Patent Publication Number: US-11032573-B2

Title: Method, apparatus and medium for decoding or encoding

Description:
PRIORITY INFORMATION 
     This is a continuation application of U.S. application Ser. No. 16/235,336 filed on Dec. 28, 2018, which claims priority from U.S. Provisional Application No. 62/670,704, filed on May 11, 2018, and U.S. Provisional Application No. 62/668,065, filed May 7, 2018, each filed in the United States Patent &amp; Trademark Office, the disclosures of which are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     Field 
     This disclosure is directed towards video compression technologies. In particular, the present disclosure is directed towards next-generation video coding technologies including video coding/decoding technologies beyond High Efficiency Video Coding (HEVC), such as Versatile Video Coding (VVC). VVC is a video coding standard that is currently under development. The disclosed subject matter may be used in the context of VVC. More specifically, an aspect of the disclosure is directed towards a method, device and computer medium for encoding or decoding based on embedding one orthonormal transform into a larger orthonormal transform. 
     Description of Related Art 
     Video coding and decoding using inter-picture prediction with motion compensation has been known for decades. Uncompressed digital video can consist of a series of pictures, each picture having a spatial dimension of, for example, 1920×1080 luminance samples and associated chrominance samples. The series of pictures can have a fixed or variable picture rate (informally also known as frame rate), of, for example 60 pictures per second or 60 hertz (Hz). Uncompressed video has significant bitrate requirements. For example, 1080p60 4:2:0 video at 8 bit per sample (1920×1080 luminance sample resolution at 60 Hz frame rate) requires close to 1.5 Gbit/s bandwidth. An hour of such video requires more than 600 GByte of storage space. 
     One purpose of video coding and decoding can be the reduction of redundancy in the input video signal, through compression. Compression can help reduce aforementioned bandwidth or storage space requirements, in some cases by two orders of magnitude or more. Both lossless and lossy compression, as well as a combination thereof can be employed. Lossless compression refers to techniques where an exact copy of the original signal can be reconstructed from the compressed original signal. When using lossy compression, the reconstructed signal may not be identical to the original signal, but the distortion between original and reconstructed signal is small enough to make the reconstructed signal useful for the intended application. In the case of video, lossy compression is widely employed. The amount of distortion tolerated depends on the application; for example, users of certain consumer streaming applications may tolerate higher distortion than users of television contribution applications. The compression ratio achievable can reflect that: higher allowable/tolerable distortion can yield higher compression ratios. 
     A person of ordinary skill generally understands video compression/decompression technology. In general, to compress video or image data, a series of functional steps are performed to result in a compressed video or image data block. Although an image, such as a 360° image (e.g., captured by a 360° camera) may be suitable for compression, for ease of reading, compression of a video data block will be explained. To generate a compressed video data block, under conventional standards (e.g., H.264, H.265) an uncompressed video sample stream that is received from a video source may be partitioned or parsed, which results in a block of samples of two or more reference pictures. 
     Bi-Prediction can relate to techniques where a prediction unit (PU), such as a predicted block of samples, can be predicted from two motion compensated blocks of samples of two or more reference pictures. Bi-prediction was first introduced into video coding standards in MPEG-1, and has also been included in other video coding technologies and standards such as MPEG-2 Part 2 (or H.262), H.264 and H.265 as well. 
     When decompressing a compressed video file, during the reconstruction of a sample of a bi-predicted PU, motion compensated and interpolated input samples from each reference block can be multiplied by a weighting factor that can be different for each reference block, and such weighted sample values of the two reference blocks can be added to generate the sample under reconstruction. Such sample can be processed further by mechanisms such as loop filtering. 
     In MPEG-1 and MPEG-2, the weighting factors can be determined based on the relative temporal distance between the picture to which the PU under reconstruction belongs to, and the two reference pictures. This is possible because, in MPEG-1 and MPEG-2, one of the two reference I or P pictures was in the “past”, and the other in the “future” (in terms of presentation order) of the B-picture under reconstruction, and because in MPEG-1 and MPEG-2, there was a well-defined timing relationship established for any picture under reconstruction in relation to its reference pictures. 
     Starting with H.264, the reference picture selection concepts for bi-predicted pictures were relaxed such that the reference pictures only needed to be earlier in decoding order, but not in presentation order. Further, the notion of time was also relaxed in that neither H.264 nor H.265 requires a constrained/fixed picture interval in the time domain. Therefore, a decoder cannot calculate weighting factors any more based on the timing information available in the bitstream. Instead, H.264 and H.265 include a “default” of 0.5 as the weighting factor for the reference samples of a bi-predicted picture. This default can be overwritten by syntax available in the slice header known as pred_weight_table( ). The default of 0.5 or the information in the pred_weight_table may apply to all bi-predicted PUs in a given slice. 
     Non-Patent Literature 1 shows the H.265/HEVC standard. However, a need for standardization of future video coding technology with a compression capability that significantly exceeds that of the HEVC standard (including its extensions) has been studied by the inventors. 
     Non-Patent Literature 2 discloses a recently launched standardization format for next-generation video coding beyond HEVC called Versatile Video Coding (VVC) with a version VTM (VVC Test Model). VVC may generally provide a large (e.g., 64-point or higher) transform core using a 10-bit integer matrix. 
     Problem(s) to be Solved 
     Different from DCT-2, of which fast methods have been extensively studied, the implementation of DST-7 is still much less efficient than DCT-2, e.g., matrix multiplication based in VTM 1 of Non-Patent Literature 2. In Non-Patent Literature 3, a method is proposed to approximate different types of DCTs and DSTs in JEM7 by applying adjustment stages to a transform in the DCT-2 family, which includes DCT-2, DCT-3, DST-2 and DST-3, and the adjustment stage refers to a matrix multiplication using a sparse matrix which requires relatively less operation counts. In Non-Patent Literature 4, a method for implementing N-point DST-7 using 2N+1 point Discrete Fourier Transform (DFT) is proposed. 
     However, additional transform types in Adaptive Multiple Transforms (AMT) increase the complexity and implementation cost(s), since additional transform cores need to be stored in on-chip memory for practical implementation, and transform cores need to be loaded when switching the transform type from one block to another. In HEVC, a smaller DCT-2 transform is a part of the larger DCT-2 transform, which supports reusing the same logic for implementing different sizes of transforms. However, with AMT, such a feature is missing, other than DCT-2, a smaller transform cannot be re-used for a larger transform, which means different logic may need to be designed for implementing different sizes of transforms. 
     The DCT-2 in HEVC is implemented using partial butterfly, which supports identical outputs between matrix multiplication and partial butterfly, i.e., dual implementation. However, half-size matrix multiplication is still needed, which is still a computation burden, for example, partial butterfly of 32-point HEVC DCT-2 requires 2×2, 4×4, 8×8 and 16×16 matrix multiplications. 
     Listing of Related Art 
     Non-Patent Literature 1: “Transform design for HEVC with 16 bit intermediate data representation”; published by: Joint Collaborative Team on Video Coding (JCT-VC) of ITU-T SG16 WP3 and ISO/IEC JTC1/SC29/WG11, 5th Meeting: Geneva, CH, Mar. 16-23, 2011 (Document: JCTVC-E243). The H.265/HEVC (High Efficiency Video Coding) standard may include Version 1 (2013), Version 2 (2014), Version 3 (2015) and Version 4 (2016) published by ITU-T VCEG (Q6/16) ISO/IEC MPEG (JTC 1/SC 29/WG 11). 
     Non-Patent Literature 2: “Algorithm description for Versatile Video Coding and Test Model 1 (VTM 1)”; published by Joint Video Experts Team (JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, 10th Meeting: San Diego, US, April 10-20, 2018 (Document: JVET-J1002-v2). 
     Non-Patent Literature 3: A. Said, H. Egilmez, V. Seregin, M. Karczewicz, “Complexity Reduction for Adaptive Multiple Transforms (AMTs) using Adjustment Stages”, Joint Video Exploration Team (JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, San Diego, US, 10-20 Apr. 2018 (Document: JVET-J0066). 
     Non-Patent Literature 4: M. Koo, J. Heo, J. Nam, N. Park, J. Lee, J. Choi, S. Yoo, H. Jang, L. Li, J. Lim, S. Paluri, M. Salehifar, S. Kim, “Description of SDR video coding technology proposal by LG Electronics,” Joint Video Exploration Team (JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, San Diego, US, 10-20 Apr. 2018 (Document: JVET-J0017). 
     SUMMARY 
     According to an aspect of the disclosure, a method of decoding or encoding may include generating a new multi-transform data structure by embedding one or more non-recursive transforms into a larger recursive transform. The method may further include receiving information regarding a target data block for encoding or decoding, the target data block for encoding or decoding being one of: a compressed video or image data block or an uncompressed video or image data block. The method of decoding or encoding may further include determining whether to use a recursive transform or a non-recursive transform for the encoding or decoding of the target data block. The method of decoding or encoding may further include, when a result of the determination is to use the recursive transform: generate the recursive transform using the multi-transform data structure and cause or transmit information that causes the target data block to be encoded or decoded using the generated recursive transform. The method of encoding or decoding may further include cause or transmit information that causes the target data block to be encoded or decoded using one of the one or more non-recursive transforms embedded in the multi-transform data structure. 
     The method of encoding or decoding may further include that at least part of the one of the one or more non-recursive transforms of the multi-transform data structure is embedded in at least part of one half of the larger recursive transform of the multi-transform data structure. 
     According to an aspect of the disclosure, an apparatus may include at least one memory configured to store computer program code and at least one processor configured to access the at least one memory and operate according to the computer program code. The computer program code may include first applying code, second applying code, third applying code, fourth applying code, fifth applying code, and sixth applying code, but is not limited in the number of code segments that may be included. 
     The first applying code may be configured to generate a new multi-transform data structure by embedding one or more non-recursive transforms into a larger recursive transform. The second applying code may be configured to cause the at least one processor to receive information regarding a target data block for encoding or decoding, the target data block for encoding or decoding being one of: a compressed video or image data block or an uncompressed video or image data block. The third applying code may be configured to cause the at least one processor to determine, for the encoding or decoding of the target data block, whether to use a recursive transform or a non-recursive transform for the encoding or decoding of the target data block. 
     The fourth applying code may be configured to cause the at least one processor to when a result of the determination is to use the recursive transform: generate the recursive transform using the multi-transform data structure and cause or transmit information that causes the target data block to be encoded or decoded using the generated recursive transform. The fifth applying code may be configured to cause the at least one processor to when a result of the determination is to use the non-recursive transform of the multi-transform data structure: cause or transmit information that causes the target data block to be encoded or decoded one of the one or more non-recursive transforms embedded in the multi-transform data structure. 
     The apparatus for encoding or decoding may further include that at least part of the one of the one or more non-recursive transforms of the multi-transform data structure is embedded in at least part of one half of the larger recursive transform of the multi-transform data structure. 
     According to an aspect of the disclosure, a non-transitory computer-readable storage medium may store instructions that cause one or more processors to execute a number of steps. The number of steps may include generating a new multi-transform data structure by embedding one or more non-recursive transforms into a larger recursive transform. The number of steps may further include receiving information regarding a target data block for encoding or decoding, the target data block for encoding or decoding may be one of: a compressed video or image data block or an uncompressed video or image data block. The number of steps may further include determining, for the encoding or decoding of the target data block, whether to use a recursive transform or a non-recursive transform for the encoding or decoding of the target data block. 
     The number of steps may further include, when a result of the determination is to use the recursive transform: generate the recursive transform using the multi-transform data structure and cause or transmit information that causes the target data block to be encoded or decoded using the generated recursive transform. The number of steps may further include, when a result of the determination is to use the non-recursive transform of the multi-transform data structure: causing or transmitting information that causes the target data block to be encoded or decoded using one of the one or more non-recursive transforms embedded in the multi-transform data structure, wherein at least part of the one of the one or more non-recursive transforms of the multi-transform data structure is embedded in at least part of one half of the larger recursive transform of the multi-transform data structure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features, the nature, and various advantages of the disclosed subject matter will be more apparent from the following detailed description and the accompanying drawings in which: 
         FIG. 1  is a schematic illustration of a simplified block diagram of a communication system in accordance with an embodiment. 
         FIG. 2  is a schematic illustration of a simplified block diagram of a streaming system in accordance with an embodiment. 
         FIG. 3  is a schematic illustration of a simplified block diagram of a decoder in accordance with an embodiment. 
         FIG. 4  is a schematic illustration of a simplified block diagram of an encoder in accordance with an embodiment. 
         FIG. 5  is a schematic illustration of an exemplary method of encoding or decoding in accordance with an embodiment. 
         FIG. 6A  is graph showing the 2 nd , 4 th , 6 th  and 8 th  basis of a 64-point DCT-2 and the 1 st , 2 nd , 3 rd  and 4 th  basis of 32-point DCT-8. 
         FIG. 6B  is a graph showing the 2 nd , 4 th , 6 th  and 8 th  basis of a 64-point DCT-2, the 1 st , 2 nd , 3 rd , and 4 th  basis of scaled 32-point DCT-8, and the flipped 1 st , 2 nd , 3 rd  and 4 th  basis of scaled 32-point DCT-8 with inverse sign. 
         FIG. 7  is a schematic illustration of a computer system in accordance with an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a simplified block diagram of a communication system ( 100 ) according to an embodiment of the present disclosure. The system ( 100 ) may include at least two terminals ( 110 ,  120 ) interconnected via a network ( 150 ). For unidirectional transmission of data, a first terminal ( 110 ) may code video data at a local location for transmission to the other terminal ( 120 ) via the network ( 150 ). The second terminal ( 120 ) may receive the coded video data of the other terminal from the network ( 150 ), decode the coded data and display the recovered video data. Unidirectional data transmission may be common in media serving applications and the like. 
       FIG. 1  illustrates a second pair of terminals ( 130 ,  140 ) provided to support bidirectional transmission of coded video that may occur, for example, during videoconferencing. For bidirectional transmission of data, each terminal ( 130 ,  140 ) may code video data captured at a local location for transmission to the other terminal via the network ( 150 ). Each terminal ( 130 ,  140 ) also may receive the coded video data transmitted by the other terminal, may decode the coded data and may display the recovered video data at a local display device. 
     In  FIG. 1 , the terminals ( 110 - 140 ) may be illustrated as a laptop  110 , a server  120 , and smartphones  130  and  140 , but the principles of the present disclosure are not so limited. Embodiments of the present disclosure find application with other equipment including but not limited to laptop computers, tablet computers, media players and/or dedicated video conferencing equipment. The network ( 150 ) represents any number of networks that convey coded video data among the terminals ( 110 - 140 ), including for example wireline and/or wireless communication networks. The communication network ( 150 ) may exchange data in circuit-switched and/or packet-switched channels. Representative networks include telecommunications networks, local area networks, wide area networks and/or the Internet. For the purposes of the present discussion, the architecture and topology of the network ( 150 ) may be immaterial to the operation of the present disclosure unless explained herein below. 
       FIG. 2  illustrates, as an example for an application for the disclosed subject matter, the placement of a video encoder and decoder in a streaming environment. The disclosed subject matter can be equally applicable to other video enabled applications, including, for example, video conferencing, digital TV, storing of compressed video on digital media including CD, DVD, memory stick and the like, and so on. 
     A streaming system may include a capture subsystem ( 213 ), which can include a video source ( 201 ), for example a digital camera, which may create an uncompressed video sample stream ( 202 ). That sample stream ( 202 ), depicted as a bold line to emphasize a high data volume when compared to encoded video bit streams, can be processed by an encoder ( 203 ) coupled to the camera ( 201 ). The encoder ( 203 ) can include hardware (e.g., a processor or circuit and memory), software, or a combination thereof to enable or implement aspects of the disclosed subject matter as described in more detail below. The encoded video bit stream ( 204 ), depicted as a thin line to emphasize the lower data volume when compared to the sample stream, can be stored on a streaming server ( 205 ) for future use. One or more streaming clients ( 206 ,  208 ) can access the streaming server ( 205 ) to retrieve copies ( 207 ,  209 ) of the encoded video bitstream ( 204 ). A client ( 206 ) can include a video decoder ( 210 ) which decodes the incoming copy of the encoded video bitstream ( 207 ) and creates an outgoing video sample stream ( 211 ) that can be rendered on a display ( 212 ) or other rendering device (not depicted). 
       FIG. 3  illustrates a functional block diagram of a video decoder ( 210 ) according to an embodiment of the present disclosure. As shown in  FIG. 3 , a receiver ( 310 ) may receive one or more codec video sequences to be decoded by the decoder ( 210 ). In the same or another embodiment, one coded video sequence may be received at a time, where the decoding of each coded video sequence is independent from other coded video sequences. The coded video sequence may be received from a channel ( 312 ), which may be a hardware/software link to a storage device which stores the encoded video data. The receiver ( 310 ) may receive the encoded video data with other data, for example, coded audio data and/or ancillary data streams, that may be forwarded to their respective using entities (not depicted). The receiver ( 310 ) may separate the coded video sequence from the other data. To combat network jitter, a buffer memory ( 315 ) may be coupled in between receiver ( 310 ) and entropy decoder/parser ( 320 ) (“parser” henceforth). When receiver ( 310 ) is receiving data from a store/forward device of sufficient bandwidth and controllability, or from an isosychronous network, the buffer ( 315 ) may not be needed, or can be small. For use on best effort packet networks such as the Internet, the buffer ( 315 ) may be required, which can be comparatively large and can advantageously be of adaptive size. 
     The video decoder ( 310 ) may include a parser ( 320 ) to reconstruct symbols ( 321 ) from the entropy coded video sequence. Categories of those symbols include information used to manage operation of the decoder ( 210 ), and potentially information to control a rendering device such as a display ( 212 ) that is not an integral part of the decoder but can be coupled to it, as was shown in  FIGS. 2 and 3 . The control information for the rendering device(s) may be in the form of Supplementary Enhancement Information (SEI messages) or Video Usability Information (VUI) parameter set fragments (not depicted). The parser ( 320 ) may parse/entropy-decode the coded video sequence received. The coding of the coded video sequence can be in accordance with a video coding technology or standard, and can follow principles well known to a person skilled in the art, including variable length coding, Huffman coding, arithmetic coding with or without context sensitivity, and so forth. The parser ( 320 ) may extract from the coded video sequence, a set of subgroup parameters for at least one of the subgroups of pixels in the video decoder, based upon at least one parameters corresponding to the group. Subgroups can include Groups of Pictures (GOPs), pictures, tiles, slices, macroblocks, Coding Units (CUs), blocks, Transform Units (TUs), Prediction Units (PUs) and so forth. The entropy decoder/parser may also extract from the coded video sequence information such as transform coefficients, quantizer parameter values, motion vectors, and so forth. 
     The parser ( 320 ) may perform entropy decoding/parsing operation on the video sequence received from the buffer ( 315 ), so as to create symbols ( 321 ). Reconstruction of the symbols ( 321 ) can involve multiple different units depending on the type of the coded video picture or parts thereof (such as: inter and intra picture, inter and intra block), and other factors. Which units are involved, and how, can be controlled by the subgroup control information that was parsed from the coded video sequence by the parser ( 320 ). The flow of such subgroup control information between the parser ( 320 ) and the multiple units below is not depicted for clarity. 
     Beyond the functional blocks already mentioned, decoder  210  can be conceptually subdivided into a number of functional units as described below. In a practical implementation operating under commercial constraints, many of these units interact closely with each other and can, at least partly, be integrated into each other. However, for the purpose of describing the disclosed subject matter, the conceptual subdivision into the functional units below is appropriate. 
     A first unit may be the scaler/inverse transform unit ( 351 ). The scaler/inverse transform unit ( 351 ) may receive a quantized transform coefficient as well as control information, including which transform to use, a block size, a quantization factor, quantization scaling matrices, etc. as symbol(s) ( 321 ) from the parser ( 320 ). It can output blocks comprising sample values that can be input into an aggregator ( 355 ). 
     In some cases, the output samples of the scaler/inverse transform ( 351 ) can pertain to an intra-coded block, which is a block that is not using predictive information from previously reconstructed pictures, but can use predictive information from previously reconstructed parts of the current picture. Such predictive information can be provided by an intra picture prediction unit ( 352 ). In some cases, the intra picture prediction unit ( 352 ) generates a block of the same size and shape of the block under reconstruction, using surrounding already reconstructed information fetched from the current (partly reconstructed) picture ( 358 ). The aggregator ( 355 ), in some cases, adds, on a per sample basis, the prediction information the intra prediction unit ( 352 ) has generated to the output sample information as provided by the scaler/inverse transform unit ( 351 ). 
     In other cases, the output samples of the scaler/inverse transform unit ( 351 ) can pertain to an inter-coded block, and a potentially motion compensated block. In such a case, a Motion Compensation Prediction unit ( 353 ) can access reference a picture memory ( 357 ) to fetch samples used for prediction. After motion compensating the fetched samples in accordance with the symbols ( 321 ) pertaining to the block, these samples can be added by the aggregator ( 355 ) to the output of the scaler/inverse transform unit (in this case called the residual samples or residual signal) so to generate output sample information. The addresses within the reference picture memory form where the motion compensation unit fetches prediction samples can be controlled by motion vectors, available to the motion compensation unit in the form of symbols ( 321 ) that can have, for example X, Y, and reference picture components. Motion compensation also can include interpolation of sample values as fetched from the reference picture memory when sub-sample exact motion vectors are in use, motion vector prediction mechanisms, and so forth. 
     The output samples of the aggregator ( 355 ) can be subject to various loop filtering techniques in the loop filter unit ( 356 ). Video compression technologies can include in-loop filter technologies that are controlled by parameters included in the coded video bitstream and made available to the loop filter unit ( 356 ) as symbols ( 321 ) from the parser ( 320 ), but can also be responsive to meta-information obtained during the decoding of previous (in decoding order) parts of the coded picture or coded video sequence, as well as responsive to previously reconstructed and loop-filtered sample values. 
     The output of the loop filter unit ( 356 ) can be a sample stream that can be output to the render device ( 212 ) as well as stored in the reference picture memory ( 357 ) for use in future inter-picture prediction. 
     Certain coded pictures, once fully reconstructed, can be used as reference pictures for future prediction. Once a coded picture is fully reconstructed and the coded picture has been identified as a reference picture (by, for example, parser ( 320 )), the current reference picture ( 358 ) can become part of the reference picture buffer ( 357 ), and a fresh current picture memory can be reallocated before commencing the reconstruction of the following coded picture. 
     The video decoder  320  may perform decoding operations according to a predetermined video compression technology that may be documented in a standard, such as ITU-T Rec. H.265. The coded video sequence may conform to a syntax specified by the video compression technology or standard being used, in the sense that it adheres to the syntax of the video compression technology or standard, as specified in the video compression technology document or standard and specifically in the profiles document therein. Also necessary for compliance can be that the complexity of the coded video sequence is within bounds as defined by the level of the video compression technology or standard. In some cases, levels restrict the maximum picture size, maximum frame rate, maximum reconstruction sample rate (measured in, for example megasamples per second), and/or maximum reference picture size. Limits set by levels can, in some cases, be further restricted through Hypothetical Reference Decoder (HRD) specifications and metadata for HRD buffer management signaled in the coded video sequence. 
     In an embodiment, the receiver ( 310 ) may receive additional (redundant) data with the encoded video. The additional data may be included as part of the coded video sequence(s). The additional data may be used by the video decoder ( 320 ) to properly decode the data and/or to more accurately reconstruct the original video data. Additional data can be in the form of, for example, temporal, spatial, or SNR enhancement layers, redundant slices, redundant pictures, forward error correction codes, and so on. 
       FIG. 4  may be a functional block diagram of a video encoder ( 203 ) according to an embodiment of the present disclosure. 
     The encoder ( 203 ) may receive video samples from a video source ( 201 ) (that is not part of the encoder) that may capture video image(s) to be coded by the encoder ( 203 ). 
     The video source ( 201 ) may provide the source video sequence to be coded by the encoder ( 203 ) in the form of a digital video sample stream that can be of any suitable bit depth (for example: 8 bit, 10 bit, 12 bit, . . . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ) and any suitable sampling structure (for example Y CrCb 4:2:0, Y CrCb 4:4:4). In a media serving system, the video source ( 201 ) may be a storage device, which may include a memory and processor, that stores previously prepared video. In a videoconferencing system, the video source ( 203 ) may include a camera that captures local image information as a video sequence. Video data may be provided as a plurality of individual pictures that impart motion when viewed in sequence. The pictures themselves may be organized as a spatial array of pixels, wherein each pixel can comprise one or more sample depending on the sampling structure, color space, etc. in use. A person skilled in the art can readily understand the relationship between pixels and samples. The description below focuses on samples. 
     According to an embodiment, the encoder ( 203 ) may code and compress the pictures of the source video sequence into a coded video sequence ( 443 ) in real time or under any other time constraints as required by the application. Enforcing appropriate coding speed is one function of controller ( 450 ). Controller controls other functional units as described below and is functionally coupled to these units. The coupling is not depicted for clarity. Parameters set by the controller can include rate control related parameters (e.g., picture skip, quantizer, lambda value of rate-distortion optimization techniques), a picture size, a group of pictures (GOP) layout, a maximum motion vector search range, and so forth. A person skilled in the art can readily identify other functions of controller ( 450 ) as they may pertain to video encoder ( 203 ) optimized for a certain system design. 
     Some video encoders operate in what a person skilled in the art readily recognizes as a “coding loop.” As an oversimplified description, a coding loop can consist of the encoding part of an encoder ( 430 ) (“source coder” henceforth), which is responsible for creating symbols based on an input picture to be coded, and a reference picture(s), and a (local) decoder ( 433 ) embedded in the encoder ( 203 ) that reconstructs the symbols to create the sample data a (remote) decoder also would create (as any compression between symbols and coded video bitstream is lossless in the video compression technologies considered in the disclosed subject matter). That reconstructed sample stream is input to the reference picture memory ( 434 ). As the decoding of a symbol stream leads to bit-exact results independent of a decoder location (local or remote), the reference picture buffer content is also bit exact between local encoder and remote encoder. In other words, the prediction part of an encoder “sees,” as reference picture samples, exactly the same sample values as a decoder would “see” when using prediction during decoding. This fundamental principle of reference picture synchronicity (and resulting drift, if synchronicity cannot be maintained, for example, because of channel errors) is well known to a person skilled in the art. 
     The operation of the “local” decoder ( 433 ) can be the same as of a “remote” decoder ( 210 ), which has already been described in detail above in conjunction with  FIG. 3 . Briefly referring also to  FIG. 3 , however, as symbols are available and en/decoding of symbols to a coded video sequence by entropy coder ( 445 ) and parser ( 320 ) can be lossless, the entropy decoding parts of decoder ( 210 ) (including channel ( 312 ), receiver ( 310 ), buffer ( 315 ), and parser ( 320 )) may not be fully implemented in local decoder ( 433 ). 
     An observation that can be made at this point is that any decoder technology except the parsing/entropy decoding that is present in a decoder also necessarily needs to be present, in substantially identical functional form, in a corresponding encoder. The description of encoder technologies are the inverse of the described decoder technologies. Only in certain areas a more detail description is required and provided below. 
     As part of its operation, the source coder ( 430 ) may perform motion compensated predictive coding, which codes an input frame predictively with reference to one or more previously-coded frames from the video sequence that were designated as “reference frames.” In this manner, the coding engine ( 432 ) codes differences between pixel blocks of an input frame and pixel blocks of reference frame(s) that may be selected as prediction reference(s) to the input frame. 
     The local video decoder ( 433 ) may decode coded video data of frames that may be designated as reference frames, based on symbols created by the source coder ( 430 ). Operations of the coding engine ( 432 ) may advantageously be lossy processes. When the coded video data may be decoded at a video decoder (not shown in  FIG. 4 ), the reconstructed video sequence typically may be a replica of the source video sequence with some errors. The local video decoder ( 433 ) replicates decoding processes that may be performed by the video decoder on reference frames and may cause reconstructed reference frames to be stored in the reference picture cache ( 434 ). In this manner, the encoder ( 203 ) may store copies of reconstructed reference frames locally that have common content as the reconstructed reference frames that will be obtained by a far-end video decoder (absent transmission errors). 
     The predictor ( 435 ) may perform prediction searches for the coding engine ( 432 ). That is, for a new frame to be coded, the predictor ( 435 ) may search the reference picture memory ( 434 ) for sample data (as candidate reference pixel blocks) or certain metadata such as reference picture motion vectors, block shapes, and so on, that may serve as an appropriate prediction reference for the new pictures. The predictor ( 435 ) may operate on a sample block-by-pixel block basis to find appropriate prediction references. In some cases, as determined by search results obtained by the predictor ( 435 ), an input picture may have prediction references drawn from multiple reference pictures stored in the reference picture memory ( 434 ). 
     The controller ( 450 ) may manage coding operations of the video coder ( 430 ), including, and for example, setting of parameters and subgroup parameters used for encoding the video data. 
     Output of all aforementioned functional units may be subjected to entropy coding in the entropy coder ( 445 ). The entropy coder may translate the symbols as generated by the various functional units into a coded video sequence, by loss-less compressing the symbols according to technologies known to a person skilled in the art as, for example Huffman coding, variable length coding, arithmetic coding, and so forth. 
     The transmitter ( 440 ) may buffer the coded video sequence(s) as created by the entropy coder ( 445 ) to prepare it for transmission via a communication channel ( 460 ), which may be a hardware/software link to a storage device which would store the encoded video data. The transmitter ( 440 ) may merge coded video data from the video coder ( 430 ) with other data to be transmitted, for example, coded audio data and/or ancillary data streams (sources not shown). 
     The controller ( 450 ) may manage operation of the encoder ( 203 ). During coding, the controller ( 450 ) may assign to each coded picture a certain coded picture type, which may affect the coding techniques that may be applied to the respective picture. For example, pictures often may be assigned as one of the following frame types from among, for example, an Intra Picture (I picture), a Predictive picture (P picture) and a Bi-directionally Predictive Picture (B Picture). 
     An Intra Picture (I picture) may be one that may be coded and decoded without using any other frame in the sequence as a source of prediction. Some video codecs allow for different types of Intra pictures, including, for example Independent Decoder Refresh Pictures. A person skilled in the art is aware of those variants of I pictures and their respective applications and features. 
     A Predictive Picture (P picture) may be one that may be coded and decoded using intra prediction or inter prediction using at most one motion vector and reference index to predict the sample values of each block. 
     A Bi-directionally Predictive Picture (B Picture) may be one that may be coded and decoded using intra prediction or inter prediction using at most two motion vectors and reference indices to predict the sample values of each block. Similarly, multiple-predictive pictures can use more than two reference pictures and associated metadata for the reconstruction of a single block. 
     Source pictures commonly may be subdivided spatially into a plurality of sample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 samples each) and coded on a block-by-block basis. Blocks may be coded predictively with reference to other (already coded) blocks as determined by the coding assignment applied to the blocks&#39; respective pictures. For example, blocks of I pictures may be coded non-predictively or they may be coded predictively with reference to already coded blocks of the same picture (spatial prediction or intra prediction). Pixel blocks of P pictures may be coded non-predictively, via spatial prediction or via temporal prediction with reference to one previously coded reference pictures. Blocks of B pictures may be coded non-predictively, via spatial prediction or via temporal prediction with reference to one or two previously coded reference pictures. 
     The video coder ( 203 ) may perform coding operations according to a predetermined video coding technology or standard, such as ITU-T Rec. H.265. In its operation, the video coder ( 203 ) may perform various compression operations, including predictive coding operations that exploit temporal and spatial redundancies in the input video sequence. The coded video data, therefore, may conform to a syntax specified by the video coding technology or standard being used. 
     In an embodiment, the transmitter ( 440 ) may transmit additional data with the encoded video. The video coder ( 430 ) may include such data as part of the coded video sequence. Additional data may comprise temporal/spatial/SNR enhancement layers, other forms of redundant data such as redundant pictures and slices, Supplementary Enhancement Information (SEI) messages, Visual Usability Information (VUI) parameter set fragments, and so on. 
     As discussed above, the coding of the coded video sequence can be in accordance with a video coding technology or standard, and can follow principles well known to a person skilled in the art, including variable length coding, Huffman coding, arithmetic coding with or without context sensitivity, and so forth. In some streaming systems, the video bitstreams ( 204 ,  207 , and  209 ) can be encoded according to certain video coding/compression standards. An example of those standards includes ITU-T HEVC Recommendation of H.265. 
     In HEVC, the primary transforms may have 4-point, 8-point, 16-point or 32-point DCT-2 transforms core matrices, and the transform core matrices may be represented using 8-bit integers, (i.e., an 8-bit transform core matrix). The HEVC transform core matrices of smaller DCT-2 are part of larger DCT-2, as shown below: 
     4×4 DCT-2 Transform 
     {64, 64, 64, 64} 
     {83, 36,−36,−83} 
     {64,−64,−64, 64} 
     {36,−83, 83,−36} 
     8×8 DCT-2 Transform 
     {64, 64, 64, 64, 64, 64, 64, 64} 
     {89, 75, 50, 18,−18,−50,−75,−89} 
     {83, 36,−36,−83,−83,−36, 36, 83} 
     {75,−18,−89,−50, 50, 89, 18,−75} 
     {64,−64,−64, 64, 64,−64,−64, 64} 
     {50,−89, 18, 75,−75,−18, 89,−50} 
     {36,−83, 83,−36,−36, 83,−83, 36} 
     {18,−50, 75,−89, 89,−75, 50,−18} 
     16×16 DCT-2 Transform 
     {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64} 
     {90 87 80 70 57 43 25 9 −9−25−43−57−70−80−87−90} 
     {89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89} 
     {87 57 9−43−80−90−70−25 25 70 90 80 43 −9−57−87} 
     {83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83} 
     {80 9−70−87−25 57 90 43−43−90−57 25 87 70 −9−80} 
     {75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75} 
     {70−43−87 9 90 25−80−57 57 80−25−90−9 87 43−70} 
     {64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64} 
     {57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57} 
     {50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50} 
     {43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43} 
     {36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36} 
     {25−70 90−80 43 9−57 87−87 57−9−43 80−90 70−25} 
     {18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18} 
     {9−25 43−57 70−80 87−90 90−87 80−70 57−43 25 −9} 
     32×32 DCT-2 Transform 
     {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64} 
     {90 90 88 85 82 78 73 67 61 54 46 38 31 22 13 4 −4−13−22−31−38−46−54−61−67−73−78−82−85−88−90−90} 
     {90 87 80 70 57 43 25 9−9−25−43−57−70−80−87−90−90−87−80−70−57−43−25−9 9 25 43 57 70 80 87 90} 
     {90 82 67 46 22−4−31−54−73−85−90−88−78−61−38−13 13 38 61 78 88 90 85 73 54 31 4−22−46−67−82−90} 
     {89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89 89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89} 
     {88 67 31−13−54−82−90−78−46−4 38 73 90 85 61 22−22−61−85−90−73−38 4 46 78 90 82 54 13−31−67−88} 
     {87 57 9−43−80−90−70−25 25 70 90 80 43−9−57−87−87−57−9 43 80 90 70 25−25−70−90−80−43 9 57 87} 
     {85 46−13−67−90−73−22 38 82 88 54−4−61−90−78−31 31 78 90 61 4−54−88−82−38 22 73 90 67 13−46−85} 
     {83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83} 
     {82 22−54−90−61 13 78 85 31−46−90−67 4 73 88 38−38−88−73−4 67 90 46−31−85−78−13 61 90 54−22−82} 
     {80 9−70−87−25 57 90 43−43−90−57 25 87 70−9−80−80−9 70 87 25−57−90−43 43 90 57−25−87−70 9 80} 
     {78 −4−82−73 13 85 67−22−88−61 31 90 54−38−90−46 46 90 38−54−90−31 61 88 22−67−85−13 73 82 4−78} 
     {75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75 75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75} 
     {73−31−90−22 78 67−38−90−13 82 61−46−88−4 85 54−54−85 4 88 46−61−82 13 90 38−67−78 22 90 31−73} 
     {70−43−87 9 90 25−80−57 57 80−25−90−9 87 43−70−70 43 87−9−90−25 80 57−57−80 25 90 9−87−43 70} 
     {67−54−78 38 85−22−90 4 90 13−88−31 82 46−73−61 61 73−46−82 31 88−13−90−4 90 22−85−38 78 54−67} 
     {64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64} 
     {61−73−46 82 31−88−13 90−4−90 22 85−38−78 54 67−67−54 78 38−85−22 90 4−90 13 88−31−82 46 73−61} 
     {57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57−57 80 25−90 9 87−43−70 70 43−87−9 90−25−80 57} 
     {54−85 −4 88−46−61 82 13−90 38 67−78−22 90−31−73 73 31−90 22 78−67−38 90−13−82 61 46−88 4 85−54} 
     {50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50 50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50} 
     {46−90 38 54−90 31 61−88 22 67−85 13 73−82 4 78−78−4 82−73−13 85−67−22 88−61−31 90−54−38 90−46} 
     {43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43−43 90−57−25 87−70−9 80−80 9 70−87 25 57−90 43} 
     {38−88 73−4−67 90−46−31 85−78 13 61−90 54 22−82 82−22−54 90−61−13 78−85 31 46−90 67 4−73 88−38} 
     {36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36} 
     {31−78 90−61 4 54−88 82−38−22 73−90 67−13−46 85−85 46 13−67 90−73 22 38−82 88−54−4 61−90 78−31} 
     {25−70 90−80 43 9−57 87−87 57−9−43 80−90 70−25−25 70−90 80−43−9 57−87 87−57 9 43−80 90−70 25} 
     {22−61 85−90 73−38−4 46−78 90−82 54−13−31 67−88 88−67 31 13−54 82−90 78−46 4 38−73 90−85 61−22} 
     {18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18 18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18} 
     {13−38 61−78 88−90 85−73 54−31 4 22−46 67−82 90−90 82−67 46−22−4 31−54 73−85 90−88 78−61 38−13} 
     {9−25 43−57 70−80 87−90 90−87 80−70 57−43 25−9 −9 25−43 57−70 80−87 90−90 87−80 70−57 43−25 9} 
     {4−13 22−31 38−46 54−61 67−73 78−82 85−88 90−90 90−90 88−85 82−78 73−67 61−54 46−38 31−22 13−4} 
     As shown above, each of the DCT-2 cores show symmetry/anti-symmetry characteristics. These symmetry/anti-symmetry characteristics include at least the following three features (Features #1-#3). Feature #1 is that even rows with index 0, 2, 4, . . . are symmetric with a symmetry point before coefficient no. N/2. Feature #2 is that odd rows with index 1, 3, 5, 7, . . . are anti-symmetric with a symmetry point before coefficient no. N/2. In addition, Feature #3 is that the N-point DCT-2 core (denoted by TN) is part of 2N-point DCT-2 (denoted by T2N) and TNx,y=T2Nx,2y, where x, y=0, 1, . . . , N−1. 
     Based on the above symmetry/anti-symmetry characteristics (the Features #1 and #2) and the relationship between N-point DCT-2 and 2N-point DCT-2 (the Feature #3), a so-called “partial butterfly” implementation may be supported to reduce the number of operation counts (multiplications, adds/subs, shifts), and identical results of matrix multiplication can be obtained using a partial butterfly implementation. More details regarding HEVC transform design can be found in Non-Patent Literature 1. 
     In this disclosure, if each of the transform bases is either symmetric or anti-symmetric, and the N-point transform is part of the 2N-point transform, it is called a “recursive transform.” Examples of a recursive transform are DCT-2, Hadamard transform, DCT-1, DST-1, and DST-2. Reordering the transform bases of one recursive transform results in another recursive transform. 
     In addition to DCT-2 and 4×4 DST-7, which have been employed in HEVC, an Adaptive Multiple Transform (AMT, or also known as Enhanced Multiple Transform (EMT), or also known as Multiple Transform Selection (MTS)) scheme has been proposed for residual coding for both inter and intra coded blocks. The AMT may use multiple selected transforms from the DCT/DST families other than the current transforms in HEVC. The newly introduced transform matrices may include DST-7, DCT-8, DST-1 and DCT-5. 
     Table 1 below shows exemplary basis functions of a selected DST/DCT. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Transform basis functions of  
               
               
                 DCT-II/V/VIII and DST-I/VII for N-point input 
               
            
           
           
               
               
            
               
                 Transform Type 
                 Basis function T i (j), i, j = 0, 1, . . . , N − 1 
               
               
                   
               
               
                 DCT-2 
                 
                   
                     
                       
                         
                           
                             T 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                         = 
                         
                           
                             ω 
                             0 
                           
                           · 
                           
                             
                               2 
                               N 
                             
                           
                           · 
                           
                             cos 
                             ( 
                             
                               
                                 π 
                                 · 
                                 i 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           where 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ω 
                             0 
                           
                         
                         = 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     2 
                                     N 
                                   
                                 
                               
                               
                                 
                                   i 
                                   = 
                                   0 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   i 
                                   ≠ 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 DCT-5 
                 
                   
                     
                       
                         
                           
                             
                               T 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           = 
                           
                             
                               ω 
                               0 
                             
                             · 
                             
                               ω 
                               1 
                             
                             · 
                             
                               
                                 2 
                                 
                                   
                                     2 
                                     ⁢ 
                                     N 
                                   
                                   - 
                                   1 
                                 
                               
                             
                             · 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       π 
                                       · 
                                       i 
                                       · 
                                       j 
                                     
                                   
                                   
                                     
                                       2 
                                       ⁢ 
                                       N 
                                     
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           where 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ω 
                             0 
                           
                         
                         = 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       2 
                                       N 
                                     
                                   
                                 
                                 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     i 
                                     ≠ 
                                     0 
                                   
                                 
                               
                             
                             , 
                             
                               
                                 ω 
                                 1 
                               
                               = 
                               
                                 { 
                                 
                                   
                                     
                                       
                                         
                                           2 
                                           N 
                                         
                                       
                                     
                                     
                                       
                                         j 
                                         = 
                                         0 
                                       
                                     
                                   
                                   
                                     
                                       1 
                                     
                                     
                                       
                                         j 
                                         ≠ 
                                         0 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 DCT-8 
                 
                   
                     
                       
                         
                           
                             T 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               4 
                               
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                                 + 
                                 1 
                               
                             
                           
                           · 
                           
                             cos 
                             ( 
                             
                               
                                 π 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               
                                 
                                   4 
                                   ⁢ 
                                   N 
                                 
                                 + 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 DST-1 
                 
                   
                     
                       
                         
                           
                             T 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               2 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                           · 
                           
                             sin 
                             ( 
                             
                               
                                 π 
                                 · 
                                 
                                   ( 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               
                                 N 
                                 + 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 DST-7 
                 
                   
                     
                       
                         
                           
                             T 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               4 
                               
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                                 + 
                                 1 
                               
                             
                           
                           · 
                           
                             sin 
                             ( 
                             
                               
                                 π 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                                 + 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     In order to keep the orthogonality of the transform matrix, the transform matrices may be quantized more accurately than the transform matrices in HEVC, with 10-bit representation instead of 8-bit in HEVC. To keep the intermediate values of the transformed coefficients within the range of 16-bit, after horizontal and after vertical transform, all the coefficients may be right shifted by 2 more bits compared to the right shift used in the current HEVC transforms. The AMT may apply to coding units (CUs) with both width and height smaller than or equal to 64, and whether AMT applies or not may be controlled by a CU level flag. When the CU level flag is equal to 0, DCT-2 may be applied in the CU to encode the residue. For luma coding block within an AMT-enabled CU, two additional flags may be signalled to identify the horizontal and vertical transform to be used. 
     For intra residue coding, due to the different residual statistics of different intra prediction modes, a mode-dependent transform candidate selection process may be used. Three transform subsets may be defined as shown in Table 2 below, and the transform subset may be selected based on the intra prediction mode, as specified in Table 3 below. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Three pre-defined transform candidate sets 
               
            
           
           
               
               
            
               
                 Transform Set 
                 Transform Candidates 
               
               
                   
               
               
                 0 
                 DST-7 DCT-8 
               
               
                 1 
                 DST-7, DST-1 
               
               
                 2 
                 DST-7, DCT-5 
               
               
                   
               
            
           
         
       
     
     With the subset concept, a transform subset may be first identified based on Table 3 using the intra prediction mode of a CU with a CU-level AMT flag that is equal to 1. After that, for each of the horizontal transform and the vertical transform, one of the two transform candidates in the identified transform subset, according to Table 2, may be selected based on being explicitly signalled with flags. 
     
       
         
           
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Selected (H)orizontal and (V)ertical transform sets for each intra prediction mode 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                   
                 Intra Mode 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
               
               
                   
               
               
                 V 
                 2 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
               
               
                 H 
                 2 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 2 
                 2 
                 2 
                 2 
               
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Intra Mode 
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                   
               
               
                 V 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
               
               
                 H 
                 2 
                 2 
                 2 
                 2 
                 2 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Intra Mode 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
               
               
                   
               
               
                 V 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 2 
                 2 
                 2 
                 2 
                 2 
                 2 
                 2 
               
               
                 H 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Intra Mode 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
               
               
                   
               
               
                 V 
                 2 
                 2 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
               
               
                 H 
                 0 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
               
               
                   
               
            
           
         
       
     
     However, for inter prediction residual, only one transform set, which consists of DST-7 and DCT-8, may be used for all inter modes and for both horizontal and vertical transforms. Among the four additional transform types, i.e., DST-7, DCT-8, DST-1 and DCT-5, the most efficiently used transform types are DST-7 and DCT-8. It is noted that DCT-8 is essentially flipping DST-7 basis left and right with sign changes, so DCT-8 and DST-7 may basically share the same transform bases. 
     The transform core, which is a matrix composed by the basis vectors, of DST-7 can be also be represented as shown below: 
     4-Point DST-7: 
     a, b, c, d 
     c, c, 0, −c 
     d, −a, −c, b, 
     b, −d, c, −a 
     8-Point DST-7: 
     a, b, c, d, e, f, g, h 
     c, f, h, e, b, −a, −d, −g 
     e, g, b, c, −h, −d, a, f 
     g, c, −d, −f, a, h, b, −e 
     h, −a, −g, b, f, −c, e, d 
     d, −h, e, −a, −c, g, −f, b 
     b, −d, f, −h, g, −e, c, −a 
     16-Point DST-7: 
     a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p 
     c, f, i, l, o, o, l, i, f, c, 0, −c, −f, −i, −l, −o 
     e, j, o, m, h, c, −b, −g, −l, −p, −k, −f, −a, d, i, n 
     g, n, l, e, −b, −i, −p, −j, −c, d, k, o, h, a, −f, −m 
     i, o, f, −c, −l, −l, −c, f, o, i, 0, −i, −o, −f, c, l 
     k, k, 0, −k, −k, 0, k, k, 0, −k, −k, 0, k, k, 0, −k 
     m, g, −f, −n, −a, l, h, −e, −o, −b, k, i, −d, −p, −c, j 
     o, c, −l, −f, i, i, −f, −l, c, o, 0, −o, −c, l, f, −i 
     p, −a, −o, b, n, −c, −m, d, l, −e, −k, f, j, −g, −i, h 
     n, −e, −i, j, d, −o, a, m, −f, −h, k, c, −p, b, l, −g 
     l, −i, −c, o, −f, −f, o, −c, −i, l, 0, −l, i, c, −o, f 
     j, −m, c, g, −p, f, d, −n, i, a, −k, l, −b, −h, o, −e 
     h, −p, i, −a, −g, o, −j, b, f, −n, k, −c, −e, m, −l, d 
     f, −l, o, −i, c, c, −i, o, −l, f, 0, −f, l, −o, i, −c 
     d, −h, l, −p, m, −i, e, −a, −c, g, −k, o, −n, j, −f, b 
     b, −d, f, −h, j, −l, n, −p, o, −m, k, −i, g, −e, c, −a 
     32-Point DST-7: 
     a, b, c, d, e, f, g, h, i, j, k l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F 
     c, f, i, l, o, r, u, x, A, D, F, C, z, w, t, q, n, k, h, e, b, −a, −d, −g, −j, −m, −p, −s, −v, −y, −B, −E 
     e, j, o, t, y, D, D, y, t, o, j, e, 0, −e, −j, −o, −t, −y, −D, −D, −y, −t, −o, −j, −e, 0, e, j, o, t, y, D 
     g, n, u, B, D, w, p, i, b, −e, −l, −s, −z, −F, −y, −r, −k, −d, c, j, q, x, E, A, t, m, f, −a, −h, −c, −v, −C 
     i, r, A, C, t, k, b, −g, −p, −y, −E, −v, −m, −d, e, n, w, F, x, o, f, −c, −l, −u, −D, −z, −q, −h, a, j, s, B 
     k, v, F, u, j, −a, −l, −w, −E, −t, −i, b, m, x, D, s, h, −c, −n, −y, −C, −r, −g, d, o, z, B, q, f, −e, −p, −A 
     m, z, z, m, 0, −m, −z, −z, −m, 0, −m, −z, −m, 0, m, z, z, m, 0, −m, −z, −z, −m, e, m, z, z, m, 0, −m, −z, −z, −m, 0, m z 
     o, D, t, e, −j, −y, −y, −j, e, t, D, o, 0, −o, −D, −t, −e, j, y, y, j, −e, −t, −D, −o, 0, o, D, t, e, −j, −y, 
     q, E, n, −o, −t, −N, −k, f, w, y, h, −i, −z, −v, −e, l, C, s, b, −o, −F, −p, a, r, D, m, −d, −u, −A, −j, g, x 
     s, A, h, −k, −D, −p, o, v, x, e, −n, −F, −m, f, y, n, b, −q, −C, −j, i, B, r, −a, −t, −z, −g, l, E, o, −d, −w 
     u, w, b, −s, −y, −d, q, A, f, −o, −C, −h, m, E, j, −k, −F, −l, i, D, n, −g, −B, −p, s, z, r, −c, −x, −t, a, v 
     w, s, −d, −A, −o, h, E, k, −l, −D, −g, p, z, c, −t, −v, a, x, r, −e, −B, −n, i, F, j, −m, −C, −f, q, y, b, −u 
     y, o, −j, −D, −e, t, t, −e, D, −j, c, y, 0, −y, −o, j, D, e, t, −t, o, D, j, −o, −y, 0, y, o, −j, −D, −e, t 
     A, k, −p, −v, e, F, f, −u, −q, j, D, a, −z, −l, o, w, −d, −E, −g, t, r, −i, −C, −b, y, m, −D, −x, c, D, h, −c 
     C, g, −v, −n, o, u, −h, −B, e, D, f, −w, −m, p, t, −i, −A, b, E, e, −x, −l, q, s, −j, −z, c, F, d, −y, −k, r 
     E, o, −B, −f, y, i, −v, −l, s, o, −p, −r, m, u, −j, −x, g, A, −d, −D, e, F, b, −C, −e, z, h, −w, −k, t, n, −q 
     F, −a, −E, b, D, −c, −C, d, B, −e, −A, f, z, −g, −y, h, x, −i, −w, j, v, −k, −u, l, t, −m, −s, n, r, −o, −q, p 
     D, −e, −y, j, t, −o, −o, t, j, −y, −e, D, o, −D, e, y, −j, −y, o, o, −t, −j, y, e, −D, 0, D, −e, −y, j, t, −o 
     B, −i, −s, r, j, −A, −a, C, −h, −t, q, k, −z, −b, D, −g, −u, p, l, −y, −c, E, −f, −v, o, m, −x, −d, F, −e, −w, n 
     z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m 
     x, −q, −g, E, −j, −n, A, −c, −u, t, d, −B, m, k, −D, f, r, −w, −a, y, −p, −h, F, −i, −o, z, −b, −v, e, e, −C, l 
     v, −u, −e, w, −t, −b, x, −n, −c, y, −r, −d, z, −q, −e, A, −p, −f, B, −o, −g, C, −n, −h, D, −m, −i, E, −l, −j, F, −k 
     t, −y, e, c, −D, j, j, −D, o, e, −y, t, 0, −t, y, −e, −o, D, −j, −j, D, −o, −e, y, −t, 0, t, −y, e, c, −D, j 
     r, −C, k, g, −y, v, −d, −n, F, −o, −c, u, −z, h, j, −B, s, −a, −q, D, −l, −f, x, −w, e, m, −E, p, b, −t, A, −i 
     p, −F, q, −a, −o, E, −r, b, n, −D, s, −c, −m, C, −t, d, l, −B, u, −e, −k, A, −v, f, j, −z, w, −g, −i, y, −x, h 
     n, −B, w, −i, −o, s, −F, r, −d, −j, −x, −A, m, a, −o, C, −v, h, f, −t, E, −q, c, k, −y, z, −l, −b, p, −D, u, −g 
     l, −x, C, −q, o, g, −c, E, −v, j, b, −n, z, −A, o, −c, −i, u, −F, t, −h, −d, p, −B, y, −m, a, k, −w, D, −r, f 
     j, −t, D, −y, o, −e, −e, o, −y, D, −t, j, 0, −j, t, −D, y, −o, e, e, −o, y, −D, t, −j, 0, j, −t, D, −y, o, −e 
     h, −p, x, −F, y, −q, i, −a, −g, o, −w, E, −z, r, −j, b, f, −n, v, −D, A, −s, k, −c, −e, m, −u, C, −B, t, −l, d 
     f, −l, r, −x, D, −C, w, −q, k, −e, −a, g, −m, s, −y, E, −B, v, −p, j, −d, −b, h, −n, t, −x, F, −A, u, −c, i, −c 
     d, −h, l, −p, t, −x, B, −F, C, −y, u, −q, m, −i, e, −a, −c, q, −k, o, −o, w, −A, E, −D, z, −v, r, −n, j, −f, b 
     b, −d, f, −h, j, −l, n, −p, r, −t, v, −x, z, −B, D, −F, E, −C, A, −y, w, −u, s, −q, o, −m, k, −i, g, −e, c, −a 
     64-Point DST-7: 
     {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl,} 
     {ab, ae, ah, ak, an, aq, at, aw, az, bc, bf, bi, bl, bo, br, bu, bx, ca, cd, cg, cj, cl, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh, be, bb, ay, av, as, ap, am, aj, ag, ad, aa, −ac, −af, −ai, −al, −ao, −ar, −au, −ax, −ba, −bd, −bg, −bj, −bm, −bp, −bs, −bv, −by, −cb, −ce, −ch, −ck,} 
     {ac, ah, am, ar, aw, bb, bg, bl, bq, by, ca, cf, ck, ci, cd, by, bt, bo, bj, be, az, au, ap, ak, af, aa, −ae, −aj, −ao, −at, −ay, −bd, −bi, −bn, −bs, −bx, −cc, −ch, −cl, −cg, −cb, −bw, −br, −bm, −bh, −bc, −ax, −as, −an, −ai, −ad, ab, ag, al, aq, av, ba, bf, bk, bp, bu, bz, ce, cj,} 
     {ad, ak, ar, ay, bf, bm, bt, ca, ch, cj, cc, by, bo, bh, ba, at, am, af, −ab, −ai, −ap, −aw, −bd, −bk, −br, −by, −cf, −cl, −ce, −bx, −bq, −bj, −bc, −ay, −ao, −ah, −aa, ag, an, au, bb, bi, bp, bw, cd, ck, cg, bz, bs, bl, be, ax, aq, aj, ac, −ae, −al, −as, −az, −bg, −bn, −bu, −cb, −ci,} 
     {ae, an, aw, bf, bo, bx, cg, ci, bz, bq, bh, ay, ap, ag, −ac, −al, −au, −bd, −bm, −by, −ce, −ck, −cb, −bs, −bj, −ba, −ar, −ai, aa, aj, as, bb, bk, bt, cc, cl, cd, bu, bl, bc, at, ak, ab, −ah, −aq, −az, −bi, −br, −ca, −cj, −cf, −bw, −bn, −be, −ay, −am, −ad, af, ao, ax, bg, bp, by, ch,} 
     {af, aq, bb, bm, bx, ci, ce, bt, bi, ax, am, ab, −aj, −au, −bf, −bq, −cb, −cl, −ca, −bp, −be, −at, −ai, ac, an, ay, bj, bu, cf, ch, bw, bl, ba, ap, ae, −ag, −ar, −bc, −bn, −by, −cj, −cd, −bs, −bh, −aw, −al, −aa, ak, ay, bg, br, cc, ck, bz, bo, bd, as, ah, −ad, −ao, −az, −bk, −by, −cg,} 
     {ag, at, bg, bt, cg, ce, br, be, ar, ae, −ai, −ay, −bi, −by, −ci, −cc, −bp, −bc, −ap, −ac, ak, ax, bk, bx, ck, ca, bn, ba, an, aa, −am, −az, −bm, −bz, −cl, −by, −bl, −ay, −al, ab, ao, bb, bo, cb, cj, bw, bj, aw, aj, −ad, −aq, −bd, −bq, −cd, −ch, −bu, −bh, −au, −ah, af, as, bf, bs, cf,} 
     {ah, aw, bl, ca, ci, bt, be, ap, aa, −ao, −bd, −bs, −ch, −cb, −bm, −ax, −ai, ag, ay, bk, bz, cj, bu, bf, aq, ab, −an, −bc, −br, −cg, −cc, −bn, −ay, −aj, af, au, bj, by, ck, by, bg, ar, ac, −am, −bb, −bq, −cf, −cd, −bo, −az, −ak, ae, at, bi, bx, cl, bw, bh, as, ad, −al, −ba, −bp, −ce,} 
     {ai, az, bq, ch, bz, bi, ar, aa, −aq, −bh, −by, −ci, −br, −ba, −aj, ah, ay, bp, cg, ca, bj, as, ab, −ap, −bg, −bx, −cj, −bs, −bb, −ak, ag, ax, bo, cf, cb, bk, at, ac, −ao, −bf, −bw, −ck, −bt, −bc, −al, af, aw, bn, ce, cc, bl, au, ad, −an, −be, −by, −cl, −bu, −bd, −am, ae, av, bm, cd,} 
     {aj, bc, by, cj, bq, ax, ae, −ao, −bh, −ca, −ce, −bl, −as, aa, at, bm, cf, bz, bg, an, −af, −ay, −br, −ck, −bu, −bb, −ai, ak, bd, bw, ci, bp, aw, ad, −ap, −bi, −cb, −cd, −bk, −ar, ab, au, bn, cg, by, bf, am, −ag, −az, −bs, −cl, −bt, −ba, −ah, al, be, bx, ch, bo, av, ac, −aq, −bj, −cc,} 
     {ak, bf, ca, cc, bh, am, −ai, −bd, −by, −ce, −bj, −ao, ag, bb, bw, cg, bl, aq, −ae, −az, −bu, −ci, −bn, −as, ac, ax, bs, ck, bp, au, −aa, −av, −bq, −cl, −br, −aw, −ab, at, bo, cj, bt, ay, ad, −ar, −bm, −ch, −by, −ba, −af, ap, bk, cf, bx, bc, ah, −an, −bi, −cd, −bz, −be, −aj, al, bg, cb,} 
     {al, bi, cf, by, ay, ab, −ay, −bs, −ci, −bl, −ao, ai, bf, cc, by, bb, ae, −as, −bp, −cl, −bo, −ar, af, bc, bz, cb, be, ah, −ap, −bm, −cj, −br, −au, ac, az, bw, ce, bh, ak, −am, −bj, −cg, −bu, −ax, −aa, aw, bt, ch, bk, an, −aj, −bg, −cd, −bx, −ba, −ad, at, bq, ck, bn, aq, −ag, −bd, −ca,} 
     {am, bl, ck, bo, ap, −aj, −bi, −ch, −br, −as, ag, bf, ce, bu, ay, −ad, −bc, −cb, −bx, −ay, aa, az, by, ca, bb, ac, −aw, −by, −cd, −be, −af, at, bs, cg, bh, ai, −aq, −bp, −cj, −bk, −al, an, bm, cl, bn, ao, −ak, −bj, −ci, −bq, −ar, ah, bg, cf, bt, au, −ae, −bd, −cc, −bw, −ax, ab, ba, bz,} 
     {an, bo, ci, bh, ag, −au, −by, −cb, −ba, aa, bb, cc, bu, at, −ah, −bi, −cj, −bn, −am, ao, bp, ch, bg, af, −ay, −bw, −ca, −az, ab, bc, cd, bt, as, −ai, −bj, −ck, −bm, −al, ap, bq, cg, bf, ae, −aw, −bx, −bz, −ay, ac, bd, ce, bs, ar, −aj, −bk, −cl, −bl, −ak, aq, br, cf, be, ad, −ax, −by,} 
     {ao, br, cd, ba, −ac, −bf, −ci, −bm, −aj, at, bw, by, ay, −ah, −bk, −ck, −bh, −ae, ay, cb, bt, aq, −am, −bp, −cf, −bc, aa, bd, cg, bo, al, −ar, −bu, −ca, −ax, af, bi, cl, bj, ag, −aw, −bz, −by, −as, ak, bn, ch, be, ab, −bb, −ce, −bq, −an, ap, bs, cc, az, −ad, −bg, −cj, −bl, −ai, au, bx,} 
     {ap, bu, by, at, −al, −bq, −cc, −ax, ah, bm, cg, bb, −ad, −bi, −ck, −bf, −aa, be, cj, bj, ae, −ba, −cf, −bn, −ai, aw, cb, br, am, −as, −bx, −by, −aq, ao, bt, bz, au, −ak, −bp, −cd, −ay, ag, bl, ch, bc, −ac, −bh, −cl, −bg, −ab, bd, ci, bk, af, −az, −ce, −bo, −aj, av, ca, bs, an, −ar, −bw,} 
     {aq, bx, bt, am, −au, −cb, −bp, −ai, ay, cf, bl, ae, −bc, −cj, −bh, −aa, bg, ck, bd, −ad, −bk, −cg, −az, ah, bo, cc, av, −al, −bs, −by, −ar, ap, bw, bu, an, −at, −ca, −bq, −aj, ax, ce, bm, af, −bb, −ci, −bi, −ab, bf, cl, be, −ac, −bj, −ch, −ba, ag, bn, cd, aw, −ak, −br, −bz, −as, ao, bv,} 
     {ar, ca, bo, af, −bd, −cl, −bc, ag, bp, bz, aq, −as, −cb, −bn, −ae, be, ck, bb, −ah, −bq, −by, −ap, at, cc, bm, ad, −bf, −cj, −ba, ai, br, bx, ao, −au, −cd, −bl, −ac, bg, ci, az, −aj, −bs, −bw, −an, av, ce, bk, ab, −bh, −ch, −ay, ak, bt, by, am, −aw, −cf, −bj, −aa, bi, cg, ax, −al, −bu,} 
     {as, cd, bj, −ab, −bm, −ca, −ap, av, cg, bg, −ae, −bp, −bx, −am, ay, cj, bd, −ah, −bs, −bu, −aj, bb, cl, ba, −ak, −by, −br, −ag, be, ci, ax, −an, −by, −bo, −ad, bh, cf, au, −aq, −cb, −bl, −aa, bk, cc, ar, −at, −ce, −bi, ac, bn, bz, ao, −aw, −ch, −bf, af, bq, bw, al, −az, −ck, −bc, ai, bt,} 
     {at, cg, be, −ai, −by, −bp, −ac, bk, ca, an, −az, −cl, −ay, ao, cb, bj, −ad, −bq, −bu, −ah, bf, cf, as, −au, −ch, −bd, aj, bw, bo, ab, −bl, −bz, −am, ba, ck, ax, −ap, −cc, −bi, ae, br, bt, ag, −bg, −ce, −ar, ay, ci, bc, −ak, −bx, −bn, −aa, bm, by, al, −bb, −cj, −aw, aq, cd, bh, −af, −bs,} 
     {au, cj, az, −ap, −ce, −be, ak, bz, bj, −af, −bu, −bo, aa, bp, bt, ae, −bk, −by, −aj, bf, cd, ao, −ba, −ci, −at, ay, ck, ay, −aq, −cf, −bd, al, ca, bi, −ag, −by, −bn, ab, bq, bs, ad, −bl, −bx, −ai, bg, cc, an, −bb, −ch, −as, aw, cl, ax, −ar, −cg, −bc, am, cb, bh, −ah, −bw, −bm, ac, br,} 
     {av, cl, au, −aw, −ck, −at, ax, cj, as, −ay, −ci, −ar, az, ch, aq, −ba, −cg, −ap, bb, cf, ao, −bc, −ce, −an, bd, cd, am, −be, −cc, −al, bf, cb, ak, −bg, −ca, −aj, bh, bz, ai, −bi, −by, −ah, bj, bx, ag, −bk, −bw, −af, bl, by, ae, −bm, −bu, −ad, bn, bt, ac, −bo, −bs, −ab, bp, br, aa, −bq,} 
     {aw, ci, ap, −bd, −cb, −ai, bk, bu, ab, −br, −bn, af, by, bg, −am, −cf, −az, at, cl, as, −ba, −ce, −al, bh, bx, ae, −bo, −bq, ac, by, bj, −aj, −cc, −bc, aq, cj, av, −ax, −ch, −ao, be, ca, ah, −bl, −bt, −aa, bs, bm, −ag, −bz, −bf, an, cg, ay, −au, −ck, −ar, bb, cd, ak, −bi, −bw, −ad, bp,} 
     {ax, cf, ak, −bk, −bs, ac, bx, bf, −ap, −ck, −as, bc, ca, af, −bp, −bn, ah, cc, ba, −au, −ci, −an, bh, by, aa, −bu, −bi, am, ch, av, −az, −cd, −ai, bm, bq, −ae, −bz, −bd, ar, cl, aq, −be, −by, −ad, br, bl, −aj, −ce, −ay, aw, cg, al, −bj, −bt, ab, bw, bg, −ao, −cj, −at, bb, cb, ag, −bo,} 
     {ay, cc, af, −br, −bj, an, ck, aq, −bg, −bu, ac, bz, bb, −av, −cf, −ai, bo, bm, −ak, −ch, −at, bd, bx, aa, −bw, −be, as, ci, al, −bl, −bp, ah, ce, aw, −ba, −ca, −ad, bt, bh, −ap, −cl, −ao, bi, bs, −ae, −cb, −az, ax, cd, ag, −bq, −bk, am, cj, ar, −bf, −by, ab, by, bc, −au, −cg, −aj, bn,} 
     {az, bz, aa, −by, −ba, ay, ca, ab, −bx, −bb, ax, cb, ac, −bw, −bc, aw, cc, ad, −by, −bd, ay, cd, ae, −bu, −be, au, ce, af, −bt, −bf, at, cf, ag, −bs, −bg, as, cg, ah, −br, −bh, ar, ch, ai, −bq, −bi, aq, ci, aj, −bp, −bj, ap, cj, ak, −bo, −bk, ao, ck, al, −bn, −bl, an, cl, am, −bm,} 
     {ba, bw, −ae, −cf, −ar, bj, bn, −an, −cj, −ai, bs, be, −aw, −ca, aa, cb, ay, −bf, −br, aj, ck, am, −bo, −bi, as, ce, ad, −bx, −az, bb, by, −af, −cg, −aq, bk, bm, −ao, −ci, −ah, bt, bd, −ax, −bz, ab, cc, au, −bg, −bq, ak, cl, al, −bp, −bh, at, cd, ac, −by, −ay, bc, bu, −ag, −ch, −ap, bl,} 
     {bb, bt, −aj, −cl, −ai, bu, ba, −bc, −bs, ak, ck, ah, −by, −az, bd, br, −al, −cj, −ag, bw, ay, −be, −bq, am, ci, af, −bx, −ax, bf, bp, −an, −ch, −ae, by, aw, −bg, −bo, ao, cg, ad, −bz, −ay, bh, bn, −ap, −cf, −ac, ca, au, −bi, −bm, aq, ce, ab, −cb, −at, bj, bl, −ar, −cd, −aa, cc, as, −bk,} 
     {bc, bq, −ao, −ce, aa, cf, an, −br, −bb, bd, bp, −ap, −cd, ab, cg, am, −bs, −ba, be, bo, −aq, −cc, ac, ch, al, −bt, −az, bf, bn, −ar, −cb, ad, ci, ak, −bu, −ay, bg, bm, −as, −ca, ae, cj, aj, −by, −ax, bh, bl, −at, −bz, af, ck, ai, −bw, −aw, bi, bk, −au, −by, ag, cl, ah, −bx, −ay, bj,} 
     {bd, bn, −at, −bx, aj, ch, aa, −cg, −ak, bw, au, −bm, −be, bc, bo, −as, −by, ai, ci, ab, −cf, −al, by, av, −bl, −bf, bb, bp, −ar, −bz, ah, cj, ac, −ce, −am, bu, aw, −bk, −bg, ba, bq, −aq, −ca, ag, ck, ad, −cd, −an, bt, ax, −bj, −bh, az, br, −ap, −cb, af, cl, ae, −cc, −ao, bs, ay, −bi,} 
     {be, bk, −ay, −bq, as, bw, −am, −cc, ag, ci, −aa, −cj, −af, cd, al, −bx, −ar, br, ax, −bl, −bd, bf, bj, −az, −bp, at, by, −an, −cb, ah, ch, −ab, −ck, −ae, ce, ak, −by, −aq, bs, aw, −bm, −bc, bg, bi, −ba, −bo, au, bu, −ao, −ca, ai, cg, −ac, −cl, −ad, cf, aj, −bz, −ap, bt, av, −bn, −bb, bh,} 
     {bf, bh, −bd, −bj, bb, bl, −az, −bn, ax, bp, −av, −br, at, bt, −ar, −by, ap, bx, −an, −bz, al, cb, −aj, −cd, ah, cf, −af, −ch, ad, cj, −ab, −cl, −aa, ck, ac, −ci, −ae, cg, ag, −ce, −ai, cc, ak, −ca, −am, by, ao, −bw, −aq, bu, as, −bs, −au, bq, aw, −bo, −ay, bm, ba, −bk, −bc, bi, be, −bg,} 
     {bg, be, −bi, −bc, bk, ba, −bm, −ay, bo, aw, −bq, −au, bs, as, −bu, −aq, bw, ao, −by, −am, ca, ak, −cc, −ai, ce, ag, −cg, −ae, ci, ac, −ck, −aa, cl, −ab, −cj, ad, ch, −af, −cf, ah, cd, −aj, −cb, al, bz, −an, −bx, ap, by, −ar, −bt, at, br, −ay, −bp, ax, bn, −az, −bl, bb, bj, −bd, −bh, bf,} 
     {bh, bb, −bn, −ay, bt, ap, −bz, −aj, cf, ad, −cl, ac, cg, −ai, −ca, ao, bu, −au, −bo, ba, bi, −bg, −bc, bm, aw, −bs, −aq, by, ak, −ce, −ae, ck, −ab, −ch, ah, cb, −an, −by, at, bp, −az, −bj, bf, bd, −bl, −ax, br, ar, −bx, −al, cd, af, −cj, aa, ci, −ag, −cc, am, bw, −as, −bq, ay, bk, −be,} 
     {bi, ay, −bs, −ao, cc, ae, −cl, af, cb, −ap, −br, az, bh, −bj, −ax, bt, an, −cd, −ad, ck, −ag, −ca, aq, bq, −ba, −bg, bk, aw, −bu, −am, ce, ac, −cj, ah, bz, −ar, −bp, bb, bf, −bl, −ay, by, al, −cf, −ab, ci, −ai, −by, as, bo, −bc, −be, bm, au, −bw, −ak, cg, aa, −ch, aj, bx, −at, −bn, bd,} 
     {bj, ay, −bx, −ah, cl, −ag, −by, au, bk, −bi, −aw, bw, ai, −ck, af, bz, −at, −bl, bh, ax, −by, −aj, cj, −ae, −ca, as, bm, −bg, −ay, bu, ak, −ci, ad, cb, −ar, −bn, bf, az, −bt, −al, ch, −ac, −cc, aq, bo, −be, −ba, bs, am, −cg, ab, cd, −ap, −bp, bd, bb, −br, −an, cf, −aa, −ce, ao, bq, −bc,} 
     {bk, as, −cc, −aa, cd, −ar, −bl, bj, at, −cb, −ab, ce, −aq, −bm, bi, au, −ca, −ac, cf, −ap, −bn, bh, av, −bz, −ad, cg, −ao, −bo, bg, aw, −by, −ae, ch, −an, −bp, bf, ax, −bx, −af, ci, −am, −bq, be, ay, −bw, −ag, cj, −al, −br, bd, az, −by, −ah, ck, −ak, −bs, bc, ba, −bu, −ai, cl, −aj, −bt, bb,} 
     {bl, ap, −ch, ag, bu, −bc, −ay, by, ac, −cd, at, bh, −bp, −al, cl, −ak, −bq, bg, au, −cc, ab, bz, −ax, −bd, bt, ah, −ci, ao, bm, −bk, −aq, cg, −af, −by, bb, az, −bx, −ad, ce, −as, −bi, bo, am, −ck, aj, br, −bf, −av, cb, −aa, −ca, aw, be, −bs, −ai, cj, −an, −bn, bj, ar, −cf, ae, bw, −ba,} 
     {bm, am, −cl, an, bl, −bn, −al, ck, −ao, −bk, bo, ak, −cj, ap, bj, −bp, −aj, ci, −aq, −bi, bq, ai, −ch, ar, bh, −br, −ah, cg, −as, −bg, bs, ag, −cf, at, bf, −bt, −af, ce, −au, −be, bu, ae, −cd, av, bd, −by, −ad, cc, −aw, −bc, bw, ac, −cb, ax, bb, −bx, −ab, ca, −ay, −ba, by, aa, −bz, az,} 
     {bn, aj, −cg, au, bc, −by, ab, by, −bf, −ar, cj, −am, −bk, bq, ag, −cd, ax, az, −cb, ae, bs, −bi, −ao, cl, −ap, −bh, bt, ad, −ca, ba, aw, −ce, ah, bp, −bl, −al, ci, −as, −be, bw, aa, −bx, bd, at, −ch, ak, bm, −bo, −ai, cf, −ay, −bb, bz, −ac, −bu, bg, aq, −ck, an, bj, −br, −af, cc, −ay,} 
     {bo, ag, −cb, bb, at, −cj, ao, bg, −bw, ab, bt, −bj, −al, cg, −aw, −ay, ce, −aj, −bl, br, ad, −by, be, aq, −cl, ar, bd, −bz, ae, bq, −bm, −ai, cd, −az, −ay, ch, −am, −bi, bu, aa, −by, bh, an, −ci, au, ba, −cc, ah, bn, −bp, −af, ca, −bc, −as, ck, −ap, −bf, bx, −ac, −bs, bk, ak, −cf, ax,} 
     {bp, ad, −bw, bi, ak, −cd, bb, ar, −ck, au, ay, −cg, an, bf, −bz, ag, bm, −bs, −aa, bt, −bl, −ah, ca, −be, −ao, ch, −ax, −ay, cj, −aq, −bc, cc, −aj, −bj, by, −ac, −bq, bo, ae, −bx, bh, al, −ce, ba, as, −cl, at, az, −cf, am, bg, −by, af, bn, −br, −ab, bu, −bk, −ai, cb, −bd, −ap, ci, −aw,} 
     {bq, aa, −br, bp, ab, −bs, bo, ac, −bt, bn, ad, −bu, bm, ae, −by, bl, af, −bw, bk, ag, −bx, bj, ah, −by, bi, ai, −bz, bh, aj, −ca, bg, ak, −cb, bf, al, −cc, be, am, −cd, bd, an, −ce, bc, ao, −cf, bb, ap, −cg, ba, aq, −ch, az, ar, −ci, ay, as, −cj, ax, at, −ck, aw, au, −cl, av,} 
     {br, −ac, −bm, bw, −ah, −bh, cb, −am, −bc, cg, −ar, −ax, cl, −aw, −as, ch, −bb, −an, cc, −bg, −ai, bx, −bl, −ad, bs, −bq, ab, bn, −by, ag, bi, −ca, al, bd, −cf, aq, ay, −ck, av, at, −ci, ba, ao, −cd, bf, aj, −by, bk, ae, −bt, bp, −aa, −bo, bu, −af, −bj, bz, −ak, −be, ce, −ap, −az, cj, −au,} 
     {bs, −af, −bh, cd, −aq, −aw, cj, −bb, −al, by, −bm, −aa, bn, −bx, ak, bc, −ci, av, ar, −ce, bg, ag, −bt, br, −ae, −bi, cc, −ap, −ax, ck, −ba, −am, bz, −bl, −ab, bo, −bw, aj, bd, −ch, au, as, −cf, bf, ah, −bu, bq, −ad, −bj, cb, −ao, −ay, cl, −az, −an, ca, −bk, −ac, bp, −by, ai, be, −cg, at,} 
     {bt, −ai, −bc, ck, −az, −al, bw, −bq, af, bf, −ch, aw, ao, −bz, bn, −ac, −bi, ce, −at, −ar, cc, −bk, −aa, bl, −cb, aq, au, −cf, bh, ad, −bo, by, −an, −ax, ci, −be, −ag, br, −by, ak, ba, −cl, bb, aj, −bu, bs, −ah, −bd, cj, −ay, −am, bx, −bp, ae, bg, −cg, ay, ap, −ca, bm, −ab, −bj, cd, −as,} 
     {bu, −al, −ax, cg, −bi, −aa, bj, −cf, aw, am, −by, bt, −ak, −ay, ch, −bh, −ab, bk, −ce, ay, an, −bw, bs, −aj, −az, ci, −bg, −ac, bl, −cd, au, ao, −bx, br, −ai, −ba, cj, −bf, −ad, bm, −cc, at, ap, −by, bq, −ah, −bb, ck, −be, −ae, bn, −cb, as, aq, −bz, bp, −ag, −bc, cl, −bd, −af, bo, −ca, ar,} 
     {by, −ao, −as, bz, −br, ak, aw, −cd, bn, −ag, −ba, ch, −bj, ac, be, −cl, bf, ab, −bi, ci, −bb, −af, bm, −ce, ax, aj, −bq, ca, −at, −an, bu, −bw, ap, ar, −by, bs, −al, −ay, cc, −bo, ah, az, −cg, bk, −ad, −bd, ck, −bg, −aa, bh, −cj, bc, ae, −bl, cf, −ay, −ai, bp, −cb, au, am, −bt, bx, −aq,} 
     {bw, −ar, −an, bs, −ca, ay, aj, −bo, ce, −az, −af, bk, −ci, bd, ab, −bg, cl, −bh, ac, bc, −ch, bl, −ag, −ay, cd, −bp, ak, au, −bz, bt, −ao, −aq, by, −bx, as, am, −br, cb, −aw, −ai, bn, −cf, ba, ae, −bj, cj, −be, −aa, bf, −ck, bi, −ad, −bb, cg, −bm, ah, ax, −cc, bq, −al, −at, by, −bu, ap,} 
     {bx, −au, −ai, bl, −cj, bg, −ad, −az, cc, −bs, ap, an, −bq, ce, −bb, −ab, be, −ch, bn, −ak, −as, by, −bz, aw, ag, −bj, cl, −bi, af, ax, −ca, bu, −ar, −al, bo, −cg, bd, −aa, −bc, cf, −bp, am, aq, −bt, cb, −ay, −ae, bh, −ck, bk, −ah, −ay, by, −bw, at, aj, −bm, ci, −bf, ac, ba, −cd, br, −ao,} 
     {by, −ax, −ad, be, −cf, br, −aq, −ak, bl, −cl, bk, −aj, −ar, bs, −ce, bd, −ac, −ay, bz, −bx, aw, ae, −bf, cg, −bq, ap, al, −bm, ck, −bj, ai, as, −bt, cd, −bc, ab, az, −ca, bw, −av, −af, bg, −ch, bp, −ao, −am, bn, −cj, bi, −ah, −at, bu, −cc, bb, −aa, −ba, cb, −by, au, ag, −bh, ci, −bo, an,} 
     {bz, −ba, ab, ax, −bw, cc, −bd, ae, au, −bt, cf, −bg, ah, ar, −bq, ci, −bj, ak, ao, −bn, cl, −bm, an, al, −bk, cj, −bp, aq, ai, −bh, cg, −bs, at, af, −be, cd, −by, aw, ac, −bb, ca, −by, az, −aa, −ay, bx, −cb, bc, −ad, −av, bu, −ce, bf, −ag, −as, br, −ch, bi, −aj, −ap, bo, −ck, bl, −am,} 
     {ca, −bd, ag, aq, −bn, ck, −bq, at, ad, −ba, bx, −cd, bg, −aj, −an, bk, −ch, bt, −aw, −aa, ax, −bu, cg, −bj, am, ak, −bh, ce, −bw, az, −ac, −au, br, −cj, bm, −ap, −ah, be, −cb, bz, −bc, af, ar, −bo, cl, −bp, as, ae, −bb, by, −cc, bf, −ai, −ao, bl, −ci, bs, −av, −ab, ay, −by, cf, −bi, al,} 
     {cb, −bg, al, aj, −be, bz, −cd, bi, −an, −ah, bc, −bx, cf, −bk, ap, af, −ba, by, −ch, bm, −ar, −ad, ay, −bt, cj, −bo, at, ab, −aw, br, −cl, bq, −ay, aa, au, −bp, ck, −bs, ax, −ac, −as, bn, −ci, bu, −az, ae, aq, −bl, cg, −bw, bb, −ag, −ao, bj, −ce, by, −bd, ai, am, −bh, cc, −ca, bf, −ak,} 
     {cc, −bj, aq, ac, −ay, bo, −ch, bx, −be, al, ah, −ba, bt, −cl, bs, −az, ag, am, −bf, by, −cg, bn, −au, ab, ar, −bk, cd, −cb, bi, −ap, −ad, aw, −bp, ci, −bw, bd, −ak, −ai, bb, −bu, ck, −br, ay, −af, −an, bg, −bz, cf, −bm, at, −aa, −as, bl, −ce, ca, −bh, ao, ae, −ax, bq, −cj, by, −bc, aj,} 
     {cd, −bm, ay, −ae, −am, bd, −bu, cl, −by, be, −an, −ad, au, −bl, cc, −ce, bn, −aw, af, al, −bc, bt, −ck, bw, −bf, ao, ac, −at, bk, −cb, cf, −bo, ax, −ag, −ak, bb, −bs, cj, −bx, bg, −ap, −ab, as, −bj, ca, −cg, bp, −ay, ah, aj, −ba, br, −ci, by, −bh, aq, aa, −ar, bi, −bz, ch, −bq, az, −ai,} 
     {ce, −bp, ba, −al, −ad, as, −bh, bw, −cl, bx, −bi, at, −ae, −ak, az, −bo, cd, −cf, bq, −bb, am, ac, −ar, bg, −by, ck, −by, bj, −au, af, aj, −ay, bn, −cc, cg, −br, bc, −an, −ab, aq, −bf, bu, −cj, bz, −bk, ay, −ag, −ai, ax, −bm, cb, −ch, bs, −bd, ao, aa, −ap, be, −bt, ci, −ca, bl, −aw, ah,} 
     {cf, −bs, bf, −as, af, ah, −au, bh, −bu, ch, −cd, bq, −bd, aq, −ad, −aj, aw, −bj, bw, −cj, cb, −bo, bb, −ao, ab, al, −ay, bl, −by, cl, −bz, bm, −az, am, aa, −an, ba, −bn, ca, −ck, bx, −bk, ax, −ak, −ac, ap, −bc, bp, −cc, ci, −by, bi, −av, ai, ae, −ar, be, −br, ce, −cg, bt, −bg, at, −ag,} 
     {cg, −by, bk, −az, ao, −ad, −ah, as, −bd, bo, −bz, ck, −cc, br, −bg, av, −ak, −aa, al, −aw, bh, −bs, cd, −cj, by, −bn, bc, −ar, ag, ae, −ap, ba, −bl, bw, −ch, cf, −bu, bj, −ay, an, −ac, −ai, at, −be, bp, −ca, cl, −cb, bq, −bf, au, −aj, −ab, am, −ax, bi, −bt, ce, −ci, bx, −bm, bb, −aq, af,} 
     {ch, −by, bp, −bg, ax, −ao, af, ad, −am, av, −be, bn, −bw, cf, −cj, ca, −br, bi, −az, aq, −ah, −ab, ak, −at, bc, −bl, bu, −cd, cl, −cc, bt, −bk, bb, −as, aj, −aa, −ai, ar, −ba, bj, −bs, cb, −ck, ce, −by, bm, −bd, au, −al, ac, ag, −ap, ay, −bh, bq, −bz, ci, −cg, bx, −bo, bf, −aw, an, −ae,} 
     {ci, −cb, bu, −bn, bg, −az, as, −al, ae, ac, −aj, aq, −ax, be, −bl, bs, −bz, cg, −ck, cd, −bw, bp, −bi, bb, −au, an, −ag, −aa, ah, −ao, ay, −bc, bj, −bq, bx, −ce, cl, −cf, by, −br, bk, −bd, aw, −ap, ai, −ab, −af, am, −at, ba, −bh, bo, −by, cc, −cj, ch, −ca, bt, −bm, bf, −ay, ar, −ak, ad,} 
     {cj, −ce, bz, −bu, bp, −bk, bf, −ba, ay, −aq, al, −ag, ab, ad, −ai, an, −as, ax, −bc, bh, −bm, br, −bw, cb, −cg, cl, −ch, cc, −bx, bs, −bn, bi, −bd, ay, −at, ao, −aj, ae, aa, −af, ak, −ap, au, −az, be, −bj, bo, −bt, by, −cd, ci, −ck, cf, −ca, by, −bq, bl, −bg, bb, −aw, ar, −am, ah, −ac,} 
     {ck, −ch, ce, −cb, by, −by, bs, −bp, bm, −bj, bg, −bd, ba, −ax, au, −ar, ao, −al, ai, −af, ac, aa, −ad, ag, −aj, am, −ap, as, −ay, ay, −bb, be, −bh, bk, −bn, bq, −bt, bw, −bz, cc, −cf, ci, −cl, cj, −cg, cd, −ca, bx, −bu, br, −bo, bl, −bi, bf, −bc, az, −aw, at, −aq, an, −ak, ah, −ae, ab,} 
     {cl, −ck, cj, −ci, ch, −cg, cf, −ce, cd, −cc, cb, −ca, bz, −by, bx, −bw, by, −bu, bt, −bs, br, −bq, bp, −bo, bn, −bm, bl, −bk, bj, −bi, bh, −bg, bf, −be, bd, −bc, bb, −ba, az, −ay, ax, −aw, ay, −au, at, −as, ar, −aq, ap, −ao, an, −am, al, −ak, aj, −ai, ah, −ag, af, −ae, ad, −ac, ab, −aa,} 
     The variables a, b c, . . . aa, ab, ac, . . . , cl can be derived based on the formulations of DST-7 shown in 
     Table, and their values can be different for different sizes of DST-7. For example, the value of “a” can be different for 4-point DST-7 and 8-point DST-7. 
     To avoid floating point operations, similar to the DCT-2 cores used in HEVC, the transform core of DST-7 may be scaled by a pre-defined factor, e.g., 64·√{square root over (N)} (corresponds to 8-bit transform core) or 256·√{square root over (N)} (corresponds to 10-bit transform core), and rounded to the nearest integer, or further tuned by an offset, e.g., +1/−1. 
     Different from DCT-2, of which fast methods have been extensively studied, the implementation of DST-7 is still much less efficient than DCT-2, e.g., matrix multiplication based in VTM 1 of Non-Patent Literature 2. 
     In Non-Patent Literature 3, a method is proposed to approximate different types of DCTs and DSTs in JEM7 by applying adjustment stages to a transform in the DCT-2 family, which includes DCT-2, DCT-3, DST-2 and DST-3, and the adjustment stage refers to a matrix multiplication using a sparse matrix which requires relatively less operation counts. In Non-Patent Literature 4, a method for implementing N-point DST-7 using 2N+1 point Discrete Fourier Transform (DFT) is proposed. 
     However, additional transform types in AMT increase the complexity and implementation cost, since additional transform cores need to be stored in on-chip memory for practical implementation, and transform cores need to be loaded when switching the transform type from one block to another. In HEVC, a smaller DCT-2 transform is a part of the larger DCT-2 transform, which supports reusing the same logic for implementing different sizes of transforms. However, with AMT, such a feature is missing, other than DCT-2, smaller transform cannot be re-used for larger transform, which means different logic may need to be designed for implementing different sizes of transform. 
     The DCT-2 in HEVC is implemented using partial butterfly, which supports identical outputs between matrix multiplication and partial butterfly, i.e., dual implementation. However, half-size matrix multiplication is still needed which is still a computation burden, for example, partial butterfly of 32-point HEVC DCT-2 requires 2×2, 4×4, 8×8 and 16×16 matrix multiplications. 
     It is observed that partial of the odd bases (odd basis refers to basis vector associated with an odd index, the first basis being associated with index 0) in DCT-2 is very similar to the scaled DCT-8 bases. In the following  FIG. 6A , the second, fourth, sixth, and eighth basis vectors of 64-point DCT-2 are shown, as indicated by solid curves. In addition, the first, second, third and fourth basis vector of 32-point DCT-8, which is scaled by a constant 
               1     2       ,         
are also shown in  FIG. 6A , as indicated by dotted curves. From  FIG. 6A  it can be seen that, the first half of the second, fourth, sixth, and eighth basis vectors of 64-point DCT-2 are very close to the first, second, third and fourth basis vectors of 32-point DCT-8 scaled by
 
     
       
         
           
             
               1 
               
                 2 
               
             
             . 
           
         
       
     
     Based on this observation, if we replace the first half of odd bases of 2N-point DCT-2 with N-point scaled DCT-8 bases, and fill the second half with the flipped DCT-8 bases plus inverse signs, then the resulting basis is very close to the original odd bases of 2N-point DCT-2, as shown in  FIG. 6B . 
     Therefore, we can derive a new orthonormal transform, namely a Compound Orthonormal Transform (COT), by embedding the N-point DCT-8 (or DST-7, since DST-7 and DCT-8 bases are symmetric to each other) into 2N-point DCT-2, and the resulting transform is still an orthonormal transform due to the symmetric/anti-symmetric characteristics of DCT-2, i.e., Feature #1 and Feature #2. 
     In this way, by embedding the N-point DCT-8/DST-7 into 2N-point DCT-2, the complexity of performing AMT is reduced in 3 aspects: (i) the logic of performing transform for DCT-2 and DCT-8/DST-7 can be shared, (ii) the on-chip memory cost for storing DCT-8 and DST-7 bases can be reduced since they are part of the DCT-2 bases and (iii) with the proposed COT, lossless fast DST-7 methods from the incorporated-by-reference &#39;076 application can be applied to further reduce the computation cost of DCT-2 since DST-7 is part of DCT-2. 
     The proposed methods may be used separately or combined in any order. Due to the aforementioned advantages of COT cores, it is proposed to use COT cores instead of traditional transform cores (such as DCT-2 and DST-7) in video and/or image compression, such as transform image or residual blocks. A 2N-point COT core may be generated by embedding N-point orthonormal transform (with transform core being denoted by T 0 , T 1 , T 2  . . . ) into another 2N-point recursive transform (with transform core being denoted by R). The resulting transform core (denoted by RT) is also an orthonormal matrix. This new resulting transform core RT (a COT core) can be used for encoding or decoding an image block or a residual block. A COT core has the following feature(s): 
     Feature #A, in one embodiment, the N-point orthonormal transform core, i.e., T 0 , T 1 , T 2  . . . , is replacing one half of the RT basis rows which are anti-symmetry, and the remaining half is derived to keep the anti-symmetry characteristics. Feature #B, in one embodiment, the N-point orthonormal transform core, i.e., T 0 , T 1 , T 2  . . . , is replacing one half of the RT basis rows which are symmetry, and the remaining half is derived to keep the symmetry characteristics. 
     Feature #C, in one embodiment, one N-point orthonormal transform is replacing one half of a selected set (denoted by set0) of RT basis rows which are symmetry, and another N-point orthonormal transform is replacing one half of a selected set (denoted by set1) of RT basis rows which are anti-symmetry, and the remaining half is derived to keep the associated symmetry and anti-symmetry characteristics. In this Feature #C example, as an example, set0 refers to the RT basis rows located within the last N rows of RT and set1 refers to the RT basis rows located within the first N rows of RT. 
     Feature #D, in one embodiment, RT is a transform from the DCT/DST families, examples are: DCT-2, DCT-3, DST-2 or DST-3. In this Feature #D example, RT may be further scaled by a constant and rounded to the nearest integer. Example values of the constant are: 64·√{square root over (N)}, 128·√{square root over (N)}, 256·√{square root over (N)}. In this Feature #D example, RT may be further scaled by a constant, rounded to the nearest integer and further adjusted by an offset. Example values of the offset include: +1/−1, +2/−2. 
     Feature #E is that, in one embodiment, T 0 , T 1 , T 2  . . . are transforms from the DCT/DST families, of which examples include: DST-7, DCT-8, DST-7 with inverse sign for selected rows/columns, DCT-8 with inverse sign for selected rows/columns, DST-4, or DCT-4. In one example of Feature #E, T 0 , T 1 , T 2  . . . may be further scaled by a constant and rounded to the nearest integer. Example values of the constant are: 64·√{square root over (N)}, 128·√{square root over (N)}, 256·√{square root over (N)}. In one example of Feature #E, T 0 , T 1 , T 2  . . . may be further scaled by a constant, rounded to the nearest integer and further adjusted by an offset. Example values of an offset include: +1/−1, +2/−2. 
     Feature #F, in one embodiment, T 0 , T 1 , T 2  . . . is a KLT. Feature #G, in one embodiment, N can be 4, 8, 16, 32 and/or 64. Feature #G, in one embodiment, T 0 , T 1 , T 2  . . . is scaled by a pre-defined constant s before being embedded into R. In one example of Feature #G, the constant s is √{square root over (2)}. 
     In Feature #I, the new resulting transform core RT may be used to replace the DCT-2 core. In the Feature #I example, as an example, only 32-point DCT-2 core and/or 64-point DCT-2 core may be replaced. 
     In Feature #J, whether the new resulting transform core RT is applied can be indicated in a VPS, SPS, PPS, or slice header, or a CTU-level flag. In one example of Feature #J, if the new resulting transform core RT is not applied, then the original 2N-point transform core R may be used. In one example of Feature #J, for coding a video sequence/picture/slice/CTU, if an additional transform type other than DCT-2, e.g., DST-7/DCT-8/DCT-4/DST-4, is used, the new resulting transform core RT may replace DCT-2 accordingly. Otherwise, if only DCT-2 is used, the new resulting transform core RT is not used. 
     Instead of embedding orthonormal transforms into a larger orthonormal transform K, it is proposed to keep the larger orthonormal transform K unchanged, but extract an orthonormal transform T from this larger orthonormal transform, i.e., the transform core T is a part of K. In one embodiment, K is a 2N-point transform core, T is an N-point transform core, and T is extracted from one half of the basis rows of RT which are anti-symmetry. In one embodiment, K is a transform from the DCT/DST families, of which examples are: DCT-2, DCT-3, DST-2 or DST-3. 
     In one embodiment, T is extracted from K for only certain transform sizes, e.g., 16-point, and/or 32-point. In one embodiment, DCT-4 is extracted from DCT-2, and DCT-4 is replacing the DCT-8 in AMT for certain block sizes, e.g., 16-point, and/or 32-point. In one embodiment, a new transform core is derived by flipping DCT-4 transform bases left and right, and this new transform core is replacing DST-7 in AMT for certain block sizes, e.g., 16-point, 32-point. 
     In one embodiment, DCT-4 is extracted from DCT-2, and DCT-4 is included as an additional transform type in AMT for certain block sizes, e.g., 4-point, 8-point, and whether DCT-4 is used depends on the block size, intra prediction mode, horizontal/vertical transform, AMT flag, AMT index. In one embodiment, a new transform core is derived by flipping DCT-4 transform bases left and right, and this new transform core is included as an additional transform type in AMT for certain block sizes, e.g., 4-point, 8-point, whether this new transform core is used depends on the block size, intra prediction mode, horizontal/vertical transform, AMT flag, AMT index. 
     For these reasons, it is proposed to use COT cores instead of traditional transform cores (such as DCT-2 and DST-7) in video and/or image compression, such as transform image or residual blocks. A COT core may be generated by doing the Kronecker product between multiple orthogonal transform cores and put them together as one larger matrix which is also an orthogonal transform core, and use it as the COT core. 
     In one embodiment, for example, given a set of N orthonormal transforms, with transform cores being denoted by T 0 , T 1 , . . . T N-1 , and an additional orthonormal transform with transform core being denoted by P, if P is divided into N matrices P 0 , P 1 , . . . P N-1 , that is, if put P 0 , P 1 , . . . P N-1  together, we will get P. A new orthonormal transform X can then be derived by putting T i ⊗P i  together as one matrix, where i=0, 1, . . . , N−1, and ⊗ denotes the Kronecker product, and use this matrix as the COT core. 
     In one embodiment, P is a Hadamard transform. In one embodiment, P is horizontally divided into N matrices P 0 , P 1 , . . . P N-1  In one embodiment, P is vertically divided into N matrices P 0 , P 1 , . . . P N-1  In one embodiment, the set of N orthonormal transform includes DCT-1˜DCT-8, DST-1˜DST-7. 
     In one example, N=2, and the set of N orthonormal transform are {T 0 =n-point DCT-2, T 1 =n-point DST-8}, P is a 2×2 Hadamard transform matrix 
                     [         1       1           1         -   1           ]     ,           
and P is vertically divided into P0=[1 1] and P1=[1 −1]. After calculating T 0 ⊗P 0  and T 1 ⊗P 1  and putting them together, we get a new 2n-point orthonormal transform X.
 
     In one embodiment, after putting T i ⊗P i  together as one matrix, the rows and/or columns are further re-ordered, which does not change the orthogonality of the new transform. 
     In one embodiment, before doing the Kronecker product, i.e., T i ⊗P i , the rows and/or columns of T i  and/or P i  are further re-ordered, which does not change the orthogonality of the new transform. 
     Referring to  FIG. 5 , in a first, simple example, a method of utilizing a Compound Orthonormal Transform may include the following steps illustrated in  FIG. 5 . Step  500 : Generate a new multi-transform data structure by embedding one or more non-recursive transforms into a larger recursive transform. Step  501 : Receive information regarding a target data block for encoding or decoding, the target data block for encoding or decoding being one of: a compressed video or image data block or an uncompressed video or image data block. The method of  FIG. 5  may also include a Step  502  that includes determining whether to use a recursive transform for the encoding or decoding of the target data block. Similarly, Step  502  may include determining, for the encoding or decoding of the target data block, whether to use a recursive transform or a non-recursive transform; 
     As shown in  FIG. 5 , if the Step  502  determination is YES (i.e., use the recursive transform), Step  503  is performed. Step  503  includes, when a result of the determination is to use the recursive transform: generate the recursive transform using the multi-transform data structure and cause or transmit information that causes the target data block to be encoded or decoded using the generated recursive transform. When the determination in Step  502  is NO (i.e., do not use the recursive transforom, or alternatively use the non-recursive transform), Step  504  is performed. Step  504  may include, when a result of the determination is to not use the recursive transform, rather use one of the one or more non-recursive transforms embedded in the multi-transform data structure. In particular, Step  504  may include causing or transmitting information that causes the target data block to be encoded or decoded using one of the one or more non-recursive transforms embedded in the multi-transform data structure. The method of  FIG. 5  may further include that at least part of the non-recursive transform of the multi-transform data structure is embedded in at least part of one half of the recursive transform of the multi-transform data structure. 
     The method illustrated in  FIG. 5  may also include processing Steps  500 - 504  being performed by one or more processors. The method illustrated in  FIG. 5  may also include storing, by the one or more processors in a memory, the multi-transform data structure in advance before or during the performing of the Steps  501 - 504 . 
     The method illustrated in  FIG. 5  may also include that the size of the non-recursive transform is M, where M=N/2, wherein N is the size of the recursive transform of the multi-transform data structure. The method of  FIG. 5  may further include that the recursive transform is one of: DCT-2, Hadamard transform, DCT-1, DST-1, or DST-2. The method of  FIG. 5  may further include that the non-recursive transform is one of: DST-7 or DCT-8. 
     The method of  FIG. 5  may further include that the recursive transform is DCT-2 and the non-recursive transform is DST-7 or DCT-8. The method of  FIG. 5  may further include that the recursive transform is DCT-2, and the method further comprises, when using the recursive DCT-2 transform of the multi-transform data structure: (i) determining the left half of the DCT-2 transform matrix based on the multi-transform data structure, and (ii) deriving the right half of the DCT-2 transform matrix according to the symmetry/anti-symmetry property of DCT-2 core in view of the left half of the DCT-2 transform matrix. 
     The method of  FIG. 5  may further include creating a resulting basis that corresponds to the original odd bases of 2N-point DCT-2 by: (i) replacing the first half of odd bases of 2N-point DCT-2 with N-point scaled DST-7 or DCT-8 bases, and (ii) filling the second half of the odd bases of the 2N-point DCT-2 with the flipped DST-7 or DCT-8 bases plus inverse signs. The method of  FIG. 5  may further include that the transform matrices are quantized with 10-bit representation. 
     The method of  FIG. 5  may further include deriving the multi-transform data structure as a new orthonormal transform, which is a Compound Orthonormal Transform (COT), by embedding an N-point non-recursive transform into a 2N-point recursive transform, which results in the new orthonormal transform which is still an orthonormal transform due to the symmetric/anti-symmetric characteristics of the recursive transform. The method of  FIG. 5  may further include deriving the multi-transform data structure as a new orthonormal transform, which is a Compound Orthonormal Transform (COT), by embedding an N-point DCT-8 or DST-7 into a 2N-point DCT-2, which results in the new orthonormal transform which is still an orthonormal transform due to the symmetric/anti-symmetric characteristics of DCT-2. 
     In one example, as the multi-transform data structure, a 64-point COT core is proposed, as shown below: 
     { 
     {aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,}, 
     {bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, −ck, −cj, −ci, −ch, −cg, −cf, −ce, −cd, −cc, −cb, −ca, −bz, −by, −bx, −bw, −by, −bu, −bt, −bs, −br, −bq, −bp, −bo, −bn, −bm, −bl, −bk, −bj, −bi, −bh, −bg, −bf,}, 
     {ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, −be, −bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as, −ar, −aq, −ap, −ap, −aq, −ar, −as, −at, −au, −av, −aw, −ax, −ay, −az, −ba, −bb, −bc, −bd, −be, be, bd, bc, bb, ba, az, ay, ax, aw, av, au, at, as, ar, aq, ap,}, 
     {bg, bj, bm, bp, bs, by, by, cb, ce, ch, ck, −cj, −cg, −cd, −ca, −bx, −bu, −br, −bo, −bl, −bi, −bf, −bh, −bk, −bn, −bq, −bt, −bw, −bz, −cc, −cf, −ci, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh, bf, bi, bl, bo, br, bu, bx, ca, cd, cg, cj, −ck, −ch, −ce, −cb, −by, −by, −bs, −bp, −bm, −bj, −bg,}, 
     {ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai, ah, ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai, ah,}, 
     {bh, bm, br, bw, cb, cg, 0, −cg, −cb, −bw, −br, −bm, −bh, −bh, −bm, −br, −bw, −cb, −cg, 0, cg, cb, bw, br, bm, bh, bh, bm, br, bw, cb, cg, −cg, −cb, −bw, −br, −bm, −bh, −bh, −bm, −br, −bw, −cb, −cg, 0, cg, cb, bw, br, bm, bh, bh, bm, br, bw, cb, cg, 0, −cg, −cb, −bw, −br, −bm, −bh,}, 
     {aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, bc, az, aw, at, aq, aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, 0, bc, az, aw, at, aq, aq, at, aw, az, bc, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, 0, bc, az, aw, at, aq,}, 
     {bi, bp, bw, cd, ck, −cf, −by, −br, −bk, −bg, −bn, −bu, −cb, −ci, ch, ca, bt, bm, bf, bl, bs, bz, cg, −cj, −cc, −by, −bo, −bh, −bj, −bq, −bx, −ce, ce, bx, bq, bj, bh, bo, by, cc, cj, −cg, −bz, −bs, −bl, −bf, −bm, −bt, −ca, −ch, ci, cb, bu, bn, bg, bk, br, by, cf, −ck, −cd, −bw, −bp, −bi,}, 
     {ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,}, 
     {bj, bs, cb, ck, −cd, −bu, −bl, −bh, −bq, −bz, −ci, cf, bw, bn, bf, bo, bx, cg, −ch, −by, −bp, −bg, −bm, −by, −ce, cj, ca, br, bi, bk, bt, cc, −cc, −bt, −bk, −bi, −br, −ca, −cj, ce, by, bm, bg, bp, by, ch, −cg, −bx, −bo, −bf, −bn, −bw, −cf, ci, bz, bq, bh, bl, bu, cd, −ck, −cb, −bs, −bj,}, 
     {ar, aw, bb, −be, −az, −au, −ap, −at, −ay, −bd, bc, ax, as, aq, ay, ba, −ba, −ay, −aq, −as, −ax, −bc, bd, ay, at, ap, au, az, be, −bb, −aw, −ar, −ar, −aw, −bb, be, az, au, ap, at, ay, bd, −bc, −ax, −as, −aq, −ay, −ba, ba, ay, aq, as, ax, bc, −bd, −ay, −at, −ap, −au, −az, −be, bb, aw, ar,}, 
     {bk, by, cg, −cf, −bu, −bj, −bl, −bw, −ch, ce, bt, bi, bm, bx, ci, −cd, −bs, −bh, −bn, −by, −cj, cc, br, bg, bo, bz, ck, −cb, −bq, −bf, −bp, −ca, ca, bp, bf, bq, cb, −ck, −bz, −bo, −bg, −br, −cc, cj, by, bn, bh, bs, cd, −ci, −bx, −bm, −bi, −bt, −ce, ch, bw, bl, bj, bu, cf, −cg, −by, −bk,}, 
     {ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al, ai, ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al, ai,}, 
     {bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl,}, 
     {as, az, −be, −ax, −aq, −au, −bb, bc, av, ap, aw, bd, −ba, −at, −ar, −ay, ay, ar, at, ba, −bd, −aw, −ap, −av, −bc, bb, au, aq, ax, be, −az, −as, −as, −az, be, ax, aq, au, bb, −bc, −av, −ap, −aw, −bd, ba, at, ar, ay, −ay, −ar, −at, −ba, bd, aw, ap, av, bc, −bb, −au, −aq, −ax, −be, az, as,}, 
     {bm, cb, −cg, −br, −bh, −bw, 0, bw, bh, br, cg, −cb, −bm, −bm, −cb, cg, br, bh, bw, 0, −bw, −bh, −br, −cg, cb, bm, bm, cb, −cg, −br, −bh, −bw, bw, bh, br, cg, −cb, −bm, −bm, −cb, cg, br, bh, bw, 0, −bw, −bh, −br, −cg, cb, bm, bm, cb, −cg, −br, −bh, −bw, 0, bw, bh, br, cg, −cb, −bm,}, 
     {ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,}, 
     {bn, ce, −cb, −bk, −bq, −ch, by, bh, bt, ck, −by, −bf, −bw, cj, bs, bi, bz, −cg, −bp, −bl, −cc, cd, bm, bo, cf, −ca, −bj, −br, −ci, bx, bg, bu, −bu, −bg, −bx, ci, br, bj, ca, −cf, −bo, −bm, −cd, cc, bl, bp, cg, −bz, −bi, −bs, −cj, bw, bf, by, −ck, −bt, −bh, −by, ch, bq, bk, cb, −ce, −bn,}, 
     {at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw, −aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw, 0, −aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw, 0, −aw, −aq, −az, bc, at,}, 
     {bo, ch, −bw, −bg, −bz, ce, bl, br, ck, −bt, −bj, −cc, cb, bi, bu, −cj, −bq, −bm, −cf, by, bf, bx, −cg, −bn, −bp, −ci, by, bh, ca, −cd, −bk, −bs, bs, bk, cd, −ca, −bh, −by, ci, bp, bn, cg, −bx, −bf, −by, cf, bm, bq, cj, −bu, −bi, −cb, cc, bj, bt, −ck, −br, −bl, −ce, bz, bg, bw, −ch, −bo,}, 
     {aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao, aj, aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao, aj,}, 
     {bp, ck, −br, −bn, −ci, bt, bl, cg, −by, −bj, −ce, bx, bh, cc, −bz, −bf, −ca, cb, bg, by, −cd, −bi, −bw, cf, bk, bu, −ch, −bm, −bs, cj, bo, bq, −bq, −bo, −cj, bs, bm, ch, −bu, −bk, −cf, bw, bi, cd, −by, −bg, −cb, ca, bf, bz, −cc, −bh, −bx, ce, bj, by, −cg, −bl, −bt, ci, bn, br, −ck, −bp,}, 
     {au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au,}, {bq, −cj, −bm, −bu, cf, bi, by, −cb, −bf, −cc, bx, bj, cg, −bt, −bn, −ck, bp, br, −ci, −bl, −by, ce, bh, bz, −ca, −bg, −cd, bw, bk, ch, −bs, −bo, bo, bs, −ch, −bk, −bw, cd, bg, ca, −bz, −bh, −ce, by, bl, ci, −br, −bp, ck, bn, bt, −cg, −bj, −bx, cc, bf, cb, −by, −bi, −cf, bu, bm, cj, −bq,}, {ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae,}, 
     {br, −cg, −bh, −cb, bw, bm, 0, −bm, −bw, cb, bh, cg, −br, −br, cg, bh, cb, −bw, −bm, 0, bm, bw, −cb, −bh, −cg, br, br, −cg, −bh, −cb, bw, bm, −bm, −bw, cb, bh, cg, −br, −br, cg, bh, cb, −bw, −bm, 0, bm, bw, −cb, −bh, −cg, br, br, −cg, −bh, −cb, bw, bm, 0, −bm, −bw, cb, bh, cg, −br,}, 
     {av, −bc, −ap, −bb, aw, au, −bd, −aq, −ba, ax, at, −be, −ar, −az, ay, as, −as, −ay, az, ar, be, −at, −ax, ba, aq, bd, −au, −aw, bb, ap, bc, −ay, −ay, bc, ap, bb, −aw, −au, bd, aq, ba, −ax, −at, be, ar, az, −ay, −as, as, ay, −az, −ar, −be, at, ax, −ba, −aq, −bd, au, aw, −bb, −ap, −bc, ay,}, 
     {bs, −cd, −bh, −ci, bn, bx, −by, −bm, cj, bi, cc, −bt, −br, ce, bg, ch, −bo, −bw, bz, bl, −ck, −bj, −cb, bu, bq, −cf, −bf, −cg, bp, by, −ca, −bk, bk, ca, −by, −bp, cg, bf, cf, −bq, −bu, cb, bj, ck, −bl, −bz, bw, bo, −ch, −bg, −ce, br, bt, −cc, −bi, −cj, bm, by, −bx, −bn, ci, bh, cd, −bs,}, 
     {ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am, ak, ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am, ak,}, 
     {bt, −ca, −bm, ch, bf, ci, −bl, −cb, bs, bu, −bz, −bn, cg, bg, cj, −bk, −cc, br, by, −by, −bo, cf, bh, ck, −bj, −cd, bq, bw, −bx, −bp, ce, bi, −bi, −ce, bp, bx, −bw, −bq, cd, bj, −ck, −bh, −cf, bo, by, −by, −br, cc, bk, −cj, −bg, −cg, bn, bz, −bu, −bs, cb, bl, −ci, −bf, −ch, bm, ca, −bt,}, 
     {aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq, aq, bc, −at, −az, aw, aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq, 0, aq, bc, −at, −az, aw, aw, −az, −at, bc, aq, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq, 0, aq, bc, −at, −az, aw,}, 
     {bu, −bx, −br, ca, bo, −cd, −bl, cg, bi, −cj, −bf, −ck, bh, ch, −bk, −ce, bn, cb, −bq, −by, bt, by, −bw, −bs, bz, bp, −cc, −bm, cf, bj, −ci, −bg, bg, ci, −bj, −cf, bm, cc, −bp, −bz, bs, bw, −by, −bt, by, bq, −cb, −bn, ce, bk, −ch, −bh, ck, bf, cj, −bi, −cg, bl, cd, −bo, −ca, br, bx, −bu,}, 
     {aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,}, 
     {by, −bu, −bw, bt, bx, −bs, −by, br, bz, −bq, −ca, bp, cb, −bo, −cc, bn, cd, −bm, −ce, bl, cf, −bk, −cg, bj, ch, −bi, −ci, bh, cj, −bg, −ck, bf, −bf, ck, bg, −cj, −bh, ci, bi, −ch, −bj, cg, bk, −cf, −bl, ce, bm, −cd, −bn, cc, bo, −cb, −bp, ca, bq, −bz, −br, by, bs, −bx, −bt, bw, bu, −by,}, 
     {ax, −aw, −ay, av, az, −au, −ba, at, bb, −as, −bc, ar, bd, −aq, −be, ap, −ap, be, aq, −bd, −ar, bc, as, −bb, −at, ba, au, −az, −ay, ay, aw, −ax, −ax, aw, ay, −ay, −az, au, ba, −at, −bb, as, bc, −ar, −bd, aq, be, −ap, ap, −be, −aq, bd, ar, −bc, −as, bb, at, −ba, −au, az, av, −ay, −aw, ax,}, 
     {bw, −br, −cb, bm, cg, −bh, 0, bh, −cg, −bm, cb, br, −bw, −bw, br, cb, −bm, −cg, bh, 0, −bh, cg, bm, −cb, −br, bw, bw, −br, −cb, bm, cg, −bh, bh, −cg, −bm, cb, br, −bw, −bw, br, cb, −bm, −cg, bh, 0, −bh, cg, bm, −cb, −br, bw, bw, −br, −cb, bm, cg, −bh, 0, bh, −cg, −bm, cb, br, −bw,}, 
     {al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj, al, al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj, al,}, 
     {bx, −bo, −cg, bf, −ch, −bn, by, bw, −bp, −cf, bg, −ci, −bm, bz, by, −bq, −ce, bh, −cj, −bl, ca, bu, −br, −cd, bi, −ck, −bk, cb, bt, −bs, −cc, bj, −bj, cc, bs, −bt, −cb, bk, ck, −bi, cd, br, −bu, −ca, bl, cj, −bh, ce, bq, −by, −bz, bm, ci, −bg, cf, bp, −bw, −by, bn, ch, −bf, cg, bo, −bx,}, 
     {ay, −at, −bd, ap, −bc, −au, ax, az, −as, −be, aq, −bb, −av, aw, ba, −ar, ar, −ba, −aw, av, bb, −aq, be, as, −az, −ax, au, bc, −ap, bd, at, −ay, −ay, at, bd, −ap, bc, au, −ax, −az, as, be, −aq, bb, av, −aw, −ba, ar, −ar, ba, aw, −av, −bb, aq, −be, −as, az, ax, −au, −bc, ap, −bd, −at, ay,}, 
     {by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by,}, 
     {af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,}, 
     {bz, −bi, cg, bs, −bp, −cj, bl, −bw, −cc, bf, −cd, −by, bm, −ck, −bo, bt, cf, −bh, ca, by, −bj, ch, br, −bq, −ci, bk, −bx, −cb, bg, −ce, −bu, bn, −bn, bu, ce, −bg, cb, bx, −bk, ci, bq, −br, −ch, bj, −by, −ca, bh, −cf, −bt, bo, ck, −bm, by, cd, −bf, cc, bw, −bl, cj, bp, −bs, −cg, bi, −bz,}, 
     {az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at, −at, aw, bc, −aq, az, az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at, 0, −at, aw, bc, −aq, az, az, −aq, bc, aw, −at, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at, 0, −at, aw, bc, −aq, az,}, 
     {ca, −bf, cb, bz, −bg, cc, by, −bh, cd, bx, −bi, ce, bw, −bj, cf, by, −bk, cg, bu, −bl, ch, bt, −bm, ci, bs, −bn, cj, br, −bo, ck, bq, −bp, bp, −bq, −ck, bo, −br, −cj, bn, −bs, −ci, bm, −bt, −ch, bl, −bu, −cg, bk, −by, −cf, bj, −bw, −ce, bi, −bx, −cd, bh, −by, −cc, bg, −bz, −cb, bf, −ca,}, 
     {am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am, −am, ah, −al, −an, ai, −ao, aj, −aj, ao, ak, −ai, an, al, −ah, am, am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai, an, al, −ah, am,}, 
     {cb, −bh, bw, cg, −bm, br, 0, −br, bm, −cg, −bw, bh, −cb, −cb, bh, −bw, −cg, bm, −br, 0, br, −bm, cg, bw, −bh, cb, cb, −bh, bw, cg, −bm, br, −br, bm, −cg, −bw, bh, −cb, −cb, bh, −bw, −cg, bm, −br, 0, br, −bm, cg, bw, −bh, cb, cb, −bh, bw, cg, −bm, br, 0, −br, bm, −cg, −bw, bh, −cb,}, 
     {ba, −aq, ax, bd, −at, au, −be, −aw, ar, −bb, −az, ap, −ay, −bc, as, −av, av, −as, bc, ay, −ap, az, bb, −ar, aw, be, −au, at, −bd, −ax, aq, −ba, −ba, aq, −ax, −bd, at, −au, be, aw, −ar, bb, az, −ap, ay, bc, −as, av, −av, as, −bc, −ay, ap, −az, −bb, ar, −aw, −be, au, −at, bd, ax, −aq, ba,}, 
     {cc, −bk, br, −cj, −by, bg, −by, −cg, bo, −bn, cf, bz, −bh, bu, ck, −bs, bj, −cb, −cd, bl, −bq, ci, bw, −bf, bx, ch, −bp, bm, −ce, −ca, bi, −bt, bt, −bi, ca, ce, −bm, bp, −ch, −bx, bf, −bw, −ci, bq, −bl, cd, cb, −bj, bs, −ck, −bu, bh, −bz, −cf, bn, −bo, cg, by, −bg, by, cj, −br, bk, −cc,}, 
     {ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,}, 
     {cd, −bn, bm, −cc, −ce, bo, −bl, cb, cf, −bp, bk, −ca, −cg, bq, −bj, bz, ch, −br, bi, −by, −ci, bs, −bh, bx, cj, −bt, bg, −bw, −ck, bu, −bf, by, −by, bf, −bu, ck, bw, −bg, bt, −cj, −bx, bh, −bs, ci, by, −bi, br, −ch, −bz, bj, −bq, cg, ca, −bk, bp, −cf, −cb, bl, −bo, ce, cc, −bm, bn, −cd,}, 
     {bb, −at, as, −ba, −bc, au, −ar, az, bd, −av, aq, −ay, −be, aw, −ap, ax, −ax, ap, −aw, be, ay, −aq, av, −bd, −az, ar, −au, bc, ba, −as, at, −bb, −bb, at, −as, ba, bc, −au, ar, −az, −bd, av, −aq, ay, be, −aw, ap, −ax, ax, −ap, aw, −be, −ay, aq, −av, bd, az, −ar, au, −bc, −ba, as, −at, bb,}, 
     {ce, −bq, bh, −by, cj, bz, −bl, bm, −ca, −ci, bu, −bg, br, −cf, −cd, bp, −bi, bw, −ck, −by, bk, −bn, cb, ch, −bt, bf, −bs, cg, cc, −bo, bj, −bx, bx, −bj, bo, −cc, −cg, bs, −bf, bt, −ch, −cb, bn, −bk, by, ck, −bw, bi, −bp, cd, cf, −br, bg, −bu, ci, ca, −bm, bl, −bz, −cj, by, −bh, bq, −ce,}, 
     {an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak, an, an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak, an,}, 
     {cf, −bt, bh, −bo, ca, ck, −by, bm, −bj, by, −ch, −cd, br, −bf, bq, −cc, −ci, bw, −bk, bl, −bx, cj, cb, −bp, bg, −bs, ce, cg, −bu, bi, −bn, bz, −bz, bn, −bi, bu, −cg, −ce, bs, −bg, bp, −cb, −cj, bx, −bl, bk, −bw, ci, cc, −bq, bf, −br, cd, ch, −by, bj, −bm, by, −ck, −ca, bo, −bh, bt, −cf,}, 
     {bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az, az, −at, aq, −aw, bc, bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az, 0, az, −at, aq, −aw, bc, bc, −aw, aq, −at, az, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az, 0, az, −at, aq, −aw, bc,}, 
     {cg, −bw, bm, −bh, br, −cb, 0, cb, −br, bh, −bm, bw, −cg, −cg, bw, −bm, bh, −br, cb, 0, −cb, br, −bh, bm, −bw, cg, cg, −bw, bm, −bh, br, −cb, cb, −br, bh, −bm, bw, −cg, −cg, bw, −bm, bh, −br, cb, 0, −cb, br, −bh, bm, −bw, cg, cg, −bw, bm, −bh, br, −cb, 0, cb, −br, bh, −bm, bw, −cg,}, 
     {ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,}, 
     {ch, −bz, br, −bj, bi, −bq, by, −cg, −ci, ca, −bs, bk, −bh, bp, −bx, cf, cj, −cb, bt, −bl, bg, −bo, bw, −ce, −ck, cc, −bu, bm, −bf, bn, −by, cd, −cd, by, −bn, bf, −bm, bu, −cc, ck, ce, −bw, bo, −bg, bl, −bt, cb, −cj, −cf, bx, −bp, bh, −bk, bs, −ca, ci, cg, −by, bq, −bi, bj, −br, bz, −ch,}, 
     {bd, −az, av, −ar, aq, −au, ay, −bc, −be, ba, −aw, as, −ap, at, −ax, bb, −bb, ax, −at, ap, −as, aw, −ba, be, bc, −ay, au, −aq, ar, −av, az, −bd, −bd, az, −av, ar, −aq, au, −ay, bc, be, −ba, aw, −as, ap, −at, ax, −bb, bb, −ax, at, −ap, as, −aw, ba, −be, −bc, ay, −au, aq, −ar, av, −az, bd,}, 
     {ci, −cc, bw, −bq, bk, −bf, bl, −br, bx, −cd, cj, ch, −cb, by, −bp, bj, −bg, bm, −bs, by, −ce, ck, cg, −ca, bu, −bo, bi, −bh, bn, −bt, bz, −cf, cf, −bz, bt, −bn, bh, −bi, bo, −bu, ca, −cg, −ck, ce, −by, bs, −bm, bg, −bj, bp, −by, cb, −ch, −cj, cd, −bx, br, −bl, bf, −bk, bq, −bw, cc, −ci,}, 
     {ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an, ao, ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an, ao,}, 
     {cj, −cf, cb, −bx, bt, −bp, bl, −bh, bg, −bk, bo, −bs, bw, −ca, ce, −ci, −ck, cg, −cc, by, −bu, bq, −bm, bi, −bf, bj, −bn, br, −by, bz, −cd, ch, −ch, cd, −bz, by, −br, bn, −bj, bf, −bi, bm, −bq, bu, −by, cc, −cg, ck, ci, −ce, ca, −bw, bs, −bo, bk, −bg, bh, −bl, bp, −bt, bx, −cb, cf, −cj,}, 
     {be, −bc, ba, −ay, aw, −au, as, −aq, ap, −ar, at, −ay, ax, −az, bb, −bd, bd, −bb, az, −ax, ay, −at, ar, −ap, aq, −as, au, −aw, ay, −ba, bc, −be, −be, bc, −ba, ay, −aw, au, −as, aq, −ap, ar, −at, ay, −ax, az, −bb, bd, −bd, bb, −az, ax, −ay, at, −ar, ap, −aq, as, −au, aw, −ay, ba, −bc, be,}, 
     {ck, −ci, cg, −ce, cc, −ca, by, −bw, bu, −bs, bq, −bo, bm, −bk, bi, −bg, bf, −bh, bj, −bl, bn, −bp, br, −bt, by, −bx, bz, −cb, cd, −cf, ch, −cj, cj, −ch, cf, −cd, cb, −bz, bx, −by, bt, −br, bp, −bn, bl, −bj, bh, −bf, bg, −bi, bk, −bm, bo, −bq, bs, −bu, bw, −by, ca, −cc, ce, −cg, ci, −ck,}, 
     } 
     where {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl} are pre-defined numbers. 
     An example value of the above pre-defined numbers (elements in the 64−point COT core) are {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 89, 88, 87, 84, 81, 77, 73, 67, 62, 55, 48, 41, 33, 25, 17, 8, 90, 90, 89, 89, 88, 87, 85, 84, 82, 81, 78, 76, 74, 71, 69, 66, 63, 60, 56, 53, 49, 46, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4}. 
     An example value of the elements in 64-point COT core is: {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9, 90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4} 
     From the above transform core, the 2-point DCT-2 can be extracted, as shown below: 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 aa 
                 aa 
               
               
                   
                 aa 
                 −aa  
               
               
                   
                   
               
            
           
         
       
     
     From the above transform core, the 4-point DCT-2 can be extracted, as shown below: 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 aa 
                  aa 
                  aa 
                  aa 
               
               
                   
                 ab 
                  ac 
                 −ac 
                 −ab 
               
               
                   
                 aa 
                 −aa 
                 −aa 
                  aa 
               
               
                   
                 ac 
                 −ab 
                  ab 
                 −ac 
               
               
                   
                   
               
            
           
         
       
     
     From the above transform core, the 8-point DCT-2 can be extracted, as shown below: 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
               
               
                   
                 ad 
                 ae 
                 af 
                 ag 
                 -ag 
                 -af 
                 -ae 
                 -ad 
               
               
                   
                 ab 
                 ac 
                 -ac 
                 -ab 
                 -ab 
                 -ac 
                 ac 
                 ab 
               
               
                   
                 ae 
                 -ag 
                 -ad 
                 -af 
                 af 
                 ad 
                 ag 
                 -ae 
               
               
                   
                 aa 
                 -aa 
                 -aa 
                 aa 
                 aa 
                 -aa 
                 -aa 
                 aa 
               
               
                   
                 af 
                 -ad 
                 ag 
                 ae 
                 -ae 
                 -ag 
                 ad 
                 -af 
               
               
                   
                 ac 
                 -ab 
                 ab 
                 -ac 
                 -ac 
                 ab 
                 -ab 
                 ac 
               
               
                   
                 ag 
                 -af 
                 ae 
                 -ad 
                 ad 
                 -ae 
                 af 
                 -ag 
               
               
                   
                   
               
            
           
         
       
     
     From the above transform core, the 16-point DCT-2 can be extracted, as shown below: 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
                 aa 
               
               
                 ah 
                 ar 
                 aj 
                 ak 
                 al 
                 am 
                 an 
                 ao 
                 -ao 
                 -an 
                 -am 
                 -al 
                 -ak 
                 -aj 
                 -ar 
                 -ah 
               
               
                 ad 
                 ae 
                 af 
                 ag 
                 -ag 
                 -af 
                 -ae 
                 -ad 
                 -ad 
                 -ae 
                 -af 
                 -ag 
                 ag 
                 af 
                 ae 
                 ad 
               
               
                 ar 
                 al 
                 ao 
                 -am 
                 -aj 
                 -ah 
                 -ak 
                 -an 
                 an 
                 ak 
                 ah 
                 aj 
                 am 
                 -ao 
                 -al 
                 -ar 
               
               
                 ab 
                 ac 
                 -ac 
                 -ab 
                 -ab 
                 -ac 
                 ac 
                 ab 
                 ab 
                 ac 
                 -ac 
                 -ab 
                 -ab 
                 -ac 
                 ac 
                 ab 
               
               
                 aj 
                 ao 
                 -ak 
                 -ar 
                 -an 
                 al 
                 ah 
                 am 
                 -am 
                 -ah 
                 -al 
                 an 
                 ar 
                 ak 
                 -ao 
                 -aj 
               
               
                 ae 
                 -ag 
                 -ad 
                 -af 
                 af 
                 ad 
                 ag 
                 -ae 
                 -ae 
                 ag 
                 ad 
                 af 
                 -af 
                 -ad 
                 -ag 
                 ae 
               
               
                 ak 
                 -am 
                 -ar 
                 ao 
                 ah 
                 an 
                 -aj 
                 -al 
                 al 
                 aj 
                 -an 
                 -ah 
                 -ao 
                 ar 
                 am 
                 -ak 
               
               
                 aa 
                 -aa 
                 -aa 
                 aa 
                 aa 
                 -aa 
                 -aa 
                 aa 
                 aa 
                 -aa 
                 -aa 
                 aa 
                 aa 
                 -aa 
                 -aa 
                 aa 
               
               
                 al 
                 -aj 
                 -an 
                 ah 
                 -ao 
                 -ar 
                 am 
                 ak 
                 -ak 
                 -am 
                 ar 
                 ao 
                 -ah 
                 an 
                 aj 
                 -al 
               
               
                 af 
                 -ad 
                 ag 
                 ae 
                 -ae 
                 -ag 
                 ad 
                 -af 
                 -af 
                 ad 
                 -ag 
                 -ae 
                 ae 
                 ag 
                 -ad 
                 af 
               
               
                 am 
                 -ah 
                 al 
                 an 
                 -ar 
                 ak 
                 ao 
                 -aj 
                 aj 
                 -ao 
                 -ak 
                 ar 
                 -an 
                 -al 
                 ah 
                 -am 
               
               
                 ac 
                 -ab 
                 ab 
                 -ac 
                 -ac 
                 ab 
                 -ab 
                 ac 
                 ac 
                 -ab 
                 ab 
                 -ac 
                 -ac 
                 ab 
                 -ab 
                 ac 
               
               
                 an 
                 -ak 
                 ah 
                 -aj 
                 am 
                 ao 
                 -al 
                 ar 
                 -ar 
                 al 
                 -ao 
                 -am 
                 aj 
                 -ah 
                 ak 
                 -an 
               
               
                 ag 
                 -af 
                 ae 
                 -ad 
                 ad 
                 -ae 
                 af 
                 -ag 
                 -ag 
                 af 
                 -ae 
                 ad 
                 -ad 
                 ae 
                 -af 
                 ag 
               
               
                 ao 
                 -an 
                 am 
                 -al 
                 ak 
                 -aj 
                 ar 
                 -ah 
                 ah 
                 -ar 
                 aj 
                 -ak 
                 al 
                 -am 
                 an 
                 -ao 
               
               
                   
               
            
           
         
       
     
     From the above transform core, the 32-point COT which replaces 32-point DCT-2 can be extracted, as shown below: 
     { 
     {aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,}, 
     {ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, −be, −bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as, −ar, −aq, −ap,}, 
     {ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai, ah,}, 
     {aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, bc, az, aw, at, aq, aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq,}, 
     {ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,}, 
     {ar, aw, bb, −be, −az, −au, −ap, −at, −ay, −bd, bc, ax, as, aq, av, ba, −ba, −av, −aq, −as, −ax, −bc, bd, ay, at, ap, au, az, be, −bb, −aw, −ar,}, 
     {ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al, ai,}, 
     {as, az, −be, −ax, −aq, −au, −bb, bc, av, ap, aw, bd, −ba, −at, −ar, −ay, ay, ar, at, ba, −bd, −aw, −ap, −av, −bc, bb, au, aq, ax, be, −az, −as,}, 
     {ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,}, 
     {at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw, −aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at,}, 
     {aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao, aj,}, 
     {au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au,}, 
     {ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae,}, 
     {av, −bc, −ap, −bb, aw, au, −bd, −aq, −ba, ax, at, −be, −ar, −az, ay, as, −as, −ay, az, ar, be, −at, −ax, ba, aq, bd, −au, −aw, bb, ap, bc, −av,}, 
     {ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am, ak,}, 
     {aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq, aq, bc, −at, −az, aw, aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw,}, 
     {aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,}, 
     {ax, −aw, −ay, av, az, −au, −ba, at, bb, −as, −bc, ar, bd, −aq, −be, ap, −ap, be, aq, −bd, −ar, bc, as, −bb, −at, ba, au, −az, −av, ay, aw, −ax,}, 
     {al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj, al,}, 
     {ay, −at, −bd, ap, −bc, −au, ax, az, −as, −be, aq, −bb, −av, aw, ba, −ar, ar, −ba, −aw, av, bb, −aq, be, as, −az, −ax, au, bc, −ap, bd, at, −ay,}, 
     {af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,}, 
     {az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at, −at, aw, bc, −aq, az, az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az,}, 
     {am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai, an, al, −ah, am,}, 
     {ba, −aq, ax, bd, −at, au, −be, −aw, ar, −bb, −az, ap, −ay, −bc, as, −av, av, −as, bc, ay, −ap, az, bb, −ar, aw, be, −au, at, −bd, −ax, aq, −ba,}, 
     {ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,}, 
     {bb, −at, as, −ba, −bc, au, −ar, az, bd, −av, aq, −ay, −be, aw, −ap, ax, −ax, ap, −aw, be, ay, −aq, av, −bd, −az, ar, −au, bc, ba, −as, at, −bb,}, 
     {an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak, an,}, 
     {bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az, az, −at, aq, −aw, bc, bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc,}, 
     {ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,}, 
     {bd, −az, av, −ar, aq, −au, ay, −bc, −be, ba, −aw, as, −ap, at, −ax, bb, −bb, ax, −at, ap, −as, aw, −ba, be, bc, −ay, au, −aq, ar, −av, az, −bd,}, 
     {ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an, ao,}, 
     {be, −bc, ba, −ay, aw, −au, as, −aq, ap, −ar, at, −av, ax, −az, bb, −bd, bd, −bb, az, −ax, av, −at, ar, −ap, aq, −as, au, −aw, ay, −ba, bc, −be,}, 
     } 
     An example value of the elements in 32-point COT core is: {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9} 
     In addition, the 16-point DCT-8 transform core can be also extracted as below, and the 16-point DST-7 transform core can be derived by flipping the 16-point DCT-8 transform core left and right plus sign changes on the odd rows. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 ap 
                 aq 
                 ar 
                 as 
                 at 
                 au 
                 av 
                 aw 
                 ax 
                 ay 
                 az 
                 ba 
                 bb 
                 bc 
                 bd 
                 be 
               
               
                 aq 
                 at 
                 aw 
                 az 
                 bc 
                 0 
                 -bc 
                 -az 
                 -aw 
                 -at 
                 -aq 
                 -aq 
                 -at 
                 -aw 
                 -az 
                 -bc 
               
               
                 ar 
                 aw 
                 bb 
                 -be 
                 -az 
                 -au 
                 -ap 
                 -at 
                 -ay 
                 -bd 
                 bc 
                 ax 
                 as 
                 aq 
                 av 
                 ba 
               
               
                 as 
                 az 
                 -be 
                 -ax 
                 -aq 
                 -au 
                 -bb 
                 bc 
                 av 
                 ap 
                 aw 
                 bd 
                 -ba 
                 -at 
                 -ar 
                 -ay 
               
               
                 at 
                 bc 
                 -az 
                 -aq 
                 -aw 
                 0 
                 aw 
                 aq 
                 az 
                 -bc 
                 -at 
                 -at 
                 -bc 
                 az 
                 aq 
                 aw 
               
               
                 au 
                 0 
                 -au 
                 -au 
                 0 
                 au 
                 au 
                 0 
                 -au 
                 -au 
                 0 
                 au 
                 au 
                 0 
                 -au 
                 -au 
               
               
                 av 
                 -bc 
                 -ap 
                 -bb 
                 aw 
                 au 
                 -bd 
                 -aq 
                 -ba 
                 ax 
                 at 
                 -be 
                 -ar 
                 -az 
                 ay 
                 as 
               
               
                 aw 
                 -az 
                 -at 
                 bc 
                 aq 
                 0 
                 -aq 
                 -bc 
                 at 
                 az 
                 -aw 
                 -aw 
                 az 
                 at 
                 -bc 
                 -aq 
               
               
                 ax 
                 -aw 
                 -ay 
                 av 
                 az 
                 -au 
                 -ba 
                 at 
                 bb 
                 -as 
                 -bc 
                 ar 
                 bd 
                 -aq 
                 -be 
                 ap 
               
               
                 ay 
                 -at 
                 -bd 
                 ap 
                 -bc 
                 -au 
                 ax 
                 az 
                 -as 
                 -be 
                 aq 
                 -bb 
                 -av 
                 aw 
                 ba 
                 -ar 
               
               
                 az 
                 -aq 
                 bc 
                 aw 
                 -at 
                 0 
                 at 
                 -aw 
                 -bc 
                 aq 
                 -az 
                 -az 
                 aq 
                 -bc 
                 -aw 
                 at 
               
               
                 ba 
                 -aq 
                 ax 
                 bd 
                 -at 
                 au 
                 -be 
                 -aw 
                 ar 
                 -bb 
                 -az 
                 ap 
                 -ay 
                 -bc 
                 as 
                 -av 
               
               
                 bb 
                 -at 
                 as 
                 -ba 
                 -bc 
                 au 
                 -ar 
                 az 
                 bd 
                 -av 
                 aq 
                 -ay 
                 -be 
                 aw 
                 -ap 
                 ax 
               
               
                 bc 
                 -aw 
                 aq 
                 -at 
                 az 
                 0 
                 -az 
                 at 
                 -aq 
                 aw 
                 -bc 
                 -bc 
                 aw 
                 -aq 
                 at 
                 -az 
               
               
                 bd 
                 -az 
                 av 
                 -ar 
                 aq 
                 -au 
                 ay 
                 -bc 
                 -be 
                 ba 
                 -aw 
                 as 
                 -ap 
                 at 
                 -ax 
                 bb 
               
               
                 be 
                 -bc 
                 ba 
                 -ay 
                 aw 
                 -au 
                 as 
                 -aq 
                 ap 
                 -ar 
                 at 
                 -av 
                 ax 
                 -az 
                 bb 
                 -bd 
               
               
                   
               
            
           
         
       
     
     In addition, the 32-point DCT-8 transform core can be also extracted as below, and the 32-point DST-7 transform core can be derived by flipping the 32-point DCT-8 transform core left and right plus sign changes on the odd rows. 
     The 32-point DCT-8 transform core is shown below: 
     { 
     {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F,}, 
     {b, e, h, k, n, q, t, w, z, C, F, −E, −B, −y, −v, −s, −p, −m, −j, −g, −d, −a, −c, −f, −i, −l, −o, −r, −u, −x, −A, −D,}, 
     {c, h, m, r, w, B, 0, −B, −w, −r, −m, −h, −c, −c, −h, −m, −r, −w, −B, 0, B, w, r, m, h, c, c, h, in, r, w, B,}, 
     {d, k, r, y, F, −A, −t, −m, −f, −b, −i, −p, −w, −D, C, v, o, h, a, g, n, u, B, −E, −x, −q, −j, −c, −e, −l, −s, −z,}, 
     {e, n, w, F, −y, −p, −g, −c, −l, −u, −D, A, r, i, a, j, s, B, −C, −t, −k, −b, −h, −q, −z, E, v, m, d, f, o, x,}, 
     {f, q, B, −A, −p, −e, −g, −r, −C, z, o, d, h, s, D, −y, −n, −c, −i, −t, −E, x, m, b, j, u, F, −w, −l, −a, −k, −v,}, 
     {g, t, 0, −t, −g, −g, −t, 0, t, g, g, t, 0, −t, −g, −g, −t, 0, t, g, g, t, 0, −t, −g, −g, −t, 0, t, g, g, t,}, 
     {h, w, −B, −m, −c, −r, 0, r, c, m, B, −w, −h, −h, −w, B, m, c, r, 0, −r, −c, −m, −B, w, h, h, w, −B, −m, −c, −r,}, 
     {i, z, −w, −f, −l, −C, t, c, o, F, −q, −a, −r, E, n, d, u, −B, −k, −g, −x, y, h, j, A, −v, −e, −m, −D, s, b, p,}, 
     {j, C, −r, −b, −u, z, g, m, F, −o, −e, −x, w, d, p, −E, −l, −h, −A, t, a, s, −B, −i, −k, −D, q, c, v, −y, −f, −n,}, 
     {k, F, −m, −i, −D, o, g, B, −q, −e, −z, s, c, x, −u, −a, −v, w, b, t, −y, −d, −r, A, f, p, −C, −h, −n, E, j, l,}, {l, −E, −h, −p, A, d, t, −w, −a, −x, s, e, B, −o, −i, −F, k, m, −D, −g, −q, z, c, u, −v, −b, −y, r, f, C, −n, −j,}, 
     {m, −B, −c, −w, r, h, 0, −h, −r, w, c, B, −m, −m, B, c, w, −r, −h, 0, h, r, −w, −c, −B, m, m, −B, −c, −w, r, h,}, 
     {n, −y, −c, −D, i, s, −t, −h, E, d, x, −o, −m, z, b, C, −j, −r, u, g, −F, −e, −w, p, l, −A, −a, −B, k, q, −v, −f,}, 
     {o, −v, −h, C, a, D, −g, −w, n, p, −u, −i, B, b, E, −f, −x, m, q, −t, −j, A, c, F, −e, −y, l, r, −s, −k, z, d,}, 
     {p, −s, −m, v, j, −y, −g, B, d, −E, −a, −F, c, C, −f, −z, i, w, −l, −t, o, q, −r, −n, u, k, −x, −h, A, e, −D, −b,}, 
     {q, −p, −r, o, s, −n, −t, m, u, −l, −v, k, w, −j, −x, i, y, −h, −z, g, A, −f, −B, e, C, −d, −D, c, E, −b, −F, a,}, 
     {r, −m, −w, h, B, −c, 0, c, −B, −h, w, m, −r, −r, m, w, −h, −B, c, 0, −c, B, h, −w, −m, r, r, −m, −w, h, B, −c,}, 
     {s, −j, −B, a, −C, −i, t, r, −k, −A, b, −D, −h, u, q, −l, −z, c, −E, −g, v, p, −m, −y, d, −F, −f, w, o, −n, −x, e,}, 
     {t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g,}, 
     {u, −d, B, n, −k, −E, g, −r, −x, a, −y, −q, h, −F, −j, o, A, −c, v, t, −e, C, m, −l, −D, f, −s, −w, b, −z, −p, i,}, 
     {v, −a, w, u, −b, x, t, −c, y, s, −d, z, r, −e, A, q, −f, B, p, −g, C, o, −h, D, n, −i, E, m, −j, F, l, −k,}, 
     {w, −c, r, B, −h, m, 0, −m, h, −B, −r, c, −w, −w, c, −r, −B, h, −m, 0, m, −h, B, r, −c, w, w, −c, r, B, −h, m,}, 
     {x, −f, m, −E, −q, b, −t, −B, j, −i, A, u, −c, p, F, −n, e, −w, −y, g, −l, D, r, −a, s, C, −k, h, −z, −v, d, −o,}, 
     {y, −i, h, −x, −z, j, −g, w, A, −k, f, −v, −B, l, −e, u, C, −m, d, −t, −D, n, −c, s, E, −o, b, −r, −F, p, −a, q,}, 
     {z, −l, c, −q, E, u, −g, h, −v, −D, p, −b, m, −A, −y, k, −d, r, −F, −t, f, −i, w, C, −o, a, −n, B, x, −j, e, −s,}, 
     {A, −o, c, −j, v, F, −t, h, −e, q, −C, −y, m, −a, l, −x, −D, r, −f, g, −s, E, w, −k, b, −n, z, B, −p, d, −i, u,}, 
     {B, −r, h, −c, m, −w, 0, w, −m, c, −h, r, −B, −B, r, −h, c, −m, w, 0, −w, m, −c, h, −r, B, B, −r, h, −c, m, −w,}, 
     {C, −u, m, −e, d, −l, t, −B, −D, v, −n, f, −c, k, −s, A, E, −w, o, −g, b, −j, r, −z, −F, x, −p, h, −a, i, −q, y,}, 
     {D, −x, r, −l, f, −a, g, −m, s, −y, E, C, −w, q, −k, e, −b, h, −n, t, −z, F, B, −v, p, −j, d, −c, i, −o, u, −A,}, 
     {E, −A, w, −s, o, −k, g, −c, b, −f, j, −n, r, −v, z, −D, −F, B, −x, t, −p, l, −h, d, −a, e, −i, m, −q, u, −y, C,}, 
     {F, −D, B, −z, x, −v, t, −r, p, −n, l, −j, h, −f, d, −b, a, −c, e, −g, i, −k, m, −o, q, −s, u, −w, y, −A, C, −E,}, 
     } 
     An example value of the elements in 32-point DCT-8 core is: {a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F}={90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4} 
     The techniques for encoding/decoding can be implemented by one or more processors that may execute computer software having computer-readable instructions that may be physically stored in one or more computer-readable media (e.g., a hard disk drive). For example,  FIG. 7  shows a computer system  700  suitable for implementing certain embodiments of the disclosed subject matter. 
     The computer software can be coded using any suitable machine code or computer language, that may be subject to assembly, compilation, linking, or like mechanisms to create code comprising instructions that can be executed directly, or through interpretation, micro-code execution, and the like, by computer central processing units (CPUs), Graphics Processing Units (GPUs), and the like. 
     The instructions can be executed on various types of computers or components thereof, including, for example, personal computers, tablet computers, servers, smartphones, gaming devices, internet of things devices, and the like. 
     The components shown in  FIG. 7  for computer system  700  are exemplary in nature and are not intended to suggest any limitation as to the scope of use or functionality of the computer software implementing embodiments of the present disclosure. Neither should the configuration of components be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary embodiment of a computer system  700 . 
     Computer system  700  may include certain human interface input devices. Such a human interface input device may be responsive to input by one or more human users through, for example, tactile input (such as: keystrokes, swipes, data glove movements), audio input (such as: voice, clapping), visual input (such as: gestures), olfactory input (not depicted). The human interface devices can also be used to capture certain media not necessarily directly related to conscious input by a human, such as audio (such as: speech, music, ambient sound), images (such as: scanned images, photographic images obtain from a still image camera), video (such as two-dimensional video, three-dimensional video including stereoscopic video). 
     Input human interface devices may include one or more of (only one of each depicted): keyboard  701 , mouse  702 , trackpad  703 , touch screen  710 , data-glove  704 , joystick  705 , microphone  706 , scanner  707 , camera  708 . 
     Computer system  700  may also include certain human interface output devices. Such human interface output devices may be stimulating the senses of one or more human users through, for example, tactile output, sound, light, and smell/taste. Such human interface output devices may include tactile output devices (for example tactile feedback by the touch-screen  710 , data-glove  704 , or joystick  705 , but there can also be tactile feedback devices that do not serve as input devices), audio output devices (such as: speakers  709 , headphones (not depicted)), visual output devices (such as screens  710  to include CRT screens, LCD screens, plasma screens, OLED screens, each with or without touch-screen input capability, each with or without tactile feedback capability—some of which may be capable to output two dimensional visual output or more than three dimensional output through means such as stereographic output; virtual-reality glasses (not depicted), holographic displays and smoke tanks (not depicted)), and printers (not depicted). 
     Computer system  700  can also include human accessible storage devices and their associated media such as optical media including CD/DVD ROM/RW  720  with CD/DVD or the like media  721 , thumb-drive  722 , removable hard drive or solid state drive  723 , legacy magnetic media such as tape and floppy disc (not depicted), specialized ROM/ASIC/PLD based devices such as security dongles (not depicted), and the like. 
     Those skilled in the art should also understand that term “computer readable media” or “computer readable medium” as used in connection with the presently disclosed subject matter does not encompass transmission media, carrier waves, or other transitory signals. 
     Computer system  700  can also include interface to one or more communication networks  755 . Networks  755  can for example be one or more of wireless, wireline, or optical. Networks  755  can further be one or more of local, wide-area, metropolitan, vehicular and industrial, real-time, or delay-tolerant, and so on. Examples of networks include local area networks such as Ethernet, wireless LANs, cellular networks to include GSM, 3G, 4G, 5G, LTE and the like, TV wireline or wireless wide area digital networks to include cable TV, satellite TV, and terrestrial broadcast TV, vehicular and industrial to include CANBus, and so forth. Certain networks  755  commonly require external network interface adapters  754  that attached to certain general purpose data ports or peripheral buses ( 749 ) (such as, for example USB ports of the computer system  700 ; others are commonly integrated into the core of the computer system  700  by attachment to a system bus as described below (for example Ethernet interface into a PC computer system or cellular network interface into a smartphone computer system). Using any of these networks  755 , computer system  700  can communicate with other entities. Such communication can be uni-directional, receive only (for example, broadcast TV), uni-directional send-only (for example CANbus to certain CANbus devices), or bi-directional, for example to other computer systems using local or wide area digital networks  755 . Certain protocols and protocol stacks can be used on each of those networks  755  and network interfaces  754  as described above. 
     Aforementioned human interface devices, human-accessible storage devices, and network interfaces can be attached to a core  740  of the computer system  700 . 
     The core  740  can include one or more Central Processing Units (CPU)  741 , Graphics Processing Units (GPU)  742 , specialized programmable processing units in the form of Field Programmable Gate Areas (FPGA)  743 , hardware accelerators for certain tasks  744 , and so forth. These devices, along with Read-only memory (ROM)  745 , Random-access memory  746 , internal mass storage such as internal non-user accessible hard drives, SSDs, and the like  747 , may be connected through a system bus  748 . In some computer systems, the system bus  748  can be accessible in the form of one or more physical plugs to enable extensions by additional CPUs, GPU, and the like. The peripheral devices can be attached either directly to the core&#39;s system bus  748 , or through a peripheral bus  749 . Architectures for a peripheral bus include PCI, USB, and the like. 
     CPUs  741 , GPUs  742 , FPGAs  743 , and accelerators  744  can execute certain instructions that, in combination, can make up the aforementioned computer code. That computer code can be stored in ROM  745  or RAM  746 . Transitional data can also be stored in RAM  746 , whereas permanent data can be stored for example, in the internal mass storage  747 . Fast storage and retrieve to any of the memory devices can be enabled through the use of cache memory, that can be closely associated with one or more CPU  741 , GPU  742 , mass storage  747 , ROM  745 , RAM  746 , and the like. 
     The computer readable media can have computer code thereon for performing various computer-implemented operations. The media and computer code can be those specially designed and constructed for the purposes of the present disclosure, or they can be of the kind well known and available to those having skill in the computer software arts. 
     As an example and not by way of limitation, the computer system having architecture  700 , and specifically the core  740  can provide functionality as a result of processor(s) (including CPUs, GPUs, FPGA, accelerators, and the like) executing software embodied in one or more tangible, computer-readable media. Such computer-readable media can be media associated with user-accessible mass storage as introduced above, as well as certain storage of the core  740  that are of non-transitory nature, such as core-internal mass storage  747  or ROM  745 . The software implementing various embodiments of the present disclosure can be stored in such devices and executed by core  740 . A computer-readable medium can include one or more memory devices or chips, according to particular needs. The software can cause the core  740  and specifically the processors therein (including CPU, GPU, FPGA, and the like) to execute particular processes or particular parts of particular processes described herein, including defining data structures stored in RAM  746  and modifying such data structures according to the processes defined by the software. In addition or as an alternative, the computer system can provide functionality as a result of logic hardwired or otherwise embodied in a circuit (for example: accelerator  744 ), which can operate in place of or together with software to execute particular processes or particular parts of particular processes described herein. Reference to software can encompass logic, and vice versa, where appropriate. Reference to a computer-readable media can encompass a circuit (such as an integrated circuit (IC)) storing software for execution, a circuit embodying logic for execution, or both, where appropriate. The present disclosure encompasses any suitable combination of hardware and software. 
     While this disclosure has described several exemplary embodiments, there are alterations, permutations, and various substitute equivalents, which fall within the scope of the disclosure. It will thus be appreciated that those skilled in the art will be able to devise numerous systems and methods which, although not explicitly shown or described herein, embody the principles of the disclosure and are thus within the spirit and scope thereof.