According to an example, for example, in the case of LFNST adopted in the VVC standard, since signaling of the LFNST index is performed after the residual coding, the encoding apparatus may know whether there is the non-zero data (significant coefficients) for all positions within the TU or CU block through the residual coding. Accordingly, the encoding apparatus may determine whether to perform signaling on the LFNST index based on the existence of the non-zero data, and the decoding apparatus may determine whether the LFNST index is parsed. When the non-zero data does not exist in the area designated in 1) and 2) above, signaling of the LFNST index is performed. 
     Meanwhile, for the adopted LFNST, the following simplification methods may be applied. 
     (i) According to an example, the number of output data for the forward LFNST may be limited to a maximum of 16. 
     In the case of (c) of  FIG.  11   , the 4×4 LFNST may be applied to two 4×4 regions adjacent to the top-left, respectively, and in this case, a maximum of 32 LFNST output data may be generated. when the number of output data for forward LFNST is limited to a maximum of 16, in the case of 4×N/N×4 (N≥16) blocks (TU or CU), the 4×4 LFNST is only applied to one 4×4 region in the top-left, the LFNST may be applied only once to all blocks of  FIG.  11   . Through this, the implementation of image coding may be simplified. 
     (ii) According to an example, zero-out may be additionally applied to a region to which the LFNST is not applied. In this document, the zero-out may mean filling values of all positions belonging to a specific region with a value of 0. That is, the zero-out can be applied to a region that is not changed due to the LFNST and maintains the result of the forward primary transformation. As described above, since the LFNST is divided into the 4×4 LFNST and the 8×8 LFNST, the zero-out can be divided into two types ((ii)-(A) and (ii)-(B)) as follows. 
     (ii)-(A) When the 4×4 LFNST is applied, a region to which the 4×4 LFNST is not applied may be zeroed out.  FIG.  13    is a diagram illustrating the zero-out in a block to which the 4×4 LFNST is applied according to an example. 
     As shown in  FIG.  13   , with respect to a block to which the 4×4 LFNST is applied, that is, for all of the blocks in (a), (b) and (c) of  FIG.  12   , the whole region to which the LFNST is not applied may be filled with zeros. 
     On the other hand, (d) of  FIG.  13    shows that when the maximum value of the number of the output data of the forward LFNST is limited to 16 according to one example, the zero-out is performed on the remaining blocks to which the 4×4 LFNST is not applied. 
     (ii)-(B) When the 8×8 LFNST is applied, a region to which the 8×8 LFNST is not applied may be zeroed out.  FIG.  14    is a diagram illustrating the zero-out in a block to which the 8×8 LFNST is applied according to an example. 
     As shown in  FIG.  14   , with respect to a block to which the 8×8 LFNST is applied, that is, for all of the blocks in (d) and (e) of  FIG.  12   , the whole region to which the LFNST is not applied may be filled with zeros. 
     (iii) Due to the zero-out presented in (ii) above, the area filled with zeros may be not same when the LFNST is applied. Accordingly, it is possible to check whether the non-zero data exists according to the zero-out proposed in (ii) over a wider area than the case of the LFNST of  FIG.  12   . 
     For example, when (ii)-(B) is applied, after checking whether the non-zero data exists where the area filled with zero values in (d) and (e) of  FIG.  12    in addition to the area filled with 0 additionally in  FIG.  14   , signaling for the LFNST index can be performed only when the non-zero data does not exist. 
     Of course, even if the zero-out proposed in (ii) is applied, it is possible to check whether the non-zero data exists in the same way as the existing LFNST index signaling. That is, after checking whether the non-zero data exists in the block filled with zeros in  FIG.  12   , the LFNST index signaling may be applied. In this case, the encoding apparatus only performs the zero out and the decoding apparatus does not assume the zero out, that is, checking only whether the non-zero data exists only in the area explicitly marked as 0 in  FIG.  12   , may perform the LFNST index parsing. 
     Various embodiments in which combinations of the simplification methods ((i), (ii)-(A), (ii)-(B), (iii)) for the LFNST are applied may be derived. Of course, the combinations of the above simplification methods are not limited to the following embodiment, and any combination may be applied to the LFNST. 
     Embodiment 
     
         
         
           
             Limit the number of output data for forward LFNST to a maximum of 16→(i) 
             When the 4×4 LFNST is applied, all areas to which the 4×4 LFNST is not applied are zero-out→(ii)-(A) 
             When the 8×8 LFNST is applied, all areas to which the 8×8 LFNST is not applied are zero-out→(ii)-(B) 
             After checking whether the non-zero data exists also the existing area filled with zero value and the area filled with zeros due to additional zero outs ((ii)-(A), (ii)-(B)), the LFNST index is signaled only when the non-zero data does not exist (iii) 
           
         
       
    
     In the case of Embodiment, when the LFNST is applied, an area in which the non-zero output data can exist is limited to the inside of the top-left 4×4 area. In more detail, in the case of  FIG.  13  ( a )  and  FIG.  14  ( a ) , the 8th position in the scan order is the last position where non-zero data can exist. In the case of  FIGS.  13  ( b ) and ( c )  and  FIG.  14  ( b ) , the 16th position in the scan order (ie, the position of the bottom-right edge of the top-left 4×4 block) is the last position where data other than 0 may exist. 
     Therefore, when the LFNST is applied, after checking whether the non-zero data exists in a position where the residual coding process is not allowed (at a position beyond the last position), it can be determined whether the LFNST index is signaled. 
     In the case of the zero-out method proposed in (ii), since the number of data finally generated when both the primary transform and the LFNST are applied, the amount of computation required to perform the entire transformation process can be reduced. That is, when the LFNST is applied, since zero-out is applied to the forward primary transform output data existing in a region to which the LFNST is not applied, there is no need to generate data for the region that become zero-out during performing the forward primary transform. Accordingly, it is possible to reduce the amount of computation required to generate the corresponding data. The additional effects of the zero-out method proposed in (ii) are summarized as follows. 
     First, as described above, the amount of computation required to perform the entire transform process is reduced. 
     In particular, when (ii)-(B) is applied, the amount of calculation for the worst case is reduced, so that the transform process can be lightened. In other words, in general, a large amount of computation is required to perform a large-size primary transformation. By applying (ii)-(B), the number of data derived as a result of performing the forward LFNST can be reduced to 16 or less. In addition, as the size of the entire block (TU or CU) increases, the effect of reducing the amount of transform operation is further increased. 
     Second, the amount of computation required for the entire transform process can be reduced, thereby reducing the power consumption required to perform the transform. 
     Third, the latency involved in the transform process is reduced. 
     The secondary transformation such as the LFNST adds a computational amount to the existing primary transformation, thus increasing the overall delay time involved in performing the transformation. In particular, in the case of intra prediction, since reconstructed data of neighboring blocks is used in the prediction process, during encoding, an increase in latency due to a secondary transformation leads to an increase in latency until reconstruction. This can lead to an increase in overall latency of intra prediction encoding. 
     However, if the zero-out suggested in (ii) is applied, the delay time of performing the primary transform can be greatly reduced when LFNST is applied, the delay time for the entire transform is maintained or reduced, so that the encoding apparatus can be implemented more simply. 
     Meanwhile, in the conventional intra prediction, a coding target block is regarded as one coding unit, and coding is performed without partition thereof. However, the ISP (Intra Sub-Partitions) coding refers to performing the intra prediction coding with the coding target block being partitioned in a horizontal direction or a vertical direction. In this case, a reconstructed block may be generated by performing encoding/decoding in units of partitioned blocks, and the reconstructed block may be used as a reference block of the next partitioned block. According to an example, in the ISP coding, one coding block may be partitioned into two or four sub-blocks and be coded, and in the ISP, intra prediction is performed on one sub-block by referring to the reconstructed pixel value of a sub-block located adjacent to the left or top side thereof. Hereinafter, the term “coding” may be used as a concept including both coding performed by the encoding apparatus and decoding performed by the decoding apparatus. 
     The ISP partitions a block predicted as luma intra into two or four sub-partitionings in a vertical direction or a horizontal direction according to the size of the block. For example, the minimum block size to which the ISP can be applied is 4×8 or 8×4. If the block size is greater than 4×8 or 8×4, the block is partitioned into four sub-partitionings. 
     When the ISP is applied, the sub-blocks are sequentially coded according to the partition type, such as, horizontally or vertically, from left to right, or from top to bottom, and coding for the next sub-block may be performed after performing up to a restoration process through inverse transform and intra prediction for one sub-block. For the leftmost or uppermost sub-block, as in a conventional intra prediction method, the reconstructed pixel of the coding block which has been already coded is referred to. Additionally, if the previous sub-block is not adjacent to each side of an internal sub-block that follows it, in order to derive reference pixels adjacent to a corresponding side, as in the conventional intra prediction method, the reconstructed pixel of an already coded adjacent coding block is referred to. 
     In the ISP coding mode, all sub-blocks may be coded with the same intra prediction mode, and a flag indicating whether or not to use the ISP coding and a flag indicating in which direction (horizontal or vertical) partition is to be performed may be signaled. At this time, the number of sub-blocks may be adjusted to 2 or 4 depending on the block shape, and when the size (width×height) of one sub-block is less than 16, the partition may not be allowed for the corresponding sub-blocks, nor the application of the ISP coding itself may be restricted. 
     Meanwhile, in the case of ISP prediction mode, one coding unit is partitioned into two or four partition blocks, that is, sub-blocks, and predicted, and the same intra prediction mode is applied to the thus partitioned two or four partition blocks. 
     As described above, both a horizontal direction (if an M×N coding unit having a horizontal length and a vertical length of M and N, respectively, is divided in the horizontal direction, it is divided into M×(N/2) blocks when divided into two, and into an M×(N/4) blocks when divided into four) and a vertical direction (if the M×N coding unit is divided in the vertical direction, it is divided into (M/2)×N blocks when divided into two, and divided into (M/4)×N blocks when divided into four) are possible as the partition direction. When partitioned in the horizontal direction, partition blocks are coded in an order from top to down, and when partitioned in the vertical direction, partition blocks are coded in an order from left to right. The currently coded partition block may be predicted by referring to the reconstructed pixel values of the top (left) partition block in the case of the horizontal (vertical) direction partition. 
     Transformation may be applied to the residual signal generated by the ISP prediction method in units of partition blocks. MTS (Multiple Transform Selection) technology based on the DST-7/DCT-8 combination as well as the existing DCT-2 may be applied to the primary transform (core transform or primary transform) based on the forward direction, and an LFNST (Low Frequency Non-Separable Transform) may be applied to a transform coefficient generated according to the primary transform to generate a final modified transform coefficient. 
     That is, LFNST may also be applied to partition blocks divided by applying the ISP prediction mode, and the same intra prediction mode is applied to the divided partition blocks as described above. Accordingly, when selecting the LFNST set derived based on the intra prediction mode, the derived LFNST set may be applied to all partition blocks. That is, the same intra prediction mode is applied to all partition blocks, and thereby the same LFNST set may be applied to all partition blocks. 
     Meanwhile, according to an example, the LFNST may be applied only to transform blocks having both a horizontal and vertical length of 4 or more. Therefore, when the horizontal or vertical length of the partition block partitioned according to the ISP prediction method is less than 4, the LFNST is not applied and the LFNST index is not signaled. Additionally, when the LFNST is applied to each partition block, the corresponding partition block may be regarded as one transform block. Of course, when the ISP prediction method is not applied, the LFNST may be applied to the coding block. 
     Application of the LFNST to each partition block is described in detail as follows. 
     According to an example, after applying the forward LFNST to an individual partition block, and after leaving only up to 16 coefficients (8 or 16) in the top-left 4×4 region according to the transform coefficient scanning order, zero-out of filling all remaining positions and regions with a value of 0 may be applied. 
     Alternatively, according to an example, when the length of one side of the partition block is 4, the LFNST is applied only to the top-left 4×4 region, and when the length of all sides of the partition block, that is, the width and height, are 8 or more, the LFNST may be applied to the remaining 48 coefficients except for a bottom-right 4×4 region in a top-left 8×8 region. 
     Alternatively, according to an example, in order to adjust the computational complexity of the worst case to 8 multiplications per sample, when each partition block is 4×4 or 8×8, only 8 transform coefficients may be output after applying the forward LFNST. That is, if the partition block is 4×4, an 8×16 matrix may be applied as a transform matrix, and if the partition block is 8×8, an 8×48 matrix may be applied as a transform matrix. 
     Meanwhile, in the current VVC standard, LFNST index signaling is performed in units of coding units. Accordingly, when the ISP prediction mode is used and the LFNST is applied to all partition blocks, then the same LFNST index value may be applied to the corresponding partition blocks. That is, when the LFNST index value is transmitted once at the coding unit level, the corresponding LFNST index may be applied to all partition blocks in the coding unit. As described above, the LFNST index value may have values of 0, 1, and 2, 0 indicates a case in which the LFNST is not applied, and 1 and 2 indicate two transform matrices present in one LFNST set when the LFNST is applied. 
     As described above, the LFNST set is determined by the intra prediction mode, and since all partition blocks in the coding unit are predicted in the same intra prediction mode in the case of the ISP prediction mode, the partition blocks may refer to the same LFNST set. 
     As another example, the LFNST index signaling is still performed in units of coding units, but in the case of the ISP prediction mode, without determining whether or not to apply the LFNST uniformly to all partition blocks, whether to apply the LFNST index value signaled at the coding unit level to each partition block or not to apply the LFNST may be determined through a separate condition. Here, the separate condition may be signaled in the form of a flag for each partition block through the bitstream, and when the flag value is 1, the LFNST index value signaled at the coding unit level may be applied, and when the flag value is 0, the LFNST may not be applied. 
     Meanwhile, in the following description, LFNST index signaling when LFNST is applied only to a luma component in the case of a single tree will be described. 
     In the current VVC specification text (JVET-Q2001-vE.docx), the LFNST index is parsed after parsing of transform coefficients is all finished, that is, after parsing of transform trees is finished. The LFNST index (lfnst_idx) is parsed after checking a value of variable LfnstDcOnly and a value of a variable LfnstZeroOutSigCoeffFlag, and the variable LfnstDcOnly and the variable UnstZeroOutSigCoeffFlag may be updated in the residual coding syntax table as shown in Table 10. 
     Accordingly, in the case of the single tree, since residual coding may be performed on all Y, Cb, and Cr components, in the case of the single tree, information on all of the Y, Cb, and Cr components may be reflected and the variable LfnstDcOnly and the variable UnstZeroOutSigCoeffFlag may be updated. For example, the value of the variable UnstZeroOutSigCoeffFlag is updated to 0 if a transform coefficient exists in a region other than the region where the LFNST applied transform coefficient can exist. In the case of the single tree, the existence for all Y, Cb, and Cr components is checked. 
     In the case of the single tree, if LFNST signaling such as the current VVC specification text is applied even for an embodiment in which LFNST is applied only to a luma component, in the case of the single tree, a transform coefficient exists even for a chroma component to which LFNST is not applied. It is checked whether or not, that is, whether a transform coefficient exists in a region other than a region in which a transform coefficient to which LFNST is applied can exist even for chroma components to which LFNST is not applied. For this reason, an unnecessary constraint arises that a non-zero transform coefficient should exist only in a region in which a transform coefficient to which LFNST is applied can exist even for a transform coefficient block of a chroma component to which LFNST is not applied. 
     Since the variable LfnstDcOnly can also be updated by all of the Y, Cb, and Cr components in the case of the single tree, even if the value of the variable LfnstDcOnly is not updated due to the Y component, it may be updated to 0 by the Cb and Cr components and the LFNST index is may be signaled. 
     That is, when LFNST index signaling according to the current standard is applied and LFNST is applied only to the luma component in the case of the single tree, even though a non-zero transform coefficient for the luma component exists in a top-left position or a DC position or does not exist at all, if a non-zero transform coefficient exists outside the top-left position of any one of the transform blocks for the chroma component, the LFNST index may be signaled. 
     Therefore, in the case of the single tree and when the LFNST is applied only to the luma component, a method of applying LFNST index signaling will be described with reference to Tables 4 to 17 below only referring to the updated variable values after performing residual coding on the luma component (e.g., LfnstDcOnly, LfnstZeroOutSigCoeffFlag). 
     On the other hand, the following LFNST index signaling scheme is possible for a configuration in which the LFNST is applied only to the luma component in the case of the single tree. 
     1. In the case of a luma separation tree, signaling the LFNST index after transform coefficient parsing for the luma component is finished 
     2. In the case of a chroma separation tree, signaling the LFNST index after transform coefficient parsing for chroma components (Cb, Cr) is finished 
     3. In the case of a single tree, signaling the LFNST index after transform coefficient parsing for the luma component is finished. That is, LFNST index signaling is possible without waiting for transform coefficient parsing for chroma components 
     Transform coefficients parsing in the 1, 2, 3 may be performed in a residual coding syntax table, and according to an example, a position of the LFNST index signaling may be moved to a transform unit level from the existing coding unit level by reflecting the 1, 2, 3. Accordingly, syntax tables for a coding unit, a transform tree, and a transform unit are shown in Tables 4 to 6. 
     
       
         
           
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Descriptor 
               
               
                   
               
             
            
               
                 coding_unit( x0, y0, cbWidth, cbHeight, cqtDepth, treeType, modeType ) { 
                   
               
               
                  ...... 
                   
               
               
                  if( cu_cbf ) { 
                   
               
               
                   ...... 
                   
               
               
                   LfnstDcOnly = 1 
                   
               
               
                   LfnstZeroOutSigCoeffFlag = 1 
                   
               
               
                   MtsZeroOutSigCoeffFlag = 1 
                   
               
               
                   lfnstWidth = ( treeType = = 
                   
               
               
                 DUAL_TREE_CHROMA ) ? cbWidth / SubWidthC 
                   
               
               
                       : ( ( IntraSubPartitionsSplitType = = 
                   
               
               
                 ISP_VER_SPLIT ) ? cbWidth / 
                   
               
               
                        NumIntraSubPartitions : cbWidth ) 
                   
               
               
                   lfnstHeight = ( treeType = = 
                   
               
               
                 DUAL_TREE_CHROMA ) ? cbHeight / SubHeightC 
                   
               
               
                       : ( ( IntraSubPartitionsSplitType = = 
                   
               
               
                 ISP_HOR_SPLIT ) ? cbHeight / 
                   
               
               
                        NumIntraSubPartitions : cbHeight ) 
                   
               
               
                     transform_tree( x0, y0, cbWidth, cbHeight, treeType, chType ) 
                   
               
               
                   if( treeType != DUAL_TREE_CHROMA &amp;&amp; lfnst_idx = = 0 &amp;&amp; 
                   
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 &amp;&amp; Max( cbWidth, 
                   
               
               
                 cbHeight ) &lt;= 32 &amp;&amp; 
                   
               
               
                    IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT &amp;&amp; 
                   
               
               
                 cu_sbt_flag = = 0 &amp;&amp; 
                   
               
               
                    MtsZeroOutSigCoeffFlag = = 1 &amp;&amp; tu_cbf_luma[ x0 ][ y0 ] ) { 
                   
               
               
                    if( ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTER &amp;&amp; 
                   
               
               
                      sps_explicit_mts_inter_enabled_flag ) | | 
                   
               
               
                      ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
                   
               
               
                      sps_explicit_mts_intra_enabled_flag ) ) ) 
                   
               
               
                      mts_idx 
                 ae(v) 
               
               
                   } 
                   
               
               
                  } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Descriptor 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 transform_tree( x0, y0, tbWidth, tbHeight, treeType, chType ) { 
               
               
                  InferTuCbfLuma = 1 
               
               
                  if( IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; !cu_sbt_flag ) { 
               
               
                   if( tbWidth &gt; MaxTbSizeY | | tbHeight &gt; MaxTbSizeY ) { 
               
               
                    verSplitFirst = ( tbWidth &gt; MaxTbSizeY &amp;&amp; tbWidth &gt; tbHeight ) ? 1 : 0 
               
               
                    trafoWidth = verSplitFirst ? (tbWidth / 2) : tbWidth 
               
               
                    trafoHeight = !verSplitFirst ? (tbHeight / 2) : tbHeight 
               
               
                    transform_tree( x0, y0, trafoWidth, trafoHeight, treeType, chType ) 
               
               
                    if( verSplitFirst ) 
               
               
                     transform_tree( x0 + trafoWidth, y0, trafoWidth, trafoHeight, treeType, chType ) 
               
               
                    else 
               
               
                     transform_tree( x0, y0 + trafoHeight, trafoWidth, trafoHeight, treeType, chType ) 
               
               
                   } else { 
               
               
                    transform_unit( x0, y0, tbWidth, tbHeight, treeType, 0, chType ) 
               
               
                   } 
               
               
                  } else if( cu_sbt_flag ) { 
               
               
                   if( !cu_sbt_horizontal_flag ) { 
               
               
                    trafoWidth = tbWidth * SbtNumFourthsTb0 / 4 
               
               
                    transform_unit( x0, y0, trafoWidth, tbHeight, treeType , 0, 0, ) 
               
               
                  transform_unit( x0 + trafoWidth, y0, tbWidth − trafoWidth, tbHeight, treeType, 1, 0 ) 
               
               
                   } else { 
               
               
                    trafoHeight = tbHeight * SbtNumFourthsTb0 / 4 
               
               
                    transform_unit( x0, y0, tbWidth, trafoHeight, treeType , 0, 0 ) 
               
               
                  transform_unit( x0, y0 + trafoHeight, tbWidth, tbHeight − trafoHeight, treeType, 1, 0 ) 
               
               
                   } 
               
               
                  } else if( IntraSubPartitionsSplitType = = ISP_HOR_SPLIT ) { 
               
               
                   trafoHeight = tbHeight / NumIntraSubPartitions 
               
               
                   for( partIdx = 0; partIdx &lt; NumIntraSubPartitions; partIdx++ ) 
               
               
                  transform_unit( x0, y0 + trafoHeight * partIdx, tbWidth, trafoHeight, treeType, partIdx, 0 
               
               
                 ) 
               
               
                  } else if( IntraSubPartitionsSplitType = = ISP_VER_SPLIT ) { 
               
               
                   trafoWidth = tbWidth / NumIntraSubPartitions 
               
               
                   for( partIdx = 0; partIdx &lt; NumIntraSubPartitions; partIdx++ ) 
               
               
                  transform_unit( x0 + trafoWidth * partIdx, y0, trafoWidth, tbHeight, treeType, partIdx, 0 
               
               
                 ) 
               
               
                  } 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 6 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  if( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                    treeType = = SINGLE_TREE &amp;&amp; subTuIndex = = 
               
               
                 NumIntraSubPartitions − 1 ) { 
               
               
                   xC = CbPosX[ chType ][ x0 ][ y0 ] 
               
               
                   yC = CbPosY[ chType ][ x0 ][ y0 ] 
               
               
                   wC = CbWdith[ chType ][ x0 ][ y0 ] / SubWidthC 
               
               
                   hC = CbHeight[ chType ][ x0 ][ y0 ] / SubHeightC 
               
               
                  } else { 
               
               
                   xC = x0 
               
               
                   yC = y0 
               
               
                   wC = tbWidth / SubWidthC 
               
               
                   hC = tbHeight / SubHeightC 
               
               
                  } 
               
               
                  chromaAvailable = treeType != DUAL_TREE_LUMA &amp;&amp; ChromaArrayType != 
               
               
                 0 &amp;&amp; 
               
               
                   ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                   ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                   subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( ( treeType = = SINGLE_TREE | | treeType = = DUAL_TREE_CHROMA ) 
               
               
                 &amp;&amp; 
               
               
                    ChromaArrayType != 0 &amp;&amp; ( IntraSubPartitionsSplitType = = 
               
               
                 ISP_NO_SPLIT &amp;&amp; 
               
               
                    ( ( subTuIndex = = 0 &amp;&amp; cu_sbt_pos_flag ) | | 
               
               
                    ( subTuIndex = = 1 &amp;&amp; !cu_sbt_pos flag ) ) ) ) | | 
               
               
                    ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                    ( subTuIndex = = NumIntraSubPartitions − 1 ) ) ) { 
               
               
                   tu_cb_coded_flag[ xC ][ yC ] 
                 ae(v) 
               
               
                   tu_cr_coded_flag[ xC ][ yC ] 
                 ae(v) 
               
               
                  } 
               
               
                  if( treeType = = SINGLE_TREE | | treeType = = DUAL_TREE_LUMA ) { 
               
               
                   if( ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; !( cu_sbt_flag &amp;&amp; 
               
               
                     ( ( subTuIndex = = 0 &amp;&amp; cu_sbt_pos_flag ) | | 
               
               
                     ( subTuIndex = = 1 &amp;&amp; !cu_sbt_pos_flag ) ) ) &amp;&amp; 
               
               
                     ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA | | 
               
               
                     ( chromaAvailable &amp;&amp; ( tu_cb_coded_flag[ xC ][ yC ] | | 
               
               
                     tu_cr_coded_flag[ xC ][ yC ] ) ) | | 
               
               
                     CbWidth[ chType ][ x0 ][ y0 ] &gt; MaxTbSizeY | | 
               
               
                     CbHeight[ chType ][ x0 ][ y0 ] &gt; MaxTbSizeY ) ) | | 
               
               
                     ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                     ( subTuIndex &lt; NumIntraSubPartitions − 1 | | !InferTuCbfLuma ) ) ) 
               
               
                    tu_y_coded_flag[ x0 ][ y0 ] 
                 ae(v) 
               
               
                   if(IntraSubPartitionsSplitType != ISP_NO_SPLIT ) 
               
               
                    InferTuCbfLuma = InferTuCbfLuma &amp;&amp; !tu_y_coded_flag[ x0 ][ y0 ] 
               
               
                  } 
               
               
                   
               
            
           
         
       
     
     The meanings of the main variables in the table are as follows. 
     1. cbWidth, cbHeight: a width and a height of a current coding block 
     2. log2TbWidth, log2TbHeight: a log value with base-2 for the width and height of a current transform block, the zero-out may be reflected and it may be reduced to a top-left region in which a non-zero coefficient may exist. 
     3. sps_lfnst_enabled_flag: A flag indicating whether to enable the LFNST. If the flag value is 0, it indicates that LFNST is not enabled, and if the flag value is 1, it indicates that LFNST is enabled. It is defined in a Sequence Parameter Set (SPS). 
     4. CuPredMode[chType][x0][y0]: a prediction mode corresponding to a variable chType and a (x0, y0) position. chType may have values of 0 and 1, where 0 represents the luma component and 1 represents the chroma component. The (x0, y0) position indicates a position on a picture, and MODE_INTRA (intra prediction) and MODE INTER (inter prediction) are possible as CuPredMode[chType][x0][y0] values. 
     5. IntraSubPartitionsSplit[x0][y0]: The contents of the (x0, y0) position are the same as in No. 4. Indicates whether ISP division is applied at the (x0, y0) position, and ISP_NO_SPLIT indicates that the coding unit corresponding to the (x0, y0) position is not divided into partition blocks. 
     6. intra_mip_flag[x0][y0]: The contents of the (x0, y0) position are the same as in No. 4. The intra_mip_flag is a flag indicating whether a Matrix-based Intra Prediction (MIP) prediction mode is applied. A flag value of 0 indicates that MIP is not applicable, and a flag value of 1 indicates that MIP is applied. 
     7. cIdx: A value of 0 indicates luma, and a value of 1 and 2 indicates Cb and Cr which are chroma components, respectively. 
     8. treeType: indicates single-tree and dual-tree (SINGLE TREE: single tree, DUAL_TREE_LUMA: dual tree for luma component, DUAL_TREE_CHROMA: dual tree for chroma component) 
     9. lastSubBlock: indicates a position in a scan order of a sub-block (Coefficient Group (CG)) in which a last non-zero coefficient is located. 0 indicates a sub-block including a DC component, and if greater than 0, it is not a sub-block including the DC component. 
     10. lastScanPos: indicates where the last significant coefficient is in a scan order within one sub-block. If the one sub-block consists of 16 positions, values from 0 to 15 are possible. 
     11. lfnst_idx[x0][y0]: LFNST index syntax element to be parsed. If it is not parsed, it is inferred as a value of 0. That is, a default value is set to 0, and indicates that LFNST is not applied. 
     12. LastSignificantCoeffX, LastSignificantCoeffY: Indicates x-coordinate and y-coordinate where the last significant coefficient is located in the transform block. The x-coordinate starts at 0 and increases from left to right, and the y-coordinate starts at 0 and increases from top to bottom. If the values of both variables are 0, it means that the last significant coefficient is located at DC. 
     13. cu_sbt_flag: A flag indicating whether SubBlock Transform (SBT) included in the current VVC standard is applicable. If the flag value is 0, it indicates that SBT is not applicable, and if the flag value is 1, Indicates that SBT is applied. 
     14. sps_explicit_mts_inter_enabled_flag, sps_explicit_mts_intra_enabled_flag: A flag indicating whether or not explicit MTS is applied to an inter CU and an intra CU, respectively. If the corresponding flag value is 0, it indicates that MTS is not enabled to the inter CU or intra CU. If the corresponding flag value is 1, it indicates that MTS is is enabled. 
     15. tu_mts_idx[x0][y0]: MTS index syntax element to be parsed. If it is not parsed, it is inferred as a value of 0. That is, a default value is set to 0, and it indicates that DCT-2 is applied to both horizontal and vertical directions. 
     The variable LfnstZeroOutSigCoeffFlag of Table 6 is 0 if there is a significant coefficient at a position where zero out is applied when LFNST is applied, otherwise it is 1. The variable LfnstZeroOutSigCoeffFlag may be set according to various conditions shown in the residual coding syntax table of Table 10. 
     The variable LfnstZeroOutSigCoeffFlag indicates whether a significant coefficient exists in a second region except for a first region at a top-left of the current block. This value is initially set to 1, and if the significant coefficient exists in the second region, the value may be changed to 0. The LFNST index can be parsed only when the initially set value of the variable LfnstZeroOutSigCoeffFlag is maintained at  1 . When determining whether the variable LfnstZeroOutSigCoeffFlag value is 1 and deriving the variable LfnstZeroOutSigCoeffFlag value, since LFNST may be applied to both the luma component and the chroma component of the current block, a color index of the current block is not determined. 
     In addition, according to an example, the variable LfnstDcOnly of Table 6 becomes 1 if all of the last significant coefficient are located in DC position (top-left position) for transform blocks in which a coded block flag (CBF, 1 if there is at least one significant coefficient in the corresponding block, 0 otherwise) having a value of 1, otherwise it becomes 0. More specifically, in the case of dual tree luma, the position of the last significant coefficient is checked with respect to one luma transform block, and in the case of dual tree chroma, the position of the last significant coefficient is checked with respect to both the transform blocks for Cb and Cr. In the case of the single tree, the position of the last significant coefficient may be checked with respect to transform blocks for luma, Cb, and Cr. 
     Instead of signaling the LFNST index after parsing all transform coefficients corresponding to one coding unit as shown in Table 4, the LFNST index may be signaled in the process of parsing the syntax table for the transform unit as shown in Table 6. 
     Here, transform coefficients corresponding to one coding unit may mean all transform coefficients for Y, Cb, and Cr components in the case of a single tree (a transform block for Y, a transform block for Cb, All transform coefficients belonging to the transform block for Cr), and they may mean transform coefficients for the Y component in the case of a luma separation tree (transform coefficients belonging to the transform block for Y), and they may mean all transform coefficients for Cr components in the case of a chroma separation tree (all transform coefficients belonging to a transform block for Cb and a transform block for Cr). 
     The transform tree is invoked from the syntax table for the coding unit as shown in Table 4, and the transform unit is invoked from the transform tree as shown in Table 5. In the case of where it is not the ISP mode, the transform unit is usually invoked only once. Specifically, if the width is greater than a maximum transform size, the maximum transform size is used as a unit to divide into several transform units in a vertical direction and code. Similarly, if the height is greater than the maximum transform size, the maximum transform size is used as a unit to divide into several transform units in a horizontal direction and code. In this way, when the coding unit is divided into several transform units, LFNST is not applied in VVC. 
     On the other hand, in the case of the ISP mode, that is, when divided into N ISP partition blocks in the horizontal or vertical direction and coded, the transform unit shown in Table 6 may be invoked as many as the number of divided partition blocks. In Table 6, subTuIndex is used as an index indicating each ISP partition block in ISP mode, and may have a value from 0 to (NumIntraSubPartitions−1). Here, NumIntraSubPartitions represents the number of the divided ISP partition blocks. 
     In the case of an ISP, since the LFNST index is signaled only once for the coding unit (CU) and the corresponding LFNST index is applied equally to all ISP partition blocks (that is, the LFNST kernel indicated by the LFNST index may be equally applied, When the LFNST index value is 0, LFNST may not be applied), according to an example of the current VVC standard, when the subTuIndex value is (NumIntraSubPartitions−1), it may be appropriate to signal the LFNST index. 
     More specifically, when a non-zero transform coefficient exists in a position other than a region where an LFNST transform coefficient (a transform coefficient generated by applying forward LFNST) may be located for all ISP partition blocks, in order to the LFNST index to be inferred as 0 without signaling, it is possible to configure that the transform coefficients for all ISP partition blocks are parsed to check whether a non-zero transform coefficient exists other than a region where the LFNST transform coefficient can be located, and then signal the LFNST index. 
     Here, when the ISP partition block is 4×4 or 8×8, LFNST transform coefficients may exist in up to 8 positions according to a forward scanning order from the top-left position of the block, and in the case of a block size to which the remaining LFNST can be applied (i.e., when the width and height are equal to or greater than 4 and not 4×4 or 8×8) LFNST transform coefficients may exist in up to 16 positions from the top-left position of the block in the forward scanning order. 
     Whether to signal the LFNST index in the ISP mode may be determined through the following conditions belonging to the conditions for determining the variable allowLfnstIndex of Table 6. 
     “(IntraSubPartitionsSplitType==ISP_NO_SPLIT∥(IntraSubPartitionsSplitType!=ISP_NO_SPLIT &amp;&amp; subTuIndex==NumIntraSubPartitions−1))” 
     In the above condition, when the variable IntraSubPartitionsSplitType is ISP_NO_SPLIT, it indicates that the ISP is not applied, and when the variable IntraSubPartitionsSplitType is not ISP_NO_SPLIT, it indicates that the ISP mode is applied. 
     If the value of the variable allowLfnstIndex in Table 6 is 1, the LFNST index may be signaled, and if the value of the variable allowLfnstIndex is 0, the LFNST index may not be signaled. In addition, in the case of the ISP mode according to the above condition, the value of the variable allowLfnstIndex may be set to 1 only when transform unit parsing for a last ISP partition block is performed (subTuIndex==NumIntraSubPartitions−1). 
     In Table 6, in the case of a luma separation tree or a single tree, the LFNST index lfnst_idx is parsed after parsing the transform skip flag and transform coefficients (residual_coding or residual_ts_coding) for the luma component. Table 7 is an excerpt of the relevant part from Table 6. 
     
       
         
           
               
             
               
                 TABLE 7 
               
               
                   
               
             
            
               
                  if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) &amp;&amp; 
               
               
                   LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                    treeType != DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 0 ] 
               
               
                 = = 0 ) 
               
               
                    lfnst_idx 
               
               
                   
               
            
           
         
       
     
     In Table 7, treeType !=DUAL_TREE_CHROMA indicates a case of a luma separation tree or a single tree, and transform_skip_flag[x0][y0][0]-0 indicates that the luma component is not coded with transform skip. In the case of the single tree, since the embodiment in which LFNST is applied only to the luma component is dealt with, only whether the transform for the luma component is skipped is checked in Table 7. 
     In addition, as shown in Tables 6 and 7, when parsing the LFNST index, the CBF, tu_y_coded_flag[x0][y0] values for the luma transform block are not directly checked, but tu_y_coded_flag[x0][y0] value is indirectly reflected in the “(IntraSubPartitionsSplitType!=ISP_NO_SPLIT∥LfnstDcOnly==0)” condition in Table 7. That is, as a result, the LFNST index may be signaled only when the CBF value for the luma component is not 0, that is, only when a non-zero transform coefficient exists in the luma transform block. 
     Since the current VVC standard guarantees that a non-zero transform coefficient exists for at least one or more of the partition blocks when the ISP mode is applied, when the condition “IntraSubPartitionsSplitType !=ISP_NO_SPLIT” is satisfied, at least one or more non-zero transform coefficients exists. 
     In addition, the variable LfnstDcOnly may be updated to 0 only when the residual_coding syntax table of Table 10 is applied. Since the residual coding for the luma component can be invoked only when a value of tu_y_coded_flag[x0][y0] is not 0 as shown in Table 6, the variable LfnstDcOnly value of 0 is only possible when the value of tu_y_coded_flag[x0][y0] is not 0. Therefore, the “LfnstDcOnly==0” condition check includes a condition that the value of tu_y_coded_flag[x0][y0] is not 0. 
     In Table 6, in the case of the chroma separation tree, the LFNST index (lfnst_idx) is parsed after parsing the transform_skip_flag and transform coefficients (residual_coding or residual_ts_coding) for the chroma components (Cb, Cr). Table 8 is an excerpt of the relevant part from Table 6. 
     
       
         
           
               
             
               
                 TABLE 8 
               
               
                   
               
             
            
               
                 if( LfnstDcOnly = = 0 &amp;&amp; LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                  treeType = = DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 1 ] 
               
               
                 = = 0 &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 2 ] = = 0 ) 
               
               
                  lfnst_idx 
               
               
                   
               
            
           
         
       
     
     In Table 8, treeType==DUAL_TREE_CHROMA indicates the case of a chroma seperation tree, and since transform_skip_flag[x0][y0][1] 0 &amp;&amp; transform_skip_flag[x0][y0][2] 0 indicates that both chroma components (Cb, Cr) are not coded with transform skip, LFNST is not applied if even one component is coded with the transform skip in the case of a chroma separation tree. 
     Similarly with Table 7, the variable LfnstDcOnly may be updated to a value of 0 only when the residual coding syntax table of Table 10 is applied, and since the residual coding for the Cb (Cr) component can be invoked only when the value tu_cb_coded_flag[x0][y0] (tu_cr_coded_flag[x0][y0]) is non-zero as shown in Table 6, so in order for the variable LfnstDcOnly value to be 0, at least one of tu_cb_coded_flag[x0][y0] and tu_cr_coded_flag[x0][y0] shouldn&#39;t be 0. Accordingly, the “LfnstDcOnly==0” condition check includes a condition that at least one of the tu_cb_coded_flag[x0][y0] value and the tu_cr_coded_flag[x0][y0] value is not 0. 
     As shown in Tables 7 and 8, the LFNST index is signaled only when the variable allowLfnstIndex value is 1, and the variable allowLfnstIndex value is determined as shown in Table 6, which is shown in Table 9 when the relevant part is extracted. 
     
       
         
           
               
             
               
                 TABLE 9 
               
               
                   
               
             
            
               
                 allowLfnstIndex = Min( lfnstWidth, lfnstHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 1 
               
               
                  &amp;&amp; 
               
               
                  CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                  ( treeType != DUAL_TREE_CHROMA | | !intra_mip_flag[ x0 ][ y0 ] | | 
               
               
                   Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                  Max( cbWidth, cbHeight ) &lt;= MaxTbSizeY &amp;&amp; 
               
               
                  ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                   ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                    subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                   
               
            
           
         
       
     
     The following is the case in which the conditions in Table 9 are classified and listed. 
     1. LFNST should be enabled (the sps_lfnst_enable_flag value is signaled as 1 through a sequence parameter set, corresponding to “sps_lfnst_enabled_flag==1” in Table 9) 
     2. Both the width and height of the transform block should be equal to or greater than 4 (in the case of ISP mode, both the width and height of the ISP partition block should be greater than or equal to 4, corresponding to “Min(lfnstWidth, lfnstHeight)&gt;=4 corresponding to” in Table 9) 
     3. Currently coded coding unit should be coded in intra prediction mode (corresponding to “CuPredMode[chType][x0][y0]==MODE_INTRA” in Table 9) 
     4. When MIP prediction mode is not applied (corresponding to treeType !=DUAL_TREE_CHROMA∥!intra_mip_flag[x0][y0] in Table 9), or when MIP prediction mode is applied, both width and height of the corresponding transform block should be at least 16 (corresponding to “Min(lfnstWidth, lfnstHeight)&gt;=16” in Table 9) 
     5. When both the width and height of the coding block for the luma component are equal to or less than a maximum transform size for the luma component (in the case of a chroma separation tree, when both the width and height when converted to the coding block for the luma component are equal to or less than the maximum transform size for the luma component, for example, when the color format is 4:2:0, the width and height of the luma component corresponding to the chroma component are twice as the width and height of the chroma component, respectively), corresponding to “Max(cbWidth, cbHeight)&lt;=MaxTbSizeY” in Table 9 
     6. When ISP is not applied, or when it is the last ISP partition block in the case of being coded in ISP mode (corresponding to the condition “IntraSubPartitionsSplitType==ISP_NO_SPLIT∥(IntraSubPartitionsSplitType !=ISP_NO_SPLIT &amp;&amp; subTuIndex==NumIntraSubPartitions−” in Table 9) 
     In Tables 7 and 8, whether the LFNST index is signaled may be determined by checking the variable LfnstDcOnly value and the variable UnstZeroOutSigCoeffFlag value. The variable LfnstDcOnly and the variable UnstZeroOutSigCoeffFlag are initialized to 1 in the coding unit syntax table of Table 4, and may be updated as shown in Table 10 in the residual coding syntax table. Residual coding may be invoked in the transform unit as shown in Table 6. 
     
       
         
           
               
               
             
               
                   
                 TABLE 10 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) { 
                   
               
               
                  if( sps_mts_enabled_flag &amp;&amp; cu_sbt_flag &amp;&amp; cIdx = = 0 &amp;&amp; 
               
               
                    log2TbWidth = = 5 &amp;&amp; log2TbHeight &lt; 6 ) 
               
               
                   log2ZoTbWidth = 4 
               
               
                  else 
               
               
                   log2ZoTbWidth = Min( log2TbWidth, 5 ) 
               
               
                  if( sps_mts_enabled_flag &amp;&amp; cu_sbt_flag &amp;&amp; cIdx = = 0 &amp;&amp; 
               
               
                    log2TbWidth &lt; 6 &amp;&amp; log2TbHeight = = 5 ) 
               
               
                   log2ZoTbHeight = 4 
               
               
                  else 
               
               
                   log2ZoTbHeight = Min( log2TbHeight, 5 ) 
               
               
                  if( log2TbWidth &gt; 0 ) 
               
               
                   last_sig_coeff_x_prefix 
                 ae(v) 
               
               
                  if( log2TbHeight &gt; 0 ) 
               
               
                   last_sig_coeff_y_prefix 
                 ae(v) 
               
               
                  if( last_sig_coeff_x_prefix &gt; 3 ) 
               
               
                   last_sig_coeff_x_suffix 
                 ae(v) 
               
               
                  if( last_sig_coeff_y_prefix &gt; 3 ) 
               
               
                   last_sig_coeff_y_suffix 
                 ae(v) 
               
               
                  log2TbWidth = log2ZoTbWidth 
               
               
                  log2TbHeight = log2ZoTbHeight 
               
               
                  remBinsPass1 = ( ( 1 &lt;&lt; ( log2TbWidth + log2TbHeight ) ) * 7 ) &gt;&gt; 2 
               
               
                  log2SbW = ( Min( log2TbWidth, log2TbHeight ) &lt; 2 ? 1 : 2 ) 
               
               
                  log2SbH = log2SbW 
               
               
                  if( log2TbWidth + log2TbHeight &gt; 3 ) { 
               
               
                   if( log2TbWidth &lt; 2 ) { 
               
               
                     log2SbW = log2TbWidth 
               
               
                     log2SbH = 4 − log2SbW 
               
               
                   } else if( log2TbHeight &lt; 2 ) { 
               
               
                     log2SbH = log2TbHeight 
               
               
                     log2SbW = 4 − log2SbH 
               
               
                   } 
               
               
                  } 
               
               
                  numSbCoeff = 1 &lt;&lt; ( log2SbW + log2SbH ) 
               
               
                  lastScanPos = numSbCoeff 
               
               
                  lastSubBlock = ( 1 &lt;&lt; ( log2TbWidth + log2TbHeight − ( log2SbW + log2SbH ) ) ) − 1 
               
               
                  do { 
               
               
                   if( lastScanPos = = 0 ) { 
               
               
                     lastScanPos = numSbCoeff 
               
               
                     lastSubBlock− − 
               
               
                   } 
               
               
                   lastScanPos− − 
               
               
                   xS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ] 
               
               
                      [ lastSubBlock ][ 0 ] 
               
               
                   yS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ] 
               
               
                      [ lastSubBlock ][ 1 ] 
               
               
                   xC = ( xS &lt;&lt; log2SbW ) + DiagScanOrder[ log2SbW ][ log2SbH ][ lastScanPos ][ 0 ] 
               
               
                   yC = ( yS &lt;&lt; log2SbH ) + DiagScanOrder[ log2SbW ][ log2SbH ][ lastScanPos ][ 1 ] 
               
               
                  } while( ( xC != LastSignificantCoeffX ) | | ( yC != LastSignificantCoeffY ) ) 
               
               
                  if( lastSubBlock = = 0 &amp;&amp; log2TbWidth &gt;= 2 &amp;&amp; log2TbHeight &gt;= 2 
               
               
                 &amp;&amp; 
               
               
                   !transform_skip_flag[ x0 ][ y0 ][ cIdx ] &amp;&amp; lastScanPos &gt; 0 ) 
               
               
                   LfnstDcOnly = 0 
               
               
                  if( ( lastSubBlock &gt; 0 &amp;&amp; log2TbWidth &gt;= 2 &amp;&amp; log2TbHeight &gt;= 2 ) | | 
               
               
                   ( lastScanPos &gt; 7 &amp;&amp; ( log2TbWidth = = 2 | | log2TbWidth = = 3 ) &amp;&amp; 
               
               
                   log2TbWidth = = log2TbHeight ) ) 
               
               
                   LfnstZeroOutSigCoeffFlag = 0 
               
               
                  QState = 0 
               
               
                  for( i = lastSubBlock; i &gt;= 0; i− − ) { 
               
               
                   startQStateSb = QState 
               
               
                   xS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ] 
               
               
                      [ i ][ 0 ] 
               
               
                   yS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ] 
               
               
                      [ i ][ 1 ] 
               
               
                   inferSbDcSigCoeffFlag = 0 
               
               
                   if( i &lt; lastSubBlock &amp;&amp; i &gt; 0 ) { 
               
               
                     coded_sub_block_flag[ xS ][ yS ] 
                 ae(v) 
               
               
                     inferSbDcSigCoeffFlag = 1 
               
               
                   } 
               
               
                   if( coded_sub_block_flag[ xS ][ yS ] &amp;&amp; ( xS &gt; 3 | | yS &gt; 3 ) &amp;&amp; cIdx = = 
               
               
                 0 ) { 
               
               
                     MtsZeroOutSigCoeffFlag = 0 
               
               
                   } 
               
               
                      ...... 
               
               
                    } 
               
               
                    ...... 
               
               
                   
               
            
           
         
       
     
     As shown in Table 10, the variable LfnstDcOnly may be set to 0 when the last non-zero transform coefficient is located in a position other than the top-left position of the corresponding transform block (which may correspond to the ISP partition block in the case of ISP mode). The value of the variable LfnstDcOnly is not updated inside the residual transform skip coding (residual_ts_coding) syntax table invoked in Table 6, and since the residual coding syntax table of Table 10 is invoked only when the corresponding transform block contains one or more non-zero transform coefficients (i.e., only invoked when the corresponding coded block flag (CBF) value is 1), summing up, the LfnstDcOnly variable may be set to 0 when any one of the non-zero transform coefficients exists in a position other than the top-left position in the transform block with the CBF value of 1 included in the current transform unit. 
     As shown in Table 6, in the case of the luma seperation tree and the single tree, since the residual coding syntax table can be invoked only for the luma component, the variable LfnstDcOnly may be updated to 0 only for the luma component, which corresponds to conforms to the embodiment in which LFNST is applied only to the luma component in the case of the luma seperation tree and the single tree. 
     In addition, in the case of the chroma separation tree as shown in Table 6, since the residual coding syntax table can be invoked only for the chroma components (Cb, Cr), the variable LfnstDcOnly may be updated to 0 only for the chroma components (i.e., in Table 6, when a treeType is DUAL_TREE_CHROMA, the transform_skip_flag parsing and the residual coding syntax table parsing for the luma component are not performed), This corresponds to an embodiment in which the LFNST index for the chroma component can be signaled in the case of the chroma separation tree. 
     In addition, as shown in Table 7, when the ISP mode is applied when the luma separation tree or single tree is applied, the condition for LfnstDcOnly may not be checked when determining whether to signal the LFNST index, and when the ISP is not applied it can be configured to signal the LFNST index only when the LfnstDcOnly value is 0. 
     As shown in Table 8, since the ISP is not applied to the chroma component in the case of the chroma separation tree, the LFNST index may be signaled only when the LfnstDcOnly value is 0. 
     As shown in Table 10, the variable LfnstZeroOutSigCoeffFlag may be updated to 0 when the last non-zero transform coefficient exists in a region in which an LFNST transform coefficient (i.e., a transform coefficient generated by applying forward LFNST) can not exist in the current transform block (which may correspond to an ISP partition block in the case of the ISP mode). 
     The variable LfnstZeroOutSigCoeffFlag value is not updated in the residual transform skip coding syntax table invoked in Table 6, and since the residual coding syntax table in Table 10 is called only when the corresponding transform block includes one or more than non-zero transform coefficient (that is, it is invoked only when the corresponding coded block flag (CBF) value is 1), collectively, the variable LfnstZeroOutSigCoeffFlag may be set to 0 when the last non-zero transform coefficient is not located in a region where an LFNST transform coefficient cannot exist in the transform block with the CBF value of 1 included in the current transform unit. 
     As shown in Table 6, in the case of the luma speration tree and the single tree, since the residual coding syntax table can be invoked only for the luma component, the variable LfnstZeroOutSigCoeffFlag is updated to 0 only for the luma component, which conforms to the embodiment in which LFNST is applied only to the luma component in the case of the luma speration tree and the single tree. 
     In addition, in the case of the chroma separation tree as shown in Table 6, since the residual coding syntax table can be invoked only for the chroma components (Cb, Cr), the variable LfnstZeroOutSigCoeffFlag may be updated to 0 only for the chroma components (e.g., in Table 6, when the treeType is DUAL_TREE_CHROMA, the transform_skip_flag parsing and the residual coding syntax table parsing for the luma component are not performed), which conforms to the embodiment in which the LFNST index is signaled for the chroma component in the case of the chroma speration tree. 
     As shown in Tables 7 and 8, the LFNST index may be signaled only when the LfnstZeroOutSigCoeffFlag value is 1. 
     LfnstDcOnly, LfnstZeroOutSigCoeffFlag, lfnstWidth, and lfnstHeight are not initialized or set in the coding unit syntax table as shown in Table 4, and may be set in the transform unit syntax table as shown in Table 11. 
     
       
         
           
               
               
             
               
                   
                 TABLE 11 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  ...... 
               
               
                   if( subTuIndex = = 0 ) { 
               
               
                   LfnstDcOnly = 1 
               
               
                   LfnstZeroOutSigCoeffFlag = 1 
               
               
                   } 
               
               
                  lfnstWidth = ( treeType = = DUAL_TREE_CHROMA ) ? cbWidth / SubWidthC 
               
               
                        : ( ( IntraSubPartitionsSplitType = = ISP_VER_SPLIT ) ? cbWidth / 
               
               
                        NumIntraSubPartitions : cbWidth ) 
               
               
                  lfnstHeight = ( treeType = = DUAL_TREE_CHROMA ) ? cbHeight / SubHeightC 
               
               
                        : ( ( IntraSubPartitionsSplitType = = ISP_HOR_SPLIT) ? cbHeight / 
               
               
                         NumIntraSubPartitions : cbHeight ) 
               
               
                  allowLfnstIndex = Min( lfnstWidth, lfnstHeight) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag 
               
               
                         = = 1 &amp;&amp; 
               
               
                         CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                         ( treeType != DUAL_TREE_CHROMA | | 
               
               
                         !intra_mip_flag[ x0 ][ y0 ] | | 
               
               
                          Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                         Max( cbWidth, cbHeight ) &lt;= MaxTbSizeY &amp;&amp; 
               
               
                         ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                          ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                           subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( tu_cbf_luma[ x0 ][ y0 ] &amp;&amp; treeType != DUAL_TREE_CHROMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 0 ] &amp;&amp; 
               
               
                     tbWidth &lt;= MaxTsSize &amp;&amp; tbHeight &lt;= MaxTsSize &amp;&amp; 
               
               
                     ( IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT ) &amp;&amp; !cu_sbt_flag ) 
               
               
                     transform_skip_flag[ x0 ][ y0 ][ 0 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ x0 ][ y0 ][ 0 ] ) 
               
               
                     residual_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                   else 
               
               
                     residual_ts_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                  } 
               
               
                   if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) 
               
               
                 &amp;&amp; 
               
               
                    LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                       treeType ! = DUAL_TREE_CHROMA &amp;&amp; 
               
               
                 transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 ) 
               
               
                       lfnst_idx 
                 ae(v) 
               
               
                  if( tu_cbf_cb[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 1 ] &amp;&amp; 
               
               
                     wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                     transform_skip_flag[ xC ][ yC ][ 1 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 1 ] ) 
               
               
                     residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   else 
               
               
                     residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; 
               
               
                      LfnstZeroOutSigCoeffFlag = = 1 ) 
               
               
                      lfnst_idx 
                 ae(v) 
               
               
                  } 
               
               
                  if( tu_cbf_cr[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA &amp;&amp; 
               
               
                   !( tu_cbf_cb[ xC ][ yC ] &amp;&amp; tu_joint_cbcr_residual_flag[ xC ][ yC ] ) ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 2 ] &amp;&amp; 
               
               
                     wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                     transform_skip_flag[ xC ][ yC ][ 2 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 2 ] ) 
               
               
                     residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                   else 
               
               
                     residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                  } 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; 
               
               
                 allowLfnstIndex &amp;&amp; 
               
               
                      treeType = = DUAL_TREE_CHROMA &amp;&amp; 
               
               
                 transform_skip_flag[ x0 ][ y0 ][ 1 ] = = 0 &amp;&amp; 
               
               
                      transform_skip_flag[ x0 ][ y0 ][ 2 ] = = 0 ) 
               
               
                      lfnst_idx 
                 ae(v) 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     As shown in Table 11, the variable LfnstDcOnly and the variable LfnstZeroOutSigCoeffFlag are initialized only when the subTuIndex value is 0, and when not in the ISP mode, 0 may always be set as the subTuIndex value as shown in Table 5. 
     In addition, in the case of the ISP mode (IntraSubPartitionsSplitType value is ISP_HOR_SPLIT or ISP_VER_SPLIT in Table 5), since the subTbIndex value can be changed from 0 to (NumIntraSubPartitions−1), LfnstDcOnly and LfnstZeroOutSigCoeffFlag need to be initialized only for a first ISP partition block (only when subTbIndex is 0). Therefore, in the case of ISP mode, LfnstDcOnly and LfnstZeroOutSigCoeffFlag are updated while processing all the transform blocks belonging to one transform unit (e.g., ISP partition blocks) one by one according to whether it is the single tree or the luma/chroma seperation tree. 
     When the transform unit syntax table is configured as shown in Table 11, the coding unit syntax table may be configured as shown in Table 12. In this case, Tables 5 and 10 may be applied as they are. 
     
       
         
           
               
               
             
               
                   
                 TABLE 12 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 coding_unit( x0, y0, cbWidth, cbHeight, cqtDepth, treeType, modeType ) { 
                   
               
               
                  ...... 
               
               
                  if( cu_cbf ) { 
               
               
                   ...... 
               
               
                   MtsZeroOutSigCoeffFlag = 1 
               
               
                   transform_tree( x0, y0, cbWidth, cbHeight, treeType, chType ) 
               
               
                   if( treeType != DUAL_TREE_CHROMA &amp;&amp; lfnst_idx = = 0 &amp;&amp; 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 &amp;&amp; Max( cbWidth, cbHeight ) 
               
               
                 &lt;= 32 &amp;&amp; 
               
               
                    IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT &amp;&amp; cu_sbt_flag 
               
               
                 = = 0 &amp;&amp; 
               
               
                    MtsZeroOutSigCoeffFlag = = 1 &amp;&amp; tu_cbf_luma[ x0 ][ y0 ] ) { 
               
               
                    if( ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTER &amp;&amp; 
               
               
                     sps_explicit_mts_inter_enabled_flag ) | | 
               
               
                     ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                     sps_explicit_mts_intra_enabled_flag ) ) ) 
               
               
                     mts_idx 
                 ae(v) 
               
               
                   } 
               
               
                  } 
               
               
                   
               
            
           
         
       
     
     In the case of applying the LFNST only to the luma component in the case of the single tree, the method of applying the scaling list in the case of the chroma component may be applied as shown in Table 13. 
     
       
         
           
               
               
             
               
                 TABLE 13 
               
               
                   
               
             
            
               
                 8.7.3 
                 Scaling process for transform coefficients 
               
            
           
           
               
            
               
                 ... 
               
               
                 The (nTbW)x(nTbH) array dz is set equal to the (nTbW)x(nTbH) array 
               
               
                 TransCoeffLevel[ xTbY ][ yTbY ][ cIdx ]. 
               
               
                 The variable applyLfnstFlag is derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If treeType is equal to SINGLE_TREE, 
               
            
           
           
               
               
            
               
                   
                 applyLfnstFlag = ( lfnst_idx &gt; 0 &amp;&amp; cIdx = = 0 ) ? 1 : 0 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, 
               
            
           
           
               
               
            
               
                   
                 applyLfnstFlag = ( lfnst_idx &gt; 0 ) ? 1 : 0 
               
            
           
           
               
            
               
                 For the derivation of the scaled transform coefficients d[ x ][ y ] with x = 0..nTbW − 1, y = 0..nTbH − 1, the 
               
               
                 following applies: 
               
            
           
           
               
               
            
               
                 - 
                 The intermediate scaling factor m[ x ][ y ] is derived as follows: 
               
               
                 - 
                 If one or more of the following conditions are true, m[ x ][ y ] is set equal to 16: 
               
            
           
           
               
               
               
            
               
                   
                  - 
                 sps_scaling_list_enabled_flag is equal to 0. 
               
               
                   
                  - 
                 pic_scaling_list_present_flag is equal to 0. 
               
               
                   
                  - 
                 transform_skip_flag[ xTbY ][ yTbY ][ cIdx ] is equal to 1. 
               
               
                   
                  - 
                 scaling_matrix_for_lfnst_disabled_flag is equal to 1 and applyLfnstFlag is not equal to 0. 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, the following applies: 
               
            
           
           
               
            
               
                 ... 
               
               
                   
               
            
           
         
       
     
     Currently, in VVC WD, a syntax element called scaling_matrix_for_lfnst_disabled_flag is defined. If scaling_matrix_for_lfnst_disabled_flag is 1, the scaling list may not be applied when LFNST is applied. If it is 0, the scaling list may be applied when LFNST is applied. 
     Here, the scaling list is a matrix designating a specific weight value for each transform coefficient position in the transform block, and the scaling list allows differential inverse quantization or quantization to be applied according to the importance of the transform coefficient by multiplying the corresponding weight value for each transform coefficient to perform inverse quantization or quantization. 
     When configuring not to apply the scaling list when LFNST is applied as shown in Table 13 (e.g., when the value of the scaling_matrix_for_lfnst_disabled_flag syntax element in Table 13 is 1), in the embodiment of applying LFNST only to the luma component in the case of the single tree, the following may be applied. 
     1. When the luma separation tree and the chroma separation tree and LFNST is applied, the scaling list may not be applied to both. It corresponds to the condition of “applyLfnstFlag=(lfnst_idx&gt;0)?1:0” in Table 13. 
     2. In the case of the single tree, when LFNST is applied, the scaling list may not be applied for the luma component and the scaling list may be applied for the chroma component. It corresponds to “applyLfnstFlag=(lfnst_idx&gt;0 &amp;&amp; cIdx==0)?1:0” in Table 13. 
     Meanwhile, in the transform process, for an embodiment in which LFNST is applied only to the luma component in the case of the single tree, the specification text may be configured as shown in Table 14. 
     
       
         
           
               
               
             
               
                 TABLE 14 
               
               
                   
               
             
            
               
                 8.7.4 
                    Transformation process for scaled transform coefficients 
               
               
                 8.7.4.1 
                    General 
               
            
           
           
               
            
               
                 Inputs to this process are: 
               
            
           
           
               
               
            
               
                 - 
                 a luma location ( xTbY, yTbY ) specifying the top-left sample of the current luma transform block 
               
               
                   
                 relative to the top-left luma sample of the current picture, 
               
               
                 - 
                 a variable nTbW specifying the width of the current transform block, 
               
               
                 - 
                 a variable nTbH specifying the height of the current transform block, 
               
               
                 - 
                 a variable cIdx specifying the colour component of the current block, 
               
               
                 - 
                 an (nTbW)x(nTbH) array d[ x ][ y ] of scaled transform coefficients with x = 0..nTbW − 1, 
               
               
                   
                 y = 0..nTbH − 1. 
               
            
           
           
               
            
               
                 Output of this process is the (nTbW)x(nTbH) array res[ x ][ y ] of residual samples with x = 0..nTbW − 1, 
               
               
                 y = 0..nTbH − 1. 
               
               
                 The variable applyLfnstFlag is derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If treeType is equal to SINGLE_TREE, 
               
               
                   
                    applyLfnstFlag = ( lfnst_idx &gt; 0 &amp;&amp; cIdx = = 0 ) ? 1 : 0 
               
               
                 - 
                 Otherwise, 
               
               
                   
                    applyLfnstFlag = ( lfnst_idx &gt; 0 ) ? 1 : 0 
               
            
           
           
               
            
               
                 When applyLfnstFlag is not equal to 0, the following applies: 
               
            
           
           
               
               
            
               
                 - 
                 The variables predModeIntra, nLfnstOutSize, log2LfnstSize, nLfnstSize, and nonZeroSize are 
               
               
                   
                 derived as follows: 
               
               
                   
                  predModeIntra = 
               
               
                   
                  ( cIdx = = 0 ) ? IntraPredModeY[ xTbY ][ yTbY ] : IntraPredModeC[ xTbY ][ yTbY ] 
               
               
                   
                   (1149) 
               
            
           
           
               
               
               
            
               
                   
                  nLfnstOutSize = ( nTbW &gt;= 8 &amp;&amp; nTbH &gt;= 8 ) ? 48 : 16 
                  (1150) 
               
               
                   
                  log2LfnstSize = ( nTbW &gt;= 8 &amp;&amp; nTbH &gt;= 8 ) ? 3 : 2 
                  (1151) 
               
               
                   
                  nLfnstSize = 1 &lt;&lt; log2LfnstSize 
                  (1152) 
               
            
           
           
               
               
            
               
                   
                  nonZeroSize = ( ( nTbW = = 4 &amp;&amp; nTbH = = 4 ) | | 
               
            
           
           
               
               
               
            
               
                   
                  ( nTbW = = 8 &amp;&amp; nTbH = = 8 ) ) ? 8 : 16 
                  (1153) 
               
            
           
           
               
               
            
               
                 - 
                 When intra_mip_flag[ xTbY ][ yTbY ] is equal to 1 and cIdx is equal to 0, predModeIntra is set equal 
               
               
                   
                 to INTRA_PLANAR. 
               
               
                 - 
                 When predModeIntra is equal to either INTRA_LT_CCLM, INTRA_L_CCLM, or 
               
               
                   
                 INTRA_T_CCLM, predModeIntra is derived as follows: 
               
               
                 - 
                 If intra_mip_flag[ xTbY + nTbW * SubWidthC / 2 ][ yTbY + nTbH * SubHeightC / 2 ] is equal to 1, 
               
               
                   
                 predModeIntra is set equal to INTRA_PLANAR. 
               
               
                 - 
                 Otherwise, if 
               
               
                   
                 CuPredMode[ 0 ][ xTbY + nTbW * SubWidthC / 2 ][ yTbY + nTbH * SubHeightC / 2 ] is equal to 
               
               
                   
                 MODE_IBC or MODE_PLT, predModeIntra is set equal to INTRA_DC. 
               
               
                 - 
                 Otherwise, predModeIntra is set equal to 
               
               
                 - 
                 IntraPredModeY[ xTbY + nTbW * SubWidthC / 2 ][ yTbY + nTbH * SubHeightC / 2 ]. 
               
               
                 - 
                 The wide angle intra prediction mode mapping process as specified in clause 8.4.5.2.6 is invoked with 
               
               
                   
                 predModeIntra, nTbW, nTbH and cIdx as inputs, and the modified predModeIntra as output. 
               
               
                 - 
                 The values of the list u[ x ] with x = 0..nonZero Size − 1 are derived as follows: 
               
               
                   
                  xC = DiagScanOrder[ 2 ][ 2 ][ x ][ 0 ] 
               
            
           
           
               
               
            
               
                   
                 (1154) 
               
            
           
           
               
               
            
               
                   
                  yC = DiagScanOrder[ 2 ][ 2 ][ x ][ 1 ] 
               
            
           
           
               
               
            
               
                   
                 (1155) 
               
            
           
           
               
               
               
            
               
                   
                  u[ x ] = d[ xC ][ yC ] 
                  (1156) 
               
            
           
           
               
               
            
               
                 - 
                 The one-dimensional low frequency non-separable transformation process as specified in 
               
               
                   
                 clause 8.7.4.2 is invoked with the input length of the scaled transform coefficients nonZeroSize, the 
               
               
                   
                 transform output length nTrS set equal to nLfnstOutSize, the list of scaled non-zero transform 
               
               
                   
                 coefficients u[ x ] with x = 0..nonZeroSize − 1, and the intra prediction mode for LFNST set selection 
               
               
                   
                 predModeIntra as inputs, and the list v[ x ] with x = 0..nLfnstOutSize − 1 as output. 
               
               
                 - 
                 The array d[ x ][ y ] with x = 0..nLfnstSize − 1, y = 0..nLfnstSize − 1 is derived as follows: 
               
               
                 - 
                 If predModeIntra is less than or equal to 34, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                    d[ x ][ y ] = ( y &lt; 4 ) ? v[ x + ( y &lt;&lt; log2LfnstSize ) ] : : 
                  (1157) 
               
            
           
           
               
               
            
               
                   
                     ( ( x &lt; 4 ) ? v[ 32 + x + ( ( y − 4 ) &lt;&lt; 2 ) ] : d[ x ][ y ] ) 
               
               
                   
                 Otherwise, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                    d[ x ][ y ] = ( x &lt; 4 ) ? v[ y + ( x &lt;&lt; log2LfnstSize ) ] : 
                  (1158) 
               
            
           
           
               
               
            
               
                   
                     ( ( y &lt; 4 ) ? v[ 32 + y + ( ( x − 4 ) &lt;&lt; 2 ) ] : d[ x ][ y ] ) 
               
            
           
           
               
            
               
                 The variable implicitMtsEnabled is derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If sps_mts_enabled_flag is equal to 1 and one of the following conditions is true, implicitMtsEnabled 
               
               
                   
                 is set equal to 1: 
               
               
                 - 
                 IntraSubPartitionsSplitType is not equal to ISP_NO_SPLIT 
               
               
                 - 
                 cu_sbt_flag is equal to 1 and Max( nTbW, nTbH ) is less than or equal to 32 
               
               
                 - 
                 sps_explicit_mts_intra_enabled_flag is equal to 0 and CuPredMode[ 0 ][ xTbY ][ yTbY ] is equal to 
               
               
                   
                 MODE_INTRA and lfnst_idx[ x0 ][ y0 ] is equal to 0 and intra_mip_flag[ x0 ][ y0 ] is equal to 0 
               
               
                 - 
                 Otherwise, implicitMtsEnabled is set equal to 0. 
               
            
           
           
               
            
               
                 The variable trTypeHor specifying the horizontal transform kernel and the variable trTypeVer specifying 
               
               
                 the vertical transform kernel are derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If one or more of the following conditions is true, trTypeHor and trTypeVer are set equal to 0. 
               
               
                 - 
                 cIdx is greater than 0 
               
               
                 - 
                 IntraSubPartitionsSplitType is not equal to ISP NO SPLIT and lfnst_idx is not equal to 0 
               
               
                 - 
                 Otherwise, if implicitMtsEnabled is equal to 1, the following applies: 
               
               
                 - 
                 If cu_sbt flag is equal to 1, trTypeHor and trTypeVer are specified in Table 38 depending on 
               
               
                   
                 cu_sbt_horizontal_flag and cu_sbt_pos_flag. 
               
               
                 - 
                 Otherwise (cu_sbt_flag is equal to 0), trTypeHor and trTypeVer are derived as follows: 
               
            
           
           
               
               
               
            
               
                   
                    trTypeHor = ( nTbW &gt;= 4 &amp;&amp; nTbW &lt;= 16 ) ? 1 : 0 
                  (1159) 
               
               
                   
                    trTypeVer = ( nTbH &gt;= 4 &amp;&amp; nTbH &lt;= 16 ) ? 1 : 0 
                  (1160) 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, trTypeHor and trTypeVer are specified in Table 37 depending on mts_idx. 
               
            
           
           
               
            
               
                 The variables nonZeroW and nonZeroH are derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If applyLfnstFlag is not equal to 0, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                  nonZeroW = ( nTbW = = 4 | | nTbH = = 4 ) ? 4 : 8 
                  (1161) 
               
               
                   
                  nonZeroH = ( nTbW = = 4 | | nTbH = = 4 ) ? 4 : 8 
                  (1162) 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                  nonZeroW = Min( nTbW, ( trTypeHor &gt; 0 ) ? 16 : 32 ) 
                  (1163) 
               
               
                   
                  nonZeroH = Min( nTbH, ( trTypeVer &gt; 0 ) ? 16 : 32 ) 
                  (1164) 
               
            
           
           
               
            
               
                 The (nTbW)x(nTbH) array r of residual samples is derived as follows: 
               
               
                   
               
            
           
         
       
     
     As shown in Table 14, since LFNST is not applied to the chroma component when LFNST is applied only to the luma component in the single tree, it can be configured not to consider zero out due to LFNST when determining the nonZeroW and nonZeroH values when processing the chroma component. It corresponds to “If applyLfnstFlag is not equal to 0, the following applies: nonZeroW=(nTbW==4∥nTbH==4)?4:8, nonZeroH=(nTbW==4∥nTbH==4)?4:8” in Table 14. 
     Hereinafter, in the case of the single tree, an embodiment of MTS index signaling when LFNST is applied only to the luma component will be described. 
     According to an example, since MTS is applied only to the luma component in the current VVC standard and whether to signal the MTS index is determined according to the LFNST index value, the MTS index may be configured to be signaled immediately after the LFNST index signaling position. 
     Specifically, the MTS index is signaled only when LFNST is not applied, that is, only when the LFNST index value is 0, and in other cases, the MTS index value is inferred to be 0. Here, the MTS index of 0 may indicate that DCT-2 is applied in both horizontal and vertical directions, and the MTS index of non-zero may indicate that DST-7 or DCT-8 is applied for each of the horizontal and vertical directions. 
     When the signaling position for the MTS index is moved immediately after the signaling for the LFNST index while applying Table 6 basically, the syntax tables for the coding unit and the transform unit may be configured as shown in Tables 15 and 16. 
     
       
         
           
               
               
             
               
                   
                 TABLE 15 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 coding_unit( x0, y0, cbWidth, cbHeight, cqtDepth, treeType, modeType ) { 
               
               
                  ...... 
               
               
                  if( cu_cbf ) { 
               
               
                   ...... 
               
               
                   LfnstDcOnly = 1 
               
               
                   LfnstZeroOutSigCoeffFlag = 1 
               
               
                   MtsZeroOutSigCoeffFlag = 1 
               
               
                   lfnstWidth = ( treeType = = DUAL_TREE_CHROMA ) ? cbWidth / SubWidthC 
               
               
                     : ( ( IntraSubPartitionsSplitType = = ISP_VER_SPLIT ) ? cbWidth / 
               
               
                      NumIntraSubPartitions : cbWidth ) 
               
               
                   lfnstHeight = ( treeType = = DUAL_TREE_CHROMA ) ? cbHeight / SubHeightC 
               
               
                     : ( ( IntraSubPartitionsSplitType = = ISP_HOR_SPLIT ) ? cbHeight / 
               
               
                      NumIntraSubPartitions : cbHeight ) 
               
               
                    transform_tree( x0, y0, cbWidth, cbHeight, treeType, chType ) 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 16 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  ...... 
               
               
                  allowLfnstIndex = Min( lfnstWidth, lfnstHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 
               
               
                       1 &amp;&amp; 
               
               
                       CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                       ( treeType != DUAL_TREE_CHROMA | | !intra_mip_flag[ x0 ][ y0 ] 
               
               
                       | | 
               
               
                        Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                       Max( cbWidth, cbHeight ) &lt;= MaxTbSizeY &amp;&amp; 
               
               
                       ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                        ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                         subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( tu_cbf_luma[ x0 ][ y0 ] &amp;&amp; treeType != DUAL_TREE_CHROMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 0 ] &amp;&amp; 
               
               
                    tbWidth &lt;= MaxTsSize &amp;&amp; tbHeight &lt;= MaxTsSize &amp;&amp; 
               
               
                    ( IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT ) &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ x0 ][ y0 ][ 0 ] ) 
               
               
                    residual_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                   else 
               
               
                    residual_ts_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                  } 
               
               
                   if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) &amp;&amp; 
               
               
                    LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                      treeType ! = DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 0 ] 
               
               
                 = = 0 ) 
               
               
                      lfnst_idx 
                 ae(v) 
               
               
                  if( treeType != DUAL_TREE_CHROMA &amp;&amp; lfnst_idx = = 0 &amp;&amp; 
               
               
                  transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 &amp;&amp; Max( cbWidth, cbHeight ) &lt;= 32 &amp;&amp; 
               
               
                  IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; cu_sbt_flag = = 0 &amp;&amp; 
               
               
                  MtsZeroOutSigCoeffFlag = = 1 &amp;&amp; tu_cbf_luma[ x0 ][ y0 ] ) { 
               
               
                    if( ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTER &amp;&amp; 
               
               
                    sps_explicit_mts_inter_enabled_flag ) | | 
               
               
                    ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                    sps_explicit_mts_intra_enabled_flag ) ) ) 
               
               
                    mts_idx 
                 ae(v) 
               
               
                   } 
               
               
                  if( tu_cbf_cb[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 1 ] &amp;&amp; 
               
               
                    wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 1 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 1 ] ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; 
               
               
                     LfnstZeroOutSigCoeffFlag = = 1 ) 
               
               
                     lfnst_idx 
                 ae(v) 
               
               
                  } 
               
               
                  if( tu_cbf_cr[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA &amp;&amp; 
               
               
                   !( tu_cbf_cb[ xC ][ yC ] &amp;&amp; tu_joint_cbcr_residual_flag[ xC ][ yC ] ) ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 2 ] &amp;&amp; 
               
               
                    wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 2 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 2 ] ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                  } 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex 
               
               
                 &amp;&amp; 
               
               
                     treeType = = DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 1 ] = = 
               
               
                 0 &amp;&amp; 
               
               
                     transform_skip_flag[ x0 ][ y0 ][ 2 ] = = 0 ) 
               
               
                     lfnst_idx 
                 ae(v) 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     In the case of the transform tree syntax table and the residual coding syntax table, Tables 5 and 10 may be applied. 
     When the signaling position for the MTS index is moved immediately after the signaling for the LFNST index based on Table 11 and the initialization for the variable MtsZeroOutSigCoeffFlag is also moved into the transform unit syntax table, the syntax table for the transform unit may be configured as Table 17. Also in this case, in the case of the transform tree syntax table and the residual coding syntax table, Tables 5 and 10 may be applied. 
     
       
         
           
               
               
             
               
                   
                 TABLE 17 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  ...... 
               
               
                   if( subTuIndex = = 0 ) { 
               
               
                   LfnstDcOnly = 1 
               
               
                   LfnstZeroOutSigCoeffFlag = 1 
               
               
                     MtsZeroOutSigCoeffFlag = 1 
               
               
                   } 
               
               
                  lfnstWidth = ( treeType = = DUAL_TREE_CHROMA ) ? cbWidth / SubWidthC 
               
               
                       : ( ( IntraSubPartitionsSplitType = = ISP_VER_SPLIT ) ? cbWidth / 
               
               
                       NumIntraSubPartitions : cbWidth ) 
               
               
                  lfnstHeight = ( treeType = = DUAL_TREE_CHROMA ) ? cbHeight / SubHeightC 
               
               
                       : ( ( IntraSubPartitionsSplitType = = ISP_HOR_SPLIT ) ? cbHeight / 
               
               
                       NumIntraSubPartitions : cbHeight ) 
               
               
                  allowLfnstIndex = Min( lfnstWidth, lfnstHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag 
               
               
                        = = 1 &amp;&amp; 
               
               
                        CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                        ( treeType != DUAL_TREE_CHROMA | | 
               
               
                        !intra_mip_flag[ x0 ][ y0 ] | | 
               
               
                         Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                        Max( cbWidth, cbHeight ) &lt;= MaxTbSizeY &amp;&amp; 
               
               
                        ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                         ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                          subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( tu_cbf_luma[ x0 ][ y0 ] &amp;&amp; treeType != DUAL_TREE_CHROMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 0 ] &amp;&amp; 
               
               
                    tbWidth &lt;= MaxTsSize &amp;&amp; tbHeight &lt;= MaxTsSize &amp;&amp; 
               
               
                    ( IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT ) &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ x0 ][ y0 ][ 0 ] ) 
               
               
                    residual_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                   else 
               
               
                    residual_ts_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                  } 
               
               
                   if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) 
               
               
                 &amp;&amp; 
               
               
                    LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                      treeType ! = DUAL_TREE_CHROMA &amp;&amp; 
               
               
                 transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 ) 
               
               
                      lfnst_idx 
                 ae(v) 
               
               
                   if( treeType != DUAL_TREE_CHROMA &amp;&amp; lfnst_idx = = 0 &amp;&amp; 
               
               
                   transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 &amp;&amp; Max( cbWidth, cbHeight ) &lt;= 
               
               
                 32 &amp;&amp; 
               
               
                   IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; cu_sbt_flag = = 0 &amp;&amp; 
               
               
                   MtsZeroOutSigCoeffFlag = = 1 &amp;&amp; tu_cbf_luma[ x0 ][ y0 ] ) { 
               
               
                     if( ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTER &amp;&amp; 
               
               
                    sps_explicit_mts_inter_enabled_flag ) | | 
               
               
                    ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                    sps_explicit_mts_intra_enabled_flag ) ) ) 
               
               
                    mts idx 
                 ae(v) 
               
               
                   } 
               
               
                  if( tu_cbf_cb[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 1 ] &amp;&amp; 
               
               
                    wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 1 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 1 ] ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; 
               
               
                     LfnstZeroOutSigCoeffFlag = = 1 ) 
               
               
                     lfnst_idx 
                 ae(v) 
               
               
                  } 
               
               
                  if( tu_cbf_cr[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA &amp;&amp; 
               
               
                   !( tu_cbf_cb[ xC ][ yC ] &amp;&amp; tu_joint_cbcr_residual_flag[ xC ][ yC ] ) ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 2 ] &amp;&amp; 
               
               
                    wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 2 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 2 ] ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                  } 
               
               
                   if( LfnstDcOnly = = 0 &amp;&amp; LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; 
               
               
                 allowLfnstIndex &amp;&amp; 
               
               
                     treeType = = DUAL_TREE_CHROMA &amp;&amp; 
               
               
                 transform_skip_flag[ x0 ][ y0 ][ 1 ] = = 0 &amp;&amp; 
               
               
                     transform_skip_flag[ x0 ][ y0 ][ 2 ] = = 0 ) 
               
               
                     lfnst_idx 
                 ae(v) 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     In Table 17, MtsZeroOutSigCoeffFlag is initialized to 1 only when subTuIndex is 0, but since the MTS index is not signaled in the case of ISP mode, in general, the condition in which subTuIndex is 0 may not be checked and MtsZeroOutSigCoeffFlag may be initialized. 
     The variable MtsZeroOutSigCoeffFlag is updated to 0 when a non-zero transform coefficient exists outside a top-left 16×16 region for the luma transform block (Table 10). Accordingly, when the transform coefficients for the luma component are parsed (e.g., when the residual coding syntax table for the luma component is performed), since the MtsZeroOutSigCoeffFlag value is derived, the MTS index may be signaled, and the MTS index may be signaled immediately after the LFNST index only when the LFNST index value is 0 as shown in Tables 16 and 17. 
     Conditions to be satisfied when signaling the MTS index in Tables 16 and 17 are as follows. 
     1. In the case of the single tree or the luma speration tree (corresponding to “treeType !=DUAL_TREE_CHROMA” in Tables 16 and 17) 
     2. The LFNST index value should be 0, that is, when LFNST is not applied (corresponding to “lfnst_idx==0” in Tables 16 and 17) 
     3. Luma component should not be coded with transform skip (corresponds to “transform_skip_flag[x0][y0][0]==0” in Tables 16 and 17) 
     4. Both the width and height of the transform block for the luma component should be equal to or less than 32 (corresponding to “Max(cbWidth, cbHeight)&lt;=32” in Tables 16 and 17) 
     5. Neither ISP nor SBT (Subblock Transform) should be applied (corresponding to “IntraSubPartitionsSplitType==ISP_NO_SPLIT &amp;&amp; cu_sbt_flag==0” in Tables 16 and 17) 
     6. For the transform block for the luma component, a non-zero transform coefficient should not exist in a region other than the top-left 16×16 region (corresponding to “MtsZeroOutSigCoeffFlag==1” in Tables 16 and 17). 
     When the MTS index is not 0, DST-7 or DCT-8 is applied in the horizontal and vertical directions, and when DST-7 or DCT-8 is applied to both the horizontal and vertical directions, a region where a non-zero transform coefficient can exist is limited to the top-left 16×16. Therefore, if the non-zero transform coefficient exists outside the top-left 16×16 region, it is certain that DCT-2 is applied in both the horizontal and vertical directions, so that the MTS index value may be inferred as 0 without signaling. 
     7. When the CBF value for the luma component is 1, that is, when at least one non-zero transform coefficient exists in the transform block for the luma component (corresponding to “tu_cbf_luma[x0][y0]]”) in Tables 16 and 17) 
     8. When the current coding unit is inter prediction and MTS is explicitly applied (corresponding to “(CuPredMode[chType][x0][y0]==MODE INTER &amp;&amp; sps_explicit_mts_inter_enabled_flag)” in Tables 16 and 17), when the current coding unit is intra prediction and MTS is explicitly applied (corresponding to “(CuPredMode[chType][x0][y0]==MODE_INTRA &amp;&amp; sps_explicit_mts_intra_enabled_flag)” in Tables 16 and 17) 
     In the previous embodiment, LFNST index signaling for a case in which LFNST is applied to a chroma component in the case of the chroma separation tree (dual tree chroma) and LFNST is not applied to a chroma component in the case of the single tree was examined. 
     Hereinafter, an LFNST index signaling scheme for a case in which LFNST is not applied to a chroma component will be described. That is, an embodiment in which LFNST is not applied to the chroma component even in the case of the chroma separation tree, that is, LFNST is not applied to the chroma component for all tree types, and LFNST is applied only to the luma component will be described. According to this embodiment, Table 6 may be changed as Table 18 and Table 16 may be changed to Table 19. 
     
       
         
           
               
               
             
               
                   
                 TABLE 18 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  if( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                    treeType = = SINGLE_TREE &amp;&amp; subTuIndex = = NumIntraSubPartitions − 1 ) { 
               
               
                   xC = CbPosX[ chType ][ x0 ][ y0 ] 
               
               
                   yC = CbPosY[ chType ][ x0 ][ y0 ] 
               
               
                   wC = CbWidth[ chType ][ x0 ][ y0 ] / SubWidthC 
               
               
                   hC = CbHeight[ chType ][ x0 ][ y0 ] / SubHeightC 
               
               
                  } else { 
               
               
                   xC = x0 
               
               
                   yC = y0 
               
               
                   wC = tbWidth / SubWidthC 
               
               
                   hC = tbHeight / SubHeightC 
               
               
                  } 
               
               
                  chromaAvailable = treeType != DUAL_TREE_LUMA &amp;&amp; ChromaArrayType != 0 
               
               
                 &amp;&amp; 
               
               
                   ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                   ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                   subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( ( treeType = = SINGLE_TREE | | treeType = = DUAL_TREE_CHROMA ) &amp;&amp; 
               
               
                    ChromaArrayType != 0 &amp;&amp; ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT 
               
               
                 &amp;&amp; 
               
               
                    ( ( subTuIndex = = 0 &amp;&amp; cu_sbt_pos flag ) | | 
               
               
                    ( subTuIndex = = 1 &amp;&amp; !cu_sbt_pos_flag ) ) ) ) | | 
               
               
                    ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                    ( subTuIndex = = NumIntraSubPartitions − 1 ) ) ) { 
               
               
                   tu_cb_coded_flag[ xC ][ yC ] 
                 ae(v) 
               
               
                   tu_cr_coded_flag[ xC ][ yC ] 
                 ae(v) 
               
               
                  } 
               
               
                  if( treeType = = SINGLE_TREE | | treeType = = DUAL_TREE_LUMA ) { 
               
               
                   if( ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; !( cu_sbt_flag &amp;&amp; 
               
               
                     ( ( subTuIndex = = 0 &amp;&amp; cu_sbt_pos_flag ) | | 
               
               
                     ( subTuIndex = = 1 &amp;&amp; !cu_sbt_pos_flag ) ) ) &amp;&amp; 
               
               
                     ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA | | 
               
               
                     ( chromaAvailable &amp;&amp; ( tu_cb_coded_flag[ xC ][ yC ] | | 
               
               
                     tu_cr_coded_flag[ xC ][ yC ] ) ) | | 
               
               
                     CbWidth[ chType ][ x0 ][ y0 ] &gt; MaxTbSizeY | | 
               
               
                     CbHeight[ chType ][ x0 ][ y0 ] &gt; MaxTbSizeY ) ) | | 
               
               
                     ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                     ( subTuIndex &lt; NumlntraSubPartitions − 1 | | ! InferTuCbfLuma ) ) ) 
               
               
                    tu_y_coded_flag[ x0 ][ y0 ] 
                 ae(v) 
               
               
                   if(IntraSubPartitionsSplitType != ISP_NO_SPLIT ) 
               
               
                    InferTuCbfLuma = InferTuCbfLuma &amp;&amp; !tu_y_coded_flag[ x0 ][ y0 ] 
               
               
                  } 
               
               
                  if( ( CbWidth[ chType ][ x0 ][ y0 ] &gt; 64 | | CbHeight[ chType ][ x0 ][ y0 ] &gt; 64 | | 
               
               
                    tu_y_coded_flag[ x0 ][ y0 ] | | ( chromaAvailable &amp;&amp; ( tu_cb_coded_flag[ xC ][ yC ] 
               
               
                 | | 
               
               
                    tu_cr_coded_flag[ xC ][ yC ] ) ) &amp;&amp; treeType != DUAL_TREE_CHROMA &amp;&amp; 
               
               
                    cu_qp_delta_enabled_flag &amp;&amp; !IsCuQpDeltaCoded ) { 
               
               
                   cu_qp_delta_abs 
                 ae(v) 
               
               
                   if( cu_qp_delta_abs ) 
               
               
                    cu_qp_delta_sign_flag 
                 ae(v) 
               
               
                  } 
               
               
                  if( ( CbWidth[ chType ][ x0 ][ y0 ] &gt; 64 | | CbHeight[ chType ][ x0 ][ y0 ] &gt; 64 | | 
               
               
                    ( chromaAvailable &amp;&amp; ( tu_cb_coded_flag[ xC ][ yC ] | | 
               
               
                    tu_cr_coded_flag[ xC ][ yC ] ) ) ) &amp;&amp; 
               
               
                    treeType != DUAL_TREE_LUMA &amp;&amp; cu_chroma_qp_offset_enabled_flag &amp;&amp; 
               
               
                    !IsCuChromaQpOffsetCoded ) { 
               
               
                   cu_chroma_qp_offset_flag 
                 ae(v) 
               
               
                   if( cu_chroma_qp_offset_flag &amp;&amp; chroma_qp_offset_list_len_minus1 &gt; 0 ) 
               
               
                    cu_chroma_qp_offset_idx 
                 ae(v) 
               
               
                  } 
               
               
                  if( sps_joint_cbcr_enabled_flag &amp;&amp; ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = 
               
               
                 MODE_INTRA 
               
               
                    &amp;&amp; ( tu_cb_coded_flag[ xC ][ yC ] | | tu_cr_coded_flag[ xC ][ yC ] ) ) | | 
               
               
                    ( tu_cb_coded_flag[ xC ][ yC ] &amp;&amp; tu_cr_coded_flag[ xC ][ yC ] ) ) &amp;&amp; 
               
               
                    chromaAvailable ) 
               
               
                   tu_joint_cbcr_residual_flag[ xC ][ yC ] 
                 ae(v) 
               
               
                   allowLfnstIndex = Min( lfnstWidth, lfnstHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 
               
               
                      1 &amp;&amp; 
               
               
                      CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                      treeType != DUAL_TREE_CHROMA &amp;&amp; ( !intra_mip_flag[ x0 ][ y0 ] 
               
               
                      | | 
               
               
                       Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                      Max( CbWidth[ chType ][ x0 ][ y0 ], CbHeight[ chType ][ x0 ][ y0 ] ) &lt;= 
               
               
                      MaxTbSizeY &amp;&amp; 
               
               
                      ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                       ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                        subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( tu_y_coded_flag[ x0 ][ y0 ] &amp;&amp; treeType != DUAL_TREE_CHROMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 0 ] &amp;&amp; 
               
               
                     tbWidth &lt;= MaxTsSize &amp;&amp; tbHeight &lt;= MaxTsSize &amp;&amp; 
               
               
                     ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT ) &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ x0 ][ y0 ][ 0 ] | | slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                   else 
               
               
                    residual_ts_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                  } 
               
               
                   if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) &amp;&amp; 
               
               
                    LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                      treeType ! = DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 0 ] 
               
               
                 = = 0 ) 
               
               
                      lfnst_idx 
                 ae(v) 
               
               
                  if( tu_cb_coded_flag[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 1 ] &amp;&amp; 
               
               
                     wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 1 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 1 ] | | slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                  } 
               
               
                  if( tu_cr_coded_flag[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA &amp;&amp; 
               
               
                    !( tu_cb_coded_flag[ xC ][ yC ] &amp;&amp; tu_joint_cbcr_residual_flag[ xC ][ yC ] ) ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 2 ] &amp;&amp; 
               
               
                     wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 2 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 2 ] | | slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                  } 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 19 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
                   
               
               
                  ...... 
               
               
                  allowLfnstIndex = Min( lfnstWidth, lfnstHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 
               
               
                        1 &amp;&amp; 
               
               
                        CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                        treeType != DUAL_TREE_CHROMA &amp;&amp; ( !intra_mip_flag[ x0 ][ y0 ] 
               
               
                        | | 
               
               
                         Min( lfnstWidth, lfnstHeight ) &gt;= 16 ) &amp;&amp; 
               
               
                        Max( CbWidth[ chType ][ x0 ][ y0 ], CbHeight[ chType ][ x0 ][ y0 ] ) &lt;= 
               
               
                        MaxTbSizeY &amp;&amp; 
               
               
                        ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                         ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                          subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  if( tu_y_coded_flag[ x0 ][ y0 ] &amp;&amp; treeType != DUAL_TREE_CHROMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 0 ] &amp;&amp; 
               
               
                      tbWidth &lt;= MaxTsSize &amp;&amp; tbHeight &lt;= MaxTsSize &amp;&amp; 
               
               
                      ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT ) &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ x0 ][ y0 ][ 0 ] | | slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                   else 
               
               
                    residual_ts_coding( x0, y0, Log2( tbWidth ), Log2( tbHeight ), 0 ) 
               
               
                  } 
               
               
                   if( ( IntraSubPartitionsSplitType != ISP_NO_SPLIT | | LfnstDcOnly = = 0 ) &amp;&amp; 
               
               
                    LfnstZeroOutSigCoeffFlag = = 1 &amp;&amp; allowLfnstIndex &amp;&amp; 
               
               
                       treeType ! = DUAL_TREE_CHROMA &amp;&amp; transform_skip_flag[ x0 ][ y0 ][ 0 ] 
               
               
                 = = 0 ) 
               
               
                       lfnst_idx 
                 ae(v) 
               
               
                   if( treeType != DUAL_TREE_CHROMA &amp;&amp; lfnst_idx = = 0 &amp;&amp; 
               
               
                    transform_skip_flag[ x0 ][ y0 ][ 0 ] = = 0 &amp;&amp; 
               
               
                        Max( CbWidth[ chType ][ x0 ][ y0 ], CbHeight[ chType ][ x0 ][ y0 ] ) &lt;= 32 
               
               
                 &amp;&amp; 
               
               
                    IntraSubPartitionsSplitType = = ISP_NO_SPLIT &amp;&amp; cu_sbt_flag = = 0 &amp;&amp; 
               
               
                    MtsZeroOutSigCoeffFlag = = 1 &amp;&amp; MtsDcOnly = = 0 ) { 
               
               
                     if( ( ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTER &amp;&amp; 
               
               
                    sps_explicit_mts_inter_enabled_flag ) | | 
               
               
                    ( CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                    sps_explicit_mts_intra_enabled_flag ) ) ) 
               
               
                    mts idx 
                 ae(v) 
               
               
                   } 
               
               
                  if( tu_cb_coded_flag[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 1 ] &amp;&amp; 
               
               
                      wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 1 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 1 ] slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 1 ) 
               
               
                  } 
               
               
                  if( tu_cr_coded_flag[ xC ][ yC ] &amp;&amp; treeType != DUAL_TREE_LUMA &amp;&amp; 
               
               
                    !( tu_cb_coded_flag[ xC ][ yC ] &amp;&amp; tu_joint_cbcr_residual_flag[ xC ][ yC ] ) ) { 
               
               
                   if( sps_transform_skip_enabled_flag &amp;&amp; !BdpcmFlag[ x0 ][ y0 ][ 2 ] &amp;&amp; 
               
               
                      wC &lt;= MaxTsSize &amp;&amp; hC &lt;= MaxTsSize &amp;&amp; !cu_sbt_flag ) 
               
               
                    transform_skip_flag[ xC ][ yC ][ 2 ] 
                 ae(v) 
               
               
                   if( !transform_skip_flag[ xC ][ yC ][ 2 ] slice_ts_residual_coding_disabled_flag ) 
               
               
                    residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                   else 
               
               
                    residual_ts_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) 
               
               
                  } 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     Since LFNST is not applied to the chroma component, in Tables 18 and 19, the part signaling the LFNST index (lfnst_idx) is deleted in the case of the chroma separation tree, and also it is changed for the variable allowLfnstIndex value in Tables 18 and 19 to have a value of 1 only if not the chroma separation tree (treeType !=DUAL_TREE_CHROMA). The remaining parts other than the changed part of Table 18 follow Table 6. In addition, the remaining parts other than the changed part of Table 19 follow Table 16. 
     In addition, since LFNST should be applied only to the luma component, the variable ApplyLfnstFlag indicating whether LFNST is applied may be set as shown in Table 20. 
     
       
         
           
               
               
             
               
                 TABLE 20 
               
               
                   
               
             
            
               
                 7.4.11.5 
                 Coding unit semantics 
               
            
           
           
               
            
               
                 ...... 
               
               
                 lfnst_idx specifies whether and which one of the two low frequency non-separable transform kernels in a 
               
               
                 selected transform set is used. lfnst_idx equal to 0 specifies that the low frequency non-separable transform is 
               
               
                 not used in the current coding unit. 
               
               
                 When lfnst_idx is not present, it is inferred to be equal to 0. 
               
               
                 The variable ApplyLfnstFlag is derived as follows: 
               
            
           
           
               
               
               
            
               
                   
                 ApplyLfnstFlag = ( lfnst_idx &gt; 0 &amp;&amp; cIdx = = 0 ) ? 1 : 0 
                 (176) 
               
            
           
           
               
            
               
                 ...... 
               
            
           
           
               
               
            
               
                 8.7.3 
                 Scaling process for transform coefficients 
               
            
           
           
               
            
               
                 Inputs to this process are: 
               
            
           
           
               
               
            
               
                 - 
                 a luma location ( xTbY, yTbY ) specifying the top-left sample of the current luma transform block relative 
               
               
                   
                 to the top-left luma sample of the current picture, 
               
               
                 - 
                 a variable nTbW specifying the transform block width, 
               
               
                 - 
                 a variable nTbH specifying the transform block height, 
               
               
                 - 
                 a variable predMode specifying the prediction mode of the coding unit, 
               
               
                 - 
                 a variable cIdx specifying the colour component of the current block. 
               
            
           
           
               
            
               
                 Output of this process is the (nTbW)x(nTbH) array d of scaled transform coefficients with elements d[ x ][ y ]. 
               
               
                 ...... 
               
               
                 For the derivation of the scaled transform coefficients d[ x ][ y ] with x = 0..nTbW − 1, y = 0..nTbH − 1, the 
               
               
                 following applies: 
               
            
           
           
               
               
            
               
                 - 
                 The intermediate scaling factor m[ x ][ y ] is derived as follows: 
               
            
           
           
               
               
               
            
               
                   
                 - 
                 If one or more of the following conditions are true, m[ x ][ y ] is set equal to 16: 
               
            
           
           
               
               
               
            
               
                   
                 - 
                 slice_explicit_scaling_list_used_flag is equal to 0. 
               
               
                   
                 - 
                 transform_skip_flag[ xTbY ][ yTbY ][ cIdx ] is equal to 1. 
               
               
                   
                 - 
                 scaling_matrix_for_lfnst_disabled_flag is equal to 1 and ApplyLfnstFlag is equal to 1. 
               
            
           
           
               
               
               
            
               
                   
                 - 
                 Otherwise, the following applies: 
               
            
           
           
               
            
               
                 ...... 
               
            
           
           
               
               
            
               
                 8.7.4 
                 Transformation process for scaled transform coefficients 
               
               
                 8.7.4.1 
                 General 
               
            
           
           
               
            
               
                 Inputs to this process are: 
               
            
           
           
               
               
            
               
                 - 
                 a luma location ( xTbY, yTbY ) specifying the top-left sample of the current luma transform block relative 
               
               
                   
                 to the top-left luma sample of the current picture, 
               
               
                 - 
                 a variable nTbW specifying the width of the current transform block, 
               
               
                 - 
                 a variable nTbH specifying the height of the current transform block, 
               
               
                 - 
                 a variable cIdx specifying the colour component of the current block, 
               
               
                 - 
                 an (nTbW)x(nTbH) array d[ x ][ y ] of scaled transform coefficients with x = 0..nTbW − 1, 
               
               
                   
                 y = 0..nTbH − 1. 
               
            
           
           
               
            
               
                 Output of this process is the (nTbW)x(nTbH) array res[ x ][ y ] of residual samples with x = 0..nTbW − 1, 
               
               
                 y = 0..nTbH − 1. 
               
               
                 When ApplyLfnstFlag is equal to 1, transform_skip_flag[ xTbY ][ yTbY ][ cIdx ] is equal to 0 and both nTbW 
               
               
                 and nTbH are greater than or equal to 4, the following applies: 
               
            
           
           
               
               
            
               
                 - 
                 The variables predModeIntra, nLfnstOutSize, log2LfnstSize, nLfnstSize, and nonZeroSize are derived as 
               
               
                   
                 follows: 
               
            
           
           
               
               
               
            
               
                   
                 predModeIntra 
                  = 
               
            
           
           
               
               
            
               
                   
                 ( cIdx = = 0 ) ? IntraPredModeY[ xTbY ][ yTbY ] : IntraPredModeC[ xTbY ][ yTbY ] 
               
               
                   
                 (1178) 
               
            
           
           
               
               
               
            
               
                   
                 nLfnstOutSize = ( nTbW &gt;= 8 &amp;&amp; nTbH &gt;= 8 ) ? 48 : 16 
                 (1179) 
               
               
                   
                 log2LfnstSize = ( nTbW &gt;= 8 &amp;&amp; nTbH &gt;= 8 ) ? 3 : 2 
                 (1180) 
               
               
                   
                 nLfnstSize = 1 &lt;&lt; log2LfnstSize 
                 (1181) 
               
               
                   
                 nonZeroSize = ( ( nTbW = = 4 &amp;&amp; nTbH = = 4 ) 
                  | | 
               
            
           
           
               
               
            
               
                   
                 ( nTbW = = 8 &amp;&amp; nTbH = = 8 ) ) ? 8 : 16 (1182) 
               
            
           
           
               
            
               
                 ...... 
               
               
                 The variables nonZeroW and nonZeroH are derived as follows: 
               
            
           
           
               
               
            
               
                 - 
                 If ApplyLfnstFlag is equal to 1 and nTbW is greater than or equal to 4 and nTbH is greater than or equal 
               
               
                   
                 to 4, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                 nonZeroW = ( nTbW = = 4 | | nTbH = = 4 ) ? 4 : 8 
                 (1190) 
               
               
                   
                 nonZeroH = ( nTbW = = 4 | | nTbH = = 4 ) ? 4 : 8 
                 (1191) 
               
            
           
           
               
               
            
               
                 - 
                 Otherwise, the following applies: 
               
            
           
           
               
               
               
            
               
                   
                 nonZeroW = Min( nTbW, ( trTypeHor &gt; 0 ) ? 16 : 32 ) 
                 (1192) 
               
               
                   
                 nonZeroH = Min( nTbH, ( trTypeVer &gt; 0 ) ? 16 : 32 ) 
                 (1193) 
               
            
           
           
               
            
               
                 ...... 
               
               
                   
               
            
           
         
       
     
     As shown in Table 20, the ApplyLfnstFlag value is modified to be 1 only for the luma component (i.e., when cIdx==0), and Table 20 includes the specification text affected by the ApplyLfnstFlag value. 
     As shown in Table 20, when the scaling list is set in the sequence parameter set so that it is not applied when LFNST is applied, that is, when scaling_matrix_for_lfnst_disabled_flag is 1, if the ApplyLfnstFlag value is 1, the scaling list is not applied (scaling_matrix_for_lfnst_disabled_flag is equal to 1 and ApplyLfnstFlag is equal to 1). 
     LFNST may be applied only when the ApplyLfnstFlag value is 1, and only when ApplyLfnstFlag is 1 in the transformation process for scaled transform coefficients clause of Table 20, LFNST-related operation (“When ApplyLfnstFlag is equal to 1” part related) is performed. 
     In addition, according to the ApplyLfnstFlag value in Table 20, a region in which a non-zero transform coefficient may exist (a horizontal length of the corresponding region is nonZeroW and the vertical length of the corresponding region is nonZeroH) may be set. Here, nTbW and nTbH mean the horizontal and vertical lengths of the currently processed transform block. 
     Since lfnstWidth and lfnstHeight appearing in Table 6, Table 16, Table 18, and Table 19 mean the horizontal length and vertical length of the transforma block to which the transform is applied, respectively (Although it can be regarded as the horizontal length and vertical length of the partition block in the case of ISP, are the same as the horizontal length tbWidth and tbHeight of the corresponding block in the transform unit syntax table), lfnstWidth and lfnstHeight may be replaced with tbWidth and tbHeight, respectively. 
     Since lfnstWidth and lfnstHeight in Tables 6 and 16 are referenced to set allowLfnstIndex, they may be changed as shown in Table 21. In addition, since lfnstWidth and lfnstHeight in Tables 18 and 19 are also referenced to set allowLfnstIndex, they may be changed as shown in Table 22. Due to these, the coding unit syntax table may be expressed as shown in Table 23 by deleting the part for setting lfnstWidth and lfnstHeight. 
     
       
         
           
               
               
             
               
                   
                 TABLE 21 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
               
               
                  ...... 
               
               
                  allowLfnstIndex = Min( tbWidth, tbHeight ) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 1 
               
               
                   &amp;&amp; 
               
               
                   CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                   ( treeType = = DUAL_TREE_CHROMA | | !intra_mip_flag[ x0 ][ y0 ] 
               
               
                   | | 
               
               
                    Min( tbWidth, tbHeight) &gt;= 16 ) &amp;&amp; 
               
               
                   Max( CbWidth[ chType ][ x0 ][ y0 ], CbHeight[ chType ][ x0 ][ y0 ] ) &lt;= 
               
               
                   MaxTbSizeY &amp;&amp; 
               
               
                   ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                    ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                     subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 22 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 transform_unit( x0, y0, tbWidth, tbHeight, treeType, subTuIndex, chType ) { 
               
               
                  ...... 
               
               
                  allowLfnstIndex = Min( tbWidth, tbHeight) &gt;= 4 &amp;&amp; sps_lfnst_enabled_flag = = 1 
               
               
                   &amp;&amp; 
               
               
                   CuPredMode[ chType ][ x0 ][ y0 ] = = MODE_INTRA &amp;&amp; 
               
               
                   treeType != DUAL_TREE_CHROMA &amp;&amp; ( !intra_mip_flag[ x0 ][ y0 ] 
               
               
                   | | 
               
               
                    Min( tbWidth, tbHeight) &gt;= 16 ) &amp;&amp; 
               
               
                   Max( CbWidth[ chType ][ x0 ][ y0 ], CbHeight[ chType ][ x0 ][ y0 ] ) &lt;= 
               
               
                   MaxTbSizeY &amp;&amp; 
               
               
                   ( IntraSubPartitionsSplitType = = ISP_NO_SPLIT | | 
               
               
                    ( IntraSubPartitionsSplitType != ISP_NO_SPLIT &amp;&amp; 
               
               
                     subTuIndex = = NumIntraSubPartitions − 1 ) ) 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 23 
               
               
                   
                   
               
               
                   
                 Descriptor 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 coding_unit( x0, y0, cbWidth, cbHeight, cqtDepth, treeType, modeType ) { 
               
               
                  ...... 
               
               
                  if( cu_coded_flag ) { 
               
               
                   ...... 
               
               
                   transform_tree( x0, y0, cbWidth, cbHeight, treeType, chType ) 
               
               
                   ...... 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     The following drawings are provided to describe specific examples of the present disclosure. Since specific terms for devices or specific terms for signals/messages/fields illustrated in the drawings are provided for illustration, technical features of the present disclosure are not limited to the specific terms used in the following drawings. 
       FIG.  15    is a flowchart illustrating an operation of a video decoding apparatus according to an embodiment of the present disclosure. 
     Each process disclosed in  FIG.  15    is based on some of details described with reference to  FIG.  5    to  FIG.  14   . Therefore, a description of specific details overlapping those described with reference to  FIG.  3    and  FIG.  5    to  FIG.  14    will be omitted or will be schematically made. 
     The decoding apparatus  300  according to an embodiment may receive information on an intra prediction mode, residual information, and an LFNST index from a bitstream (S 1510 ). 
     Specifically, the decoding apparatus  300  may decode information on quantized transform coefficients for the current block from the bitstream and may derive quantized transform coefficients for a target block based on the information on the quantized transform coefficients for the current block. Information on the quantized transform coefficients for the target block may be included in a sequence parameter set (SPS) or a slice header and may include at least one of information on whether an RST is applied, information on a reduced factor, information on a minimum transform size for applying an RST, information on a maximum transform size for applying an RST, an inverse RST size, and information on a transform index indicating any one of transform kernel matrices included in a transform set. 
     The decoding apparatus may further receive information on an intra prediction mode for the current block and information on whether an ISP is applied to the current block. The decoding apparatus may receive and parse flag information indicating whether to apply ISP coding or an ISP mode, thereby deriving whether the current block is split into a predetermined number of sub-partition transform blocks. Here, the current block may be a coding block. Further, the decoding apparatus may derive the size and number of split sub-partition blocks through flag information indicating a direction in which the current block is split. 
     The decoding apparatus  300  may derive transform coefficients by performing inverse quantization on residual information for the current block, that is, quantized transform coefficients (S 1520 ). 
     The derived transform coefficients may be arranged according to a inverse diagonal scan order in units of 4×4 blocks, and transform coefficients within the 4×4 block may also be arranged according to the inverse diagonal scan order. That is, transform coefficients on which inverse quantization has been performed may be arranged according to the inverse scan order applied in a video codec such as VVC or HEVC. 
     The decoding apparatus may derive a residual sample for the current block from transform coefficients based on LFNST or MTS. The decoding apparatus may derive a residual sample based on the existing inverse primary transform, not the MTS, after applying the LFNST. Alternatively, the decoding apparatus may derive the residual sample based on the MTS or the existing inverse primary transform without applying the LFNST. 
     When the LFNST is applied to the current block, the decoding apparatus may parse the LFNST index related to the LFNST matrix in the transform unit level for the current block after the transform coefficients for the current block are derived, and the LFNST index may be applied only to the luma component of the current block (S 1530 ). 
     The LFNST is a non-separated transform that applies a transform without separating the coefficients in a specific direction, unlike the primary transform that separates and transforms the transform target coefficients in a vertical or horizontal direction. This non-separated transform may be a low-frequency non-separated transform that applies a forward transform only to a low-frequency region rather than the entire block region. 
     LFNST index information may be received as syntax information, and the syntax information may be received as a binarized bin string including 0 and 1. 
     The syntax element of the LFNST index according to the present embodiment may indicate whether an inverse LFNST or an inverse non-separated transform is applied and any one of a transform kernel matrix included in the transform set, and when the transform set includes two transform kernel matrices, the syntax element of the transform index may have three values. 
     That is, according to an embodiment, the syntax element value for the LFNST index may include 0 indicating a case in which the inverse LFNST is not applied to the target block, 1 indicating a first transform kernel matrix among the transform kernel matrices and 2 indicating a second transform kernel matrix among transform kernel matrices. 
     In order to determine whether to parse the LFNST index for the current block, the decoding apparatus may examine a tree type of the current block and whether ISP is applied, and may derive various variables. 
     According to an example, if the tree type of the current block is a single tree type or a dual tree luma, the LFNST index may be signaled after a transform coefficient for the luma component of the current block is derived. 
     Also, when the ISP mode is applied to the current block, the decoding apparatus may parse the LFNST index after a transform coefficient for the last partition block of the current block is derived. 
     On the other hand, if the tree type of the current block is the dual tree chroma, the LFNST is not applied to the chroma component of the current block, and the LFNST index is not signaled either. 
     That is, according to an example, LFNST may be applied only to the luma component of the current block and LFNST may not be applied to the chroma component. 
     In addition, according to an example, the decoding apparatus may parse the LFNST index based on a first variable indicating whether the significant coefficient exists in a position other than the DC component of the current block and a second variable indicating whether a significant coefficient exists in a second region excluding a top-left first region of the current block. 
     The first variable indicating whether the significant coefficient exists in a position other than the DC component of the current block may be expressed as a variable LfnstDcOnly, and it becomes 0 if a non-zero coefficient exists at a position that is not a DC component for at least one transform block in one coding unit, and it becomes 1 if non-zero coefficients do not exist in positions that are not the DC component for all transform blocks in one coding unit. In this disclosure, the DC component refers to (0, 0) or the top-left position with respect to the 2D component. 
     Several transform blocks may exist within one coding unit. For example, in the case of a chroma component, transform blocks for Cb and Cr may exist, and in the case of a single tree type, transform blocks for luma, Cb, and Cr may exist. According to an example, if a non-zero coefficient other than the DC component position is found even in one of the transform blocks configuring the current coding block, the value of the variable LnfstDcOnly may be set to 0. 
     On the other hand, if non-zero coefficients do not exist in the transform block, since residual coding is not performed on the corresponding transform block, the value of the variable LfnstDcOnly is not changed by the corresponding transform block. Therefore, if a non-zero coefficient does not exist in positions that are not DC components of the transform block, the value of the variable LfnstDcOnly is not changed and the previous value is maintained. The variable LfnstDcOnly value is initially initialized to 1, and if no component in the current coding unit updates the variable LfnstDcOnly value to 0, it maintains the value 1 as it is. If even one of the transform blocks constituting the corresponding coding unit updates the variable LfnstDcOnly value to 0, the LfnstDcOnly value is finally maintained at 0. 
     Meanwhile, such a variable LfnstDcOnly may be derived based on an individual transform_skip_flag value for a color component of the current block. The transform_skip_flag for the current block may be signaled for each color component. 
     According to an example, since LFNST may be applied only to the luma component, if the tree type of the current block is single-tree or dual-tree luma, the variable LfnstDcOnly may be derived based on the value of the transform_skip_flag for the luma component. 
     The variable LfnstDcOnly may be initially set to 1 in the coding unit level or the transform unit level of the current block, and may be changed to 0 at the residual coding level based on the transform_skip_flag value and the position of the transform coefficient. 
     The second variable may be expressed as a variable LfnstZeroOutSigCoeffFlag, and indicates whether a significant coefficient exists in the second region except for the top-left first region of the current block. The value of the second variable may be initially set to 1, and if a significant coefficient exists in the second region, the value may be changed to 0. The LFNST index may be parsed only when the value of the initially set variable LfnstZeroOutSigCoeffFlag is maintained at 1. 
     According to an example, the variable LfnstZeroOutSigCoeffFlag may be initially set to 1 at the coding unit level or the transform unit level of the current block, and may be updated to 0 at the residual coding level based on the position of the transform coefficient. 
     Meanwhile, as described above, in the case of a luma block to which the intra sub-partition (ISP) mode can be applied, the LFNST index may be parsed without deriving the variable LfnstDcOnly. 
     Specifically, when the ISP mode is applied and the transform_skip_flag for the luma component, that is, in case that the transform_skip_flag[x0][y0][0] value is inferred to be 0, when the tree type of the current block is single tree or dual tree for luma, the LFNST index may be signaled regardless of the value of the variable LfnstDcOnly. 
     The decoding apparatus may parse the LFNST index based on the fact that the variable LfnstDcOnly, which is the first variable, is updated to 0 and the variable LfnstZeroOutSigCoeffFlag, which is the second variable, is maintained as 1. 
     According to another example, the LFNST index may be signaled at the coding unit level for the current block. That is, when the LFNST is applied only to the luma component of the current block, the LFNST index may be signaled at the coding unit level only when this condition is satisfied. 
     Thereafter, the decoding apparatus may derive the modified transform coefficients from the transform coefficients based on the LFNST index and the LFNST matrix for LFNST (S 1540 ). 
     The decoding apparatus may determine an LFNST set including LFNST matrices based on the intra prediction mode derived from the information on the intra prediction mode, and may select any one of a plurality of LFNST matrices based on the LFNST set and the LFNST index. 
     Here, the same LFNST set and the same LFNST index may be applied to sub-partition transform blocks into which the current block is split. That is, since the same intra prediction mode is applied to the sub-partition transform blocks, the LFNST set determined based on the intra prediction mode may also be equally applied to all of the sub-partition transform blocks. In addition, since the LFNST index is signaled in the coding unit level, the same LFNST matrix may be applied to the sub-partition transform blocks into which the current block is split. 
     As described above, a transform set may be determined according to an intra prediction mode for a transform block to be transformed, and an inverse LFNST may be performed based on a transform kernel matrix, that is, any one of the LFNST matrices, included in the transform set indicated by the LFNST index. The matrix applied to the inverse LFNST may be called an inverse LFNST matrix or an LFNST matrix, and is referred to by any term as long as the matrix is the transpose of the matrix used for the forward LFNST. 
     In an example, the inverse LFNST matrix may be a non-square matrix in which the number of columns is less than the number of rows. 
     According to an example, the decoding apparatus may derive residual samples for the current block based on a primary inverse transform of the modified transform coefficient (S 1550 ). 
     Here, as the primary inverse transform, a general separable transform may be used, or the foregoing MTS may be used. 
     When the MTS is applied to the current block, the decoding apparatus may receive and parse the MTS index related to the MTS matrix, and the MTS index may also be signaled immediately after signaling of the LFNST index in the transform unit level. According to an example, the MTS index may be signaled immediately after signaling of the LFNST index only for the luma component. 
     The MTS index may be signaled based on a third variable indicating whether the significant coefficient exists in a fourth region other than the top-left third region of the current block, and the third variable may be initialized in the transform unit level. 
     The third variable may be expressed as MtsZeroOutSigCoeffFlag, initially set to 1, and this value may be changed in the residual coding of Table 10. 
     According to an example, the MTS index may be signaled when the prediction mode of the current block is intra prediction and the LFNST index is 0. 
     Subsequently, the decoding apparatus  300  may generate reconstructed samples based on the residual samples for the current block and prediction samples for the current block. 
     The following drawings are provided to describe specific examples of the present disclosure. Since specific terms for devices or specific terms for signals/messages/fields illustrated in the drawings are provided for illustration, technical features of the present disclosure are not limited to the specific terms used in the following drawings. 
       FIG.  16    is a flowchart illustrating an operation of a video encoding apparatus according to an embodiment of the present disclosure. 
     Each process disclosed in  FIG.  16    is based on some of details described with reference to  FIG.  5    to  FIG.  14   . Therefore, a description of specific details overlapping those described with reference to  FIG.  2    and  FIG.  5    to  FIG.  14    will be omitted or will be schematically made. 
     The encoding apparatus  200  according to an embodiment may derive prediction samples for a current block based on an intra prediction mode applied to the current block (S 1610 ). 
     When an ISP is applied to the current block, the encoding apparatus may perform prediction by each sub-partition transform block. 
     The encoding apparatus may determine whether to apply ISP coding or an ISP mode to the current block, that is, a coding block, and may determine a direction in which the current block is split and may derive the size and number of split subblocks according to a determination result. 
     The same intra prediction mode may be applied to sub-partition transform blocks into which the current block is split, and the encoding apparatus may derive a prediction sample for each sub-partition transform block. That is, the encoding apparatus sequentially performs intra prediction, for example, horizontally or vertically, or from left to right or from top to bottom, according to the split form of the sub-partition transform blocks. For the leftmost or uppermost subblock, a reconstructed pixel of a coding block already coded is referred to as in a conventional intra prediction method. Further, for each side of a subsequent internal sub-partition transform block, which is not adjacent to a previous sub-partition transform block, to derive reference pixels adjacent to the side, a reconstructed pixel of an adjacent coding block already coded is referred to as in the conventional intra prediction method. 
     The encoding apparatus  200  may derive residual samples for the current block based on the prediction samples (S 1620 ). 
     The encoding apparatus  200  may derive transform coefficients for the current block by applying at least one of an LFNST or an MTS to the residual samples and may arrange the transform coefficients according to a predetermined scan order. 
     The encoding apparatus may derive a transform coefficient for the current block based on a primary transform or a secondary transform on the residual sample, and according to an example, LFNST may be applied only to a luma component of the current block (S 1630 ). 
     The primary transform may be performed through a plurality of transform kernels as in the MTS, in which case a transform kernel may be selected based on the intra prediction mode. 
     When MTS is applied to the current block, the encoding apparatus may encode the MTS index related to the MTS matrix, and may configure image information so that the MTS index is signaled immediately after signaling of the LFNST index at the transform unit level. The MTS index may be signaled immediately after signaling of the LFNST index only for the luma component. 
     The MTS index may be signaled based on a third variable indicating whether the significant coefficient exists in a fourth region other than the top-left third region of the current block, and the third variable may be initialized at the transform unit level. 
     The third variable may be expressed as MtsZeroOutSigCoeffFlag, initially set to 1, and this value may be changed in the residual coding of Table 10. 
     According to an example, the MTS index may be signaled when the prediction mode of the current block is intra prediction and the LFNST index is 0. 
     The encoding apparatus  200  determines whether to perform a quadratic transform or a non-separate transform, specifically LFNST, on transform coefficients for a current block and derives modified transform coefficients by applying LFNST to the transform coefficients. 
     The LFNST is a non-separable transform in which a transform is applied to coefficients without separating the coefficients in a specific direction, unlike a primary transform of vertically or horizontally separating coefficients to be transformed and transforming the same. This non-separable transform may be a low-frequency non-separable transform of applying transform only to a low-frequency region rather than the entire target block to be transformed. 
     In order to determine whether LFNST is applied to the current block, the encoding apparatus may examine the tree type of the current block and whether ISP is applied, and may derive various variables. 
     According to an example, if the tree type of the current block is a single tree type or a dual tree luma, image information may be configured such that the LFNST index is signaled after a transform coefficient for the luma component of the current block is derived. 
     In addition, when the ISP mode is applied to the current block, the encoding apparatus may apply the LFNST index after a transform coefficient for the last partition block of the current block is derived. 
     Alternatively, if the tree type of the current block is dual tree chroma, the LFNST is not applied to the chroma component of the current block, and the LFNST index is not encoded either. 
     In addition, according to an example, the decoding apparatus may apply the LFNST based on a first variable indicating whether the significant coefficient exists in a position other than the DC component of the current block and a second variable indicating whether a significant coefficient exists in a second region excluding the top-left first region of the current block. 
     the first variable indicating whether the significant coefficient exists in a position other than the DC component of the current block may be expressed as a variable LfnstDcOnly, and it becomes 1 if a non-zero coefficient exists at a position that is not a DC component for at least one transform block in one coding unit, and it becomes if non-zero coefficients do not exist in positions that are not the DC component for all transform blocks in one coding unit. In this document, the DC component refers to (0, 0) or the top-left position with respect to the 2D component. 
     Several transform blocks may exist within one coding unit. For example, in the case of a chroma component, transform blocks for Cb and Cr may exist, and in the case of a single tree type, transform blocks for luma, Cb, and Cr may exist. According to an example, if a non-zero coefficient other than the DC component position is found even in one of the transform blocks constituting the current coding block, the value of the variable LnfstDcOnly may be set to 0. 
     On the other hand, if non-zero coefficients do not exist in the transform block, since residual coding is not performed on the corresponding transform block, the value of the variable LfnstDcOnly is not changed by the corresponding transform block. Therefore, if a non-zero coefficient does not exist in positions that are not the DC component of the transform block, the value of the variable LfnstDcOnly is not changed and the previous value is maintained. The variable LfnstDcOnly value is initially initialized to 1, and if no component in the current coding unit updates the variable LfnstDcOnly value to 0, it maintains the value 1 as it is. If even one of the transform blocks constituting the corresponding coding unit updates the variable LfnstDcOnly value to 0, the LfnstDcOnly value is finally maintained at 0. 
     Meanwhile, such a variable LfnstDcOnly may be derived based on an individual transform_skip_flag value for a color component of the current block. The transform_skip_flag for the current block may be signaled for each color component. 
     According to an example, since LFNST may be applied only to the luma component, if the tree type of the current block is single-tree or dual-tree luma, the variable LfnstDcOnly may be derived based on the value of the transform_skip_flag for the luma component. 
     The variable LfnstDcOnly may be initially set to 1 at the coding unit level or the transform unit level of the current block, and may be changed to 0 at the residual coding level based on the transform_skip_flag value and the position of the transform coefficient. 
     The second variable may be expressed as a variable LfnstZeroOutSigCoeffFlag, and indicates whether a significant coefficient exists in the second area except for the top-left first region of the current block. The value of the second variable may be initially set to 1, and if a significant coefficient exists in the second region, the value may be changed to 0. The LFNST can be applied only when the value of the initially set variable LfnstZeroOutSigCoeffFlag is maintained at 1. 
     According to an example, the variable LfnstZeroOutSigCoeffFlag may be initially set to 1 at the coding unit level or the transform unit level of the current block, and may be updated to 0 at the residual coding level based on the position of the transform coefficient. 
     Meanwhile, as described above, a luma block to which the intra sub-partition (ISP) mode can be applied, may apply the LFNST index without deriving the variable LfnstDcOnly. 
     Specifically, when the ISP mode is applied and the transform_skip_flag for the luma component, that is, in case that the transform_skip_flag[x0][y0][0] value is 0, when the tree type of the current block is single tree or dual tree for luma, the LFNST index may be applied regardless of the value of the variable LfnstDcOnly. 
     The encoding apparatus may apply LFNST based on the fact that the variable LfnstDcOnly, which is the first variable, is updated to 0 and the variable LfnstZeroOutSigCoeffFlag, which is the second variable, is maintained as 1. 
     The encoding apparatus may derive the first variable and the second variable by applying a plurality of LFNST matrices to the transform coefficients. 
     The encoding apparatus may derive a variable after applying LFNST to each LFNST matrix candidate, or in a state in which LFNST is not applied when LFNST is not applied. 
     The encoding apparatus may compare the RD values only between LFNST matrices in which the variable LfnstDcOnly value is 0 and the variable LfnstZeroOutSigCoeffFlag is 1 by applying the plurality of LFNST candidates, that is, the LFNST matrix. 
     The encoding apparatus may select the most optimal LFNST matrix and derive modified transform coefficients based on the selected LFNST matrix. 
     A transform set may be determined according to an intra prediction mode of a transform block to be transformed, and a matrix applied to LFNST has a transforce relationship with a matrix used for inverse LFNST. 
     In one example, the LFNST matrix may be a non-square matrix in which the number of rows is less than the number of columns. 
     When LFNST is applied to the current block, the encoding device configures the image information so that the LFNST index related to the LFNST matrix is signaled at the transform unit level for the current block after the transform coefficient for the current block is derived in the decoding device, and the LFNST index may be applied to the luma component of the current block (S 1640 ). 
     According to another example, the LFNST index may be signaled at the coding unit level for the current block. That is, when the LFNST is applied only to the luma component of the current block, the LFNST index may be signaled at the coding unit level only when this condition is satisfied. 
     The encoding apparatus may derive quantized transform coefficients by performing quantization based on the modified transform coefficients for the current block, and may encode and output image information including the LFNST index (S 1650 ). 
     The encoding apparatus may generate residual information including information on the quantized transform coefficients. The residual information may include the above-described conversion related information/syntax element. The encoding apparatus may encode image/video information including residual information and output the encoded image/video information in the form of a bitstream. 
     More specifically, the encoding apparatus  200  may generate the information on the quantized transform coefficients and may encode the quantized information on the generated transform coefficients. 
     A syntax element of the LFNST index according to the present embodiment may indicate any one of whether the (inverse) LFNST is applied and any one of LFNST matrices included in the LFNST set, and when the LFNST set includes two transform kernel matrices, the syntax element of the LFNST index may have three values. 
     According to an example, when the split tree structure of the current block is a dual-tree type, the LFNST index may be encoded for each of a luma block and a chroma block. 
     According to an embodiment, the values of the syntax element of the transform index may include 0 indicating that no (inverse) LFNST is applied to the current block, 1 indicating a first LFNST matrix among the LFNST matrices, and 2 indicating a second LFNST matrix among the LFNST matrices. 
     In the present disclosure, at least one of quantization/dequantization and/or transform/inverse transform may be omitted. When quantization/dequantization is omitted, a quantized transform coefficient may be referred to as a transform coefficient. When transform/inverse transform is omitted, the transform coefficient may be referred to as a coefficient or a residual coefficient, or may still be referred to as a transform coefficient for consistency of expression. 
     Further, in the present disclosure, a quantized transform coefficient and a transform coefficient may be referred to as a transform coefficient and a scaled transform coefficient, respectively. In this case, residual information may include information on a transform coefficient(s), and the information on the transform coefficient(s) may be signaled through a residual coding syntax. Transform coefficients may be derived based on the residual information (or information on the transform coefficient(s)), and scaled transform coefficients may be derived through inverse transform (scaling) of the transform coefficients. Residual samples may be derived based on the inverse transform (transform) of the scaled transform coefficients. These details may also be applied/expressed in other parts of the present disclosure. 
     In the above-described embodiments, the methods are explained on the basis of flowcharts by means of a series of steps or blocks, but the present disclosure is not limited to the order of steps, and a certain step may be performed in order or step different from that described above, or concurrently with another step. Further, it may be understood by a person having ordinary skill in the art that the steps shown in a flowchart are not exclusive, and that another step may be incorporated or one or more steps of the flowchart may be removed without affecting the scope of the present disclosure. 
     The above-described methods according to the present disclosure may be implemented as a software form, and an encoding apparatus and/or decoding apparatus according to the disclosure may be included in a device for image processing, such as, a TV, a computer, a smartphone, a set-top box, a display device or the like. 
     When embodiments in the present disclosure are embodied by software, the above-described methods may be embodied as modules (processes, functions or the like) to perform the above-described functions. The modules may be stored in a memory and may be executed by a processor. The memory may be inside or outside the processor and may be connected to the processor in various well-known manners. The processor may include an application-specific integrated circuit (ASIC), other chipset, logic circuit, and/or a data processing device. The memory may include a read-only memory (ROM), a random access memory (RAM), a flash memory, a memory card, a storage medium, and/or other storage device. That is, embodiments described in the present disclosure may be embodied and performed on a processor, a microprocessor, a controller or a chip. For example, function units shown in each drawing may be embodied and performed on a computer, a processor, a microprocessor, a controller or a chip. 
     Further, the decoding apparatus and the encoding apparatus to which the present disclosure is applied, may be included in a multimedia broadcasting transceiver, a mobile communication terminal, a home cinema video device, a digital cinema video device, a surveillance camera, a video chat device, a real time communication device such as video communication, a mobile streaming device, a storage medium, a camcorder, a video on demand (VoD) service providing device, an over the top (OTT) video device, an Internet streaming service providing device, a three-dimensional (3D) video device, a video telephony video device, and a medical video device, and may be used to process a video signal or a data signal. For example, the over the top (OTT) video device may include a game console, a Blu-ray player, an Internet access TV, a Home theater system, a smartphone, a Tablet PC, a digital video recorder (DVR) and the like. 
     In addition, the processing method to which the present disclosure is applied, may be produced in the form of a program executed by a computer, and be stored in a computer-readable recording medium. Multimedia data having a data structure according to the present disclosure may also be stored in a computer-readable recording medium. The computer-readable recording medium includes all kinds of storage devices and distributed storage devices in which computer-readable data are stored. The computer-readable recording medium may include, for example, a Blu-ray Disc (BD), a universal serial bus (USB), a ROM, a PROM, an EPROM, an EEPROM, a RAM, a CD-ROM, a magnetic tape, a floppy disk, and an optical data storage device. Further, the computer-readable recording medium includes media embodied in the form of a carrier wave (for example, transmission over the Internet). In addition, a bitstream generated by the encoding method may be stored in a computer-readable recording medium or transmitted through a wired or wireless communication network. Additionally, the embodiments of the present disclosure may be embodied as a computer program product by program codes, and the program codes may be executed on a computer by the embodiments of the present disclosure. The program codes may be stored on a computer-readable carrier. 
     The claims described herein may be combined in various ways. For example, the technical features of the method claims of the present specification may be combined and implemented as an apparatus, and the technical features of the apparatus claims of the present specification may be combined and implemented as a method. In addition, the technical features of the method claim of the present specification and the technical features of the apparatus claim may be combined to be implemented as an apparatus, and the technical features of the method claim of the present specification and the technical features of the apparatus claim may be combined and implemented as a method.