Patent Publication Number: US-2023156241-A1

Title: Method and apparatus for three-dimensional (3d)-tree coding for neural network model compression

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation application of U.S. application Ser. No. 17/536,455, filed on Nov. 29, 2021, which is a continuation of U.S. application Ser. No. 17/081,158, filed on Oct. 27, 2020, now U.S. Pat. No. 11,234,024 issued on Jan. 25, 2022, which claims priority from U.S. Provisional Patent Application No. 62/939,054, filed on Nov. 22, 2019, U.S. Provisional Patent Application No. 62/940,427, filed on Nov. 26, 2019, U.S. Provisional Patent Application No. 62/957,699, filed on Jan. 6, 2020, U.S. Provisional Patent Application No. 62/957,691, filed on Jan. 6, 2020, U.S. Provisional Patent Application No. 62/975,485, filed on Feb. 12, 2020, and U.S. Provisional Patent Application No. 62/994,660, filed on Mar. 25, 2020, in the U.S. Patent and Trademark Office, the disclosures of which are incorporated herein by reference in their entireties. 
    
    
     BACKGROUND 
     Success of Deep Neural Networks (DNNs) in a large range of video applications such as semantic classification, target detection/recognition, target tracking, video quality enhancement, etc. poses a need for compressing DNN models. Therefore, the Motion Picture Experts Group (MPEG) is actively working on the Coded Representation of Neural Network standard (NNR) that is used to encode DNN models to save both storage and computation. 
     SUMMARY 
     According to embodiments, a method of three-dimensional (3D)-Tree coding for neural network model compression, is performed by at least one processor, and includes reshaping a four-dimensional (4D) parameter tensor of a neural network into a 3D parameter tensor of the neural network, the 3D parameter tensor comprising a convolution kernel size, an input feature size, and an output feature size, partitioning the 3D parameter tensor along a plane that is formed by the input feature size and the output feature size into 3D coding tree units (CTU3Ds), partitioning each of the CTU3Ds into a plurality of 3D coding units (CU3Ds) recursively until a predetermined depth, using a quad-tree, constructing a 3D tree for each of the plurality of CU3Ds, and entropy encoding each of a plurality of values of a plurality of nodes of the 3D tree. 
     According to embodiments, an apparatus for adaptive block partitioning for neural network model compression, includes at least one memory configured to store program code, and at least one processor configured to read the program code and operate as instructed by the program code. The program code includes reshaping code configured to cause the at least one processor to reshape a four-dimensional (4D) parameter tensor of a neural network into a 3D parameter tensor of the neural network, the 3D parameter tensor comprising a convolution kernel size, an input feature size, and an output feature size, first partitioning code configured to cause the at least one processor to partition the 3D parameter tensor along a plane that is formed by the input feature size and the output feature size, into 3D coding tree units (CTU3Ds), second partitioning code configured to cause the at least one processor to partition each of the CTU3Ds into a plurality of 3D coding units (CU3Ds) recursively until a maximum depth, using a quad-tree, first constructing code configured to cause the at least one processor to construct a 3D-Tree for each of the plurality of CU3Ds, and first entropy encoding code configured to cause the at least one processor to entropy encode each of a plurality of values of a plurality of nodes of the 3D tree. 
     According to embodiments, a non-transitory computer-readable medium stores instructions that, when executed by at least one processor for adaptive block partitioning for neural network model compression, cause the at least one processor to reshape a four-dimensional (4D) parameter tensor of a neural network into a 3D parameter tensor of the neural network, the 3D parameter tensor comprising a convolution kernel size, an input feature size, and an output feature size, partition the 3D parameter tensor along a plane that is formed by the input feature size and the output feature size, into 3D coding tree units (CTU3Ds), partition each of the CTU3Ds into a plurality of 3D coding units (CU3Ds) recursively until a maximum depth, using a quad-tree, construct a 3D-Tree for each of the plurality of CU3Ds, and entropy encode each of a plurality of values of a plurality of nodes of the 3D tree. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a diagram of a GEPM/GEPP partition method. 
         FIG.  2    is a diagram of an environment in which methods, apparatuses and systems described herein may be implemented, according to embodiments. 
         FIG.  3    is a block diagram of example components of one or more devices of  FIG.  2   . 
         FIG.  4    is a functional block diagram of a system for neural network model compression, according to embodiments. 
         FIG.  5    is a diagram of two examples of an adaptive CTU3D/3D coding unit (CU3D) partition using a raster scan at a vertical direction, according to embodiments. 
         FIG.  6    is a diagram of an example of a 3D-Octree structure with three depths, according to embodiments. 
         FIG.  7    is a flowchart of a method of 3D-Tree coding for neural network model compression, according to embodiments. 
         FIG.  8    is a block diagram of an apparatus for 3D-Tree coding for neural network model compression, according to embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     This disclosure is related to neural network model compression. To be more specific, methods and apparatuses described herein are related to 3D-Tree coding for neural network model compression. 
     Scan Order 
     In the compression of neural networks for multimedia content description and analysis, if a dimension of a weight tensor is more than two (such as a convolution layer), this weight tensor is reshaped to a two-dimensional (2D) tensor. No reshape is performed if the dimension of weight tensor is no more than two (such as a fully connected layer or a bias layer). 
     The encoding method scans weight coefficients in a row-first manner from left to right and scans rows from top to bottom. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 quant_weight_tensor( dimensions, maxNumNoRem ) { 
               
               
                  dim = Size( dimensions ) 
               
               
                  for( i = TensorIterator( dim );!TensorIteratorEnd( i, dimensions ); i = 
               
               
                 TensorIteratorNext( i, dimensions ) { 
               
               
                   quant_weight( i, maxNumNoRem ) 
               
               
                  } 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     Quantization 
     In the compression of neural networks for multimedia content description and analysis, nearest neighbour quantization is applied in a uniform way to each weight coefficient in weight matrices. A fixed step size is applied. Reconstructed values in a decoded matrix are integer multiples of the step size. The step size is defined as a 32-bit floating number. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
             
            
               
                   
                 step_size( ) { 
                   
               
               
                   
                  step_size 
                 flt(32) 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     step_size is the quantization step size. 
     Entropy Coding 
     In the compression of neural networks for multimedia content description and analysis, each quantized weight level is encoded according to the following procedure employing an integer parameter maxNumNoRem: 
     In a first step, a binary syntax element sig_flag is encoded for the quantized weight level, which specifies whether a corresponding level is equal to zero. If the sig_flag is equal to one, a further binary syntax element sign_flag is encoded. A bin indicates if a current weight level is positive or negative. Next, a unary sequence of bins is encoded, followed by a fixed length sequence as follows: 
     A variable k is initialized with zero and X is initialized with 1&lt;&lt;k. A syntax element abs_level_greater_X is encoded, which indicates that an absolute value of the quantized weight level is greater than X. If abs_level_greater_X is equal to 1 and if X is greater than maxNumNoRem, the variable k is increased by 1. Afterwards, 1&lt;&lt;k is added to X and a further abs_level_greater_X is encoded. This procedure is continued until an abs_level_greater_X is equal to O. Now, X must be one of values (X, X−1, . . . X−(1&lt;&lt;k)+1). A code of length k is encoded, which points to values in a list that is an absolute quantized weight level. 
     Context modeling corresponds to associating three type of flags sig flag, sign_flag, and abs_level_greater_X with context models. In this way, flags with similar statistical behavior may be associated with the same context model so that a probability estimator (inside of the context model) can adapt to underlying statistics. 
     The context modeling of the presented approach is as follows: 
     Three context models are distinguished for the sig_flag, depending on whether a neighboring quantized weight level to the left is zero, smaller than zero, or larger than zero. 
     Three other context models are distinguished for the sign_flag depending on whether the neighboring quantized weight level to the left is zero, smaller than zero, or larger than zero. 
     For the abs_level_greater_X flags, each X uses, either one or two separate context models. If X&lt;=maxNumNoRem, two context models are distinguished depending on the sign_flag. If X&gt;maxNumNoRem, only one context model is used. 
     
       
         
           
               
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                 quant_weight( i, maxNumNoRem ) { 
                   
               
               
                  QuantWeight[i] = 0 
                   
               
               
                   sig _flag 
                 ae(v) 
               
               
                  if( sig_flag ) { 
                   
               
               
                   QuantWeight[i]++ 
                   
               
               
                    sign _flag 
                 ae(v) 
               
               
                   j = −1 
                   
               
               
                   do { 
                   
               
               
                    j++ 
                   
               
               
                     abs _level_greater_x  [   j   ]   
                 ae(v) 
               
               
                    QuantWeight[i] += abs_level_greater_x[j] 
                   
               
               
                   } while( abs_level_greater_x[j] == 1 &amp;&amp; j &lt;  
                   
               
               
                   maxNumNoRem ) 
                   
               
               
                   if( j == maxNumNoRem ) { 
                   
               
               
                    RemBits = 0 
                   
               
               
                    j = −1 
                   
               
               
                    do { 
                   
               
               
                     j++ 
                   
               
               
                      abs _level_greater_x2 [   j   ]   
                 ae(v) 
               
               
                     if( abs_level_greater_x2[j] ) { 
                   
               
               
                      RemBits++ 
                   
               
               
                      QuantWeight[i] += 1 &lt;&lt; RemBits 
                   
               
               
                     } 
                   
               
               
                    } while( abs_level_greater_x2[j] ) 
                   
               
               
                      abs _remainder 
                 uab(RemBits) 
               
               
                       QuantWeight[i] += abs_remainder 
                   
               
               
                      } 
                   
               
               
                      QuantWeight[i] = sign_flag ? -QuantWeight[i] : 
                   
               
               
                       QuantWeight[i] 
                   
               
               
                  } 
                   
               
               
                 } 
               
               
                   
               
               
                   sig _flag specifies whether a quantized weight QuantWeight[i] is nonzero. A sig_flag equal to 0 indicates that QuantWeight[i] is zero. 
               
               
                   sign _flag specifies whether the quantized weight QuantWeight[i] is positive or negative. A sign_flag equal to 1 indicates that QuantWeight[i] is negative. 
               
               
                   abs _level_greater_x [   j   ]  indicates whether an absolute level of QuantWeight[i] is greater j + 1. 
               
               
                   abs _level_greater_x2 [   j   ]  includes an unary part of an exponential golomb remainder. 
               
               
                   abs _remainder indicates a fixed length remainder. 
               
            
           
         
       
     
     Inference operation for deep learning system uses matrix multiplication intensively so a high-performance matrix multiplication library (GEMM) is the key for inference operation. Depending on a size of a left-hand-side (lhs) matrix and a right-hand-side (rhs) matrix, two GEMM routines (GEPP/GEBP, GEPM/GEBP) are recognized by the industry over the last decade as the optimal GEMM solution. As shown in  FIG.  1   , both methods partition the lhs matrix and the rhs matrix recursively to make the best use of different characteristics of off-chip memory (such as Double Data Rate (DDR)) and on-chip memory (such as multi-level cache) in modern computing platform, and the lhs matrix is usually stored in a column-major order to achieve the optimal memory access pattern. The lhs matrix is usually transposed to achieve the optimal memory access pattern. Some newer GEMM routines (such as QNNPACK) are optimized for neural networks designed for mobile and edge devices, are a variation of either a GEPP routine or a GEPM routine, and follow a similar matrix blocking/partitioning method. 
     A matrix scan order in the NNR is defined as a row-first manner from left to right and rows from top to bottom. This scan order does not match with a scan order required by the inference operation, as the inference operation must buffer an excessive size of weight coefficients before starts the operation. For example, when the inference operation is performed for a first fully-connect layer of VGG16, given that a matrix size of this layer is 25088×4096, a buffer that can store Nx25088 coefficients has to be reserved to perform a GEMM routine. If N=64 for a normal GEMM operation, a buffer size will be 1.5 MB even if coefficients are represented by an 8-bit integer instead of a 32-bit floating number, but such a buffer size is too high especially for mobile and edge devices. 
     Further, entropy coding in the NNR is performed on a quantized weight coefficient directly. The NNR did not consider a local distribution after a weight tensor is partitioned to non-overlapping 2D/3D coding tree units (CTUs)/3D coding tree units (CTU3Ds). Most weight coefficients are zeros after a sparse/prune operation. 
       FIG.  2    is a diagram of an environment  200  in which methods, apparatuses and systems described herein may be implemented, according to embodiments. As shown in  FIG.  2   , the environment  200  may include a user device  210 , a platform  220 , and a network  230 . Devices of the environment  200  may interconnect via wired connections, wireless connections, or a combination of wired and wireless connections. 
     The user device  210  includes one or more devices capable of receiving, generating, storing, processing, and/or providing information associated with platform  220 . For example, the user device  210  may include a computing device (e.g., a desktop computer, a laptop computer, a tablet computer, a handheld computer, a smart speaker, a server, etc.), a mobile phone (e.g., a smart phone, a radiotelephone, etc.), a wearable device (e.g., a pair of smart glasses or a smart watch), or a similar device. In some implementations, the user device  210  may receive information from and/or transmit information to the platform  220 . 
     The platform  220  includes one or more devices as described elsewhere herein. In some implementations, the platform  220  may include a cloud server or a group of cloud servers. In some implementations, the platform  220  may be designed to be modular such that software components may be swapped in or out. As such, the platform  220  may be easily and/or quickly reconfigured for different uses. 
     In some implementations, as shown, the platform  220  may be hosted in a cloud computing environment  222 . Notably, while implementations described herein describe the platform  220  as being hosted in the cloud computing environment  222 , in some implementations, the platform  220  may not be cloud-based (i.e., may be implemented outside of a cloud computing environment) or may be partially cloud-based. 
     The cloud computing environment  222  includes an environment that hosts the platform  220 . The cloud computing environment  222  may provide computation, software, data access, storage, etc. services that do not require end-user (e.g., the user device  210 ) knowledge of a physical location and configuration of system(s) and/or device(s) that hosts the platform  220 . As shown, the cloud computing environment  222  may include a group of computing resources  224  (referred to collectively as “computing resources  224 ” and individually as “computing resource  224 ”). 
     The computing resource  224  includes one or more personal computers, workstation computers, server devices, or other types of computation and/or communication devices. In some implementations, the computing resource  224  may host the platform  220 . The cloud resources may include compute instances executing in the computing resource  224 , storage devices provided in the computing resource  224 , data transfer devices provided by the computing resource  224 , etc. In some implementations, the computing resource  224  may communicate with other computing resources  224  via wired connections, wireless connections, or a combination of wired and wireless connections. 
     As further shown in  FIG.  2   , the computing resource  224  includes a group of cloud resources, such as one or more applications (“APPs”)  224 - 1 , one or more virtual machines (“VMs”)  224 - 2 , virtualized storage (“VSs”)  224 - 3 , one or more hypervisors (“HYPs”)  224 - 4 , or the like. 
     The application  224 - 1  includes one or more software applications that may be provided to or accessed by the user device  210  and/or the platform  220 . The application  224 - 1  may eliminate a need to install and execute the software applications on the user device  210 . For example, the application  224 - 1  may include software associated with the platform  220  and/or any other software capable of being provided via the cloud computing environment  222 . In some implementations, one application  224 - 1  may send/receive information to/from one or more other applications  224 - 1 , via the virtual machine  224 - 2 . 
     The virtual machine  224 - 2  includes a software implementation of a machine (e.g., a computer) that executes programs like a physical machine. The virtual machine  224 - 2  may be either a system virtual machine or a process virtual machine, depending upon use and degree of correspondence to any real machine by the virtual machine  224 - 2 . A system virtual machine may provide a complete system platform that supports execution of a complete operating system (“OS”). A process virtual machine may execute a single program, and may support a single process. In some implementations, the virtual machine  224 - 2  may execute on behalf of a user (e.g., the user device  210 ), and may manage infrastructure of the cloud computing environment  222 , such as data management, synchronization, or long-duration data transfers. 
     The virtualized storage  224 - 3  includes one or more storage systems and/or one or more devices that use virtualization techniques within the storage systems or devices of the computing resource  224 . In some implementations, within the context of a storage system, types of virtualizations may include block virtualization and file virtualization. Block virtualization may refer to abstraction (or separation) of logical storage from physical storage so that the storage system may be accessed without regard to physical storage or heterogeneous structure. The separation may permit administrators of the storage system flexibility in how the administrators manage storage for end users. File virtualization may eliminate dependencies between data accessed at a file level and a location where files are physically stored. This may enable optimization of storage use, server consolidation, and/or performance of non-disruptive file migrations. 
     The hypervisor  224 - 4  may provide hardware virtualization techniques that allow multiple operating systems (e.g., “guest operating systems”) to execute concurrently on a host computer, such as the computing resource  224 . The hypervisor  224 - 4  may present a virtual operating platform to the guest operating systems, and may manage the execution of the guest operating systems. Multiple instances of a variety of operating systems may share virtualized hardware resources. 
     The network  230  includes one or more wired and/or wireless networks. For example, the network  230  may include a cellular network (e.g., a fifth generation (5G) network, a long-term evolution (LTE) network, a third generation (3G) network, a code division multiple access (CDMA) network, etc.), a public land mobile network (PLMN), a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a telephone network (e.g., the Public Switched Telephone Network (PSTN)), a private network, an ad hoc network, an intranet, the Internet, a fiber optic-based network, or the like, and/or a combination of these or other types of networks. 
     The number and arrangement of devices and networks shown in  FIG.  2    are provided as an example. In practice, there may be additional devices and/or networks, fewer devices and/or networks, different devices and/or networks, or differently arranged devices and/or networks than those shown in  FIG.  2   . Furthermore, two or more devices shown in  FIG.  2    may be implemented within a single device, or a single device shown in  FIG.  2    may be implemented as multiple, distributed devices. Additionally, or alternatively, a set of devices (e.g., one or more devices) of the environment  200  may perform one or more functions described as being performed by another set of devices of the environment  200 . 
       FIG.  3    is a block diagram of example components of one or more devices of  FIG.  2   . The device  300  may correspond to the user device  210  and/or the platform  220 . As shown in  FIG.  3   , device  300  may include a bus  310 , a processor  320 , a memory  330 , a storage component  340 , an input component  350 , an output component  360 , and a communication interface  370 . 
     The bus  310  includes a component that permits communication among the components of the device  300 . The processor  320  is implemented in hardware, firmware, or a combination of hardware and software. The processor  320  is a central processing unit (CPU), a graphics processing unit (GPU), an accelerated processing unit (APU), a microprocessor, a microcontroller, a digital signal processor (DSP), a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), or another type of processing component. In some implementations, the processor  320  includes one or more processors capable of being programmed to perform a function. The memory  330  includes a random access memory (RAM), a read only memory (ROM), and/or another type of dynamic or static storage device (e.g., a flash memory, a magnetic memory, and/or an optical memory) that stores information and/or instructions for use by the processor  320 . 
     The storage component  340  stores information and/or software related to the operation and use of the device  300 . For example, the storage component  340  may include a hard disk (e.g., a magnetic disk, an optical disk, a magneto-optic disk, and/or a solid state disk), a compact disc (CD), a digital versatile disc (DVD), a floppy disk, a cartridge, a magnetic tape, and/or another type of non-transitory computer-readable medium, along with a corresponding drive. 
     The input component  350  includes a component that permits the device  300  to receive information, such as via user input (e.g., a touch screen display, a keyboard, a keypad, a mouse, a button, a switch, and/or a microphone). Additionally, or alternatively, the input component  350  may include a sensor for sensing information (e.g., a global positioning system (GPS) component, an accelerometer, a gyroscope, and/or an actuator). The output component  360  includes a component that provides output information from the device  300  (e.g., a display, a speaker, and/or one or more light-emitting diodes (LEDs)). 
     The communication interface  370  includes a transceiver-like component (e.g., a transceiver and/or a separate receiver and transmitter) that enables the device  300  to communicate with other devices, such as via a wired connection, a wireless connection, or a combination of wired and wireless connections. The communication interface  370  may permit the device  300  to receive information from another device and/or provide information to another device. For example, the communication interface  370  may include an Ethernet interface, an optical interface, a coaxial interface, an infrared interface, a radio frequency (RF) interface, a universal serial bus (USB) interface, a Wi-Fi interface, a cellular network interface, or the like. 
     The device  300  may perform one or more processes described herein. The device  300  may perform these processes in response to the processor  320  executing software instructions stored by a non-transitory computer-readable medium, such as the memory  330  and/or the storage component  340 . A computer-readable medium is defined herein as a non-transitory memory device. A memory device includes memory space within a single physical storage device or memory space spread across multiple physical storage devices. 
     Software instructions may be read into the memory  330  and/or the storage component  340  from another computer-readable medium or from another device via the communication interface  370 . When executed, software instructions stored in the memory  330  and/or the storage component  340  may cause the processor  320  to perform one or more processes described herein. Additionally, or alternatively, hardwired circuitry may be used in place of or in combination with software instructions to perform one or more processes described herein. Thus, implementations described herein are not limited to any specific combination of hardware circuitry and software. 
     The number and arrangement of components shown in  FIG.  3    are provided as an example. In practice, the device  300  may include additional components, fewer components, different components, or differently arranged components than those shown in  FIG.  3   . Additionally, or alternatively, a set of components (e.g., one or more components) of the device  300  may perform one or more functions described as being performed by another set of components of the device  300 . 
       FIG.  4    is a functional block diagram of a system  400  for neural network model compression, according to embodiments. 
     As shown in  FIG.  4   , the system  400  includes a parameter reduction module  405 , a parameter approximation module  410 , a reconstruction module  415 , an encoder  420 , and a decoder  425 . 
     The parameter reduction module  405  reduces a set of parameters of an input neural network, to obtain an output neural network. The neural network may include the parameters and an architecture as specified by a deep learning framework. 
     For example, the parameter reduction module  405  may sparsify (set weights to zero) and/or prune away connections of the neural network. In another example, the parameter reduction module  405  may perform matrix decomposition on parameter tensors of the neural network into a set of smaller parameter tensors. The parameter reduction module  405  may perform these methods in cascade, for example, may first sparsify the weights and then decompose a resulting matrix. 
     The parameter approximation module  410  applies parameter approximation techniques on parameter tensors that are extracted from the output neural network that is obtained from the parameter reduction module  405 . For example, the techniques may include any one or any combination of quantization, transformation and prediction. The parameter approximation module  410  outputs first parameter tensors that are not modified by the parameter approximation module  410 , second parameter tensors that are modified or approximated by the parameter approximation module  410 , and respective metadata to be used to reconstruct original parameter tensors that are not modified by the parameter approximation module  410 , from the modified second parameter tensors. 
     The reconstruction module  415  reconstructs the original parameter tensors from the modified second parameter tensors that are obtained from the parameter approximation module  410  and/or the decoder  425 , using the respective metadata that is obtained from the parameter approximation module  410  and/or the decoder  425 . The reconstruction module  415  may reconstruct the output neural network, using the reconstructed original parameter tensors and the first parameter tensors. 
     The encoder  420  may perform entropy encoding on the first parameter tensors, the second parameter tensors and the respective metadata that are obtained from the parameter approximation module  410 . This information may be encoded into a bitstream to the decoder  425 . 
     The decoder  425  may decode the bitstream that is obtained from the encoder  420 , to obtain the first parameter tensors, the second parameter tensors and the respective metadata. 
     The system  400  may be implemented in the platform  220 , and one or more modules of  FIG.  4    may be performed by a device or a group of devices separate from or including the platform  220 , such as the user device  210 . 
     Methods and apparatuses for 3D-Tree coding for neural network model compression will now be described in detail. 
     CTU3D Partition 
     If an lhs tensor is stored in a column-major order, or after a transpose of a row-major tensor, a dimension of a weight tensor is usually 4 for a convolution layer with a layout of [R][S][C][K], 2 for a fully-connected layer with a layout of [C][K], and 1 for a bias and batch normal layer. R/S is a convolution kernel size, C is an input feature size and K is an output feature size. 
     In embodiments, for the convolution layer, a 2D [R][S] dimension is reshaped to an 1D [RS] dimension so that the four-dimensional (4D) tensor [R][S][C][K] is reshaped to a 3D tensor [RS][C][K]. The fully-connected layer is treated as a special case of the 3D tensor with R=S=1. 
     As the kernel size RS is usually much smaller than C/K, the 3D tensor [RS][C][K] is partitioned along a [C][K] plane with non-overlapping smaller blocks (CTU3D). Each CTU3D has a shape of [RS][ctu3d_height][ctu3d_width], where ctu3d_height=max_ctu3d_height, ctu3d_width=max_ctu3d_width, and max_ctu3d_height/max_ctu3d_width is encoded in a model header. For a CTU3D that is located at the right and/or bottom of the tensor, its ctu3d_height is a remainder of C/max_ctu3d_height, and its ctu3d_width is a remainder of K/max_ctu3d_width. 
     As shown in  FIG.  5   , for a CTU3D  505 , at the right and/or bottom of a tensor, a parent CU3D node  510  at a given depth may not have all 4 child nodes. For the CU3D  510  that is located at the right and/or bottom of a tensor, cu3d_height is a remainder of max_ctu3d_height/max_cu3d_height, and cu3d_width is a remainder of max_ctu3d_width/max_cu3d_width. 
     In further embodiments, a square shape partition is used so that max_ctu3d_height=max_ctu3d_width, and a variable max_ctu3d_size is used to represent both max_ctu3d_height and max_ctu3d_width. max_ctu3d_size is defined as 2**N, and a value of N is 8, 16, 32, 64. 
     To facilitate an on-chip memory requirement in an inference operation, in embodiments, a flag is defined to indicate whether there is limit for a total CTU3D size for layers with different kernel sizes. The flag that equals to 0 indicates that ctu3d_height/ctu3d_width is kept unchanged regardless of the kernel size, and in this case, a size of a CTU3D for the convolution layer is RS times bigger than a size of a CTU3D for the fully-connected layer. The flag that equals to 1 indicates that ctu3d_height/ctu3d_width is scaled based on the kernel size. For example, ctu3d_height=ctu3d_width=int(ctu3d height*ctu3d_width/R/S). 
     While any scan order can be used to scan and process CTU3Ds in a 3D tensor, in embodiments, they are scanned and processed using a raster scan order at either a horizontal direction (SCAN_CK) or a vertical direction (SCAN_KC). 
     An example of corresponding syntax tables is listed below in Tables 4-6: 
     
       
         
           
               
               
             
               
                 TABLE 4 
               
               
                   
               
             
            
               
                 nnr( ) { 
                   
               
               
                  . . . 
                   
               
               
                  layer_header( ) 
                   
               
               
                  if(enable_max_ctu3d_size){ 
                   
               
               
                   max_ctu3d_height = max_ctu3d_width=int( 
                 embodi- 
               
               
                   max_ctu3d_size*max_ctu3d_size/R/S), or 
                 ment 1 
               
               
                   max_ctu3d_height= max_ctu3d_width=(2**(bitdepth(int( 
                 embodi- 
               
               
                   max_ctu3d_size * max_ctu3d_size /R/S))-1) 
                 ment 2 
               
               
                  } 
                   
               
               
                  if(layer_scan_order==SCAN_CK){ 
                   
               
               
                   for(c=0;c&lt;C;c+=max_ctu3d_height){ 
                   
               
               
                    for(k=0;k&lt;K;k+=max_ctu3d_width) { 
                   
               
               
                     ctu3d_height=min(max_ctu3d_height,C-c); 
                   
               
               
                     ctu3d_width=min(max_ctu3d_width,K-k); 
                   
               
               
                     last_ctu3d_flag=(max_ctu3d_height&gt;=C-c &amp;&amp; 
                   
               
               
                     max_ctu3d_width&gt;=K-k)?1:0 
                   
               
               
                     ctu3d(c,k,ctu3d_height,ctu3d_width) 
                   
               
               
                     end_of_layer(last_ctu3d_flag) 
                   
               
               
                    } 
                   
               
               
                   } 
                   
               
               
                  }else if(layer_scan_order==SCAN_KC){ 
                   
               
               
                   for(k=0;k&lt;K;k+=max_ctu3d_width) { 
                   
               
               
                    for(c=0;c&lt;C;c+=max_ctu3d_height){ 
                   
               
               
                     ctu3d_height=min(max_ctu3d_height,C-c); 
                   
               
               
                     ctu3d_width=min(max_ctu3d_width,K-k); 
                   
               
               
                     last_ctu3d_flag=(max_ctu3d_height&gt;=C-c &amp;&amp; 
                   
               
               
                     max_ctu3d_width&gt;=K-k)?1:0 
                   
               
               
                     ctu3d(c,k,ctu3d_height,ctu3d_width) 
                   
               
               
                     end_of_layer(last_ctu3d_flag) 
                   
               
               
                    } 
                   
               
               
                   } 
                   
               
               
                  } 
                   
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 5 
               
               
                   
                   
               
             
            
               
                   
                 nnr_header( ) { 
               
               
                   
                  ...... 
               
               
                   
                  enable_max_ctu3d_size 
               
               
                   
                  max_ctu3d_idx 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     enable_max_ctu3d_size being 0 indicates that ctu3d_height/ctu3d_width is kept unchanged regardless of the kernel size, and enable_max_ctu3d_size being 1 indicates that ctu3d_height/ctu3d_width is scaled based on the kernel size. 
     max_ctu3d_idx is in the following equation: 
       max_ctu_3d_size=(max_ctu3d_idx==0)?64:(max_ctu3d_idx==1)?32:(max_ctu3d_idx==2)?16:8   (1)
 
     
       
         
           
               
               
             
               
                   
                 TABLE 6 
               
               
                   
                   
               
             
            
               
                   
                 layer_header( ) { 
               
               
                   
                  ...... 
               
               
                   
                  layer_scan_order 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     layer_scan_order being 0 indicates the raster scan order at a horizontal direction, and layer_scan_order being 1 indicates the raster scan order at a vertical direction. 
     Adaptive CU3D Partition 
     In embodiments, a CTU/CU adaptive partitioning method that is used in video coding standards is used. 
     A simplified blocking structure is used, where a CTU3D/CU3D is partitioned to smaller CU3Ds recursively using a quad-tree structure until a maximum recursive depth is reached. Starting from a CTU3D node, this quad-tree of a CU3D is scanned and processed using a depth-first quad-tree scan order. Child nodes under the same parent node are scanned and processed using a raster scan order at either a horizontal direction or a vertical direction. 
     For a CU3D at a given quad-tree depth, a max_cu3d_height/max_cu3d_width of these CU3Ds is calculated using Equations (2) and (3) below, and a maximum recursive depth is reached when both max_cu3d_height and max_cu3d_width is smaller than or equal to a predefined threshold. This threshold can either be included in a bitstream explicitly, or can be a predefined number (such as 8) so it can be inferred by a decoder implicitly. 
       max_cu3d_height=max_ctu3d_height&gt;&gt;depth   (2)
 
       max_cu3d_width=max_ctu3d_width&gt;&gt;depth   (3)
 
     In further embodiments, a square shape partition is used so that max_ctu3d_height=max_ctu3d_width. For a CU3D at a given quad-tree depth, a max_cu3d_size of these CU3Ds is calculated using Equation (4) below, and a maximum recursive depth is reached when max_cu3d_size is smaller than or equal to a predefined threshold. This threshold can either be included in a bitstream explicitly, or can be a predefined number (such as 8) so it can be inferred by a decoder implicitly. 
       max_cu3d_size=max_ctu3d_size&gt;&gt;depth   (4)
 
     As shown in  FIG.  5   , for a CTU3D  505 , at the right and/or bottom of a tensor, a parent CU3D node  510  at a given depth may not have all 4 child nodes. For the CU3D  510  that is located at the right and/or bottom of a tensor, cu3d_height is a remainder of max_ctu3d_height/max_cu3d_height, and cu3d_width is a remainder of max_ctu3d_width/max_cu3d_width. 
     In further embodiments, a Rate-Distortion (RD) based encoding algorithm is used to decide whether a parent CU3D is split to multiple smaller child CU3Ds. The parent CU3D is split to the multiple smaller child CU3Ds if a combined RD of these smaller child CU3Ds is smaller than a RD from the parent CU3D. Otherwise, the parent CU3D is not split. A split flag is defined to record this splitting decision. 
     An example of a corresponding syntax table is listed below in Tables 7 and 8: 
     
       
         
           
               
               
             
               
                   
                 TABLE 7 
               
               
                   
                   
               
             
            
               
                   
                 ctu3d(...) { 
               
               
                   
                  ...... 
               
               
                   
                  cu3d(0,0,0) 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 8 
               
               
                   
                   
               
             
            
               
                   
                 cu3d(depth,y_idx,x_idx){ 
               
               
                   
                  ...... 
               
               
                   
                  if(cu3d does not exist) 
               
               
                   
                   return 
               
               
                   
                  if(depth&lt;ctu3d_depth−1){ 
               
               
                   
                   split_flag 
               
               
                   
                   if(split_flag){ 
               
               
                   
                    cu3d(depth+1,(y_idx&lt;&lt;1),(x_idx&lt;&lt;1)) 
               
               
                   
                    cu3d(depth+1,(y_idx&lt;&lt;1)+1,(x_idx&lt;&lt;1)) 
               
               
                   
                    cu3d(depth+1,(y_idx&lt;&lt;1),(x_idx&lt;&lt;1)+1) 
               
               
                   
                    cu3d(depth+1,(y_idx&lt;&lt;1)+1,(x_idx&lt;&lt;1)+1) 
               
               
                   
                    return 
               
               
                   
                   } 
               
               
                   
                  } 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     split_flag is a flag to indicate if a parent CU3D is split to 4 smaller child CU3Ds. 
     Kernel Plane Reorder 
     As discussed above, a simplified blocking structure is used, in which a CTU3D/CU3D is partitioned to a smaller CU3D recursively, using a quad-tree structure until a maximum recursive depth is reached. 
     In further embodiments, to facilitate a better compression operation, an RS number of 2D planes within a CU3D or CTU3D can be reordered along an [RS] axis. Reorder indices are encoded in a CU3D header so that each CU3D has its own reorder index, or encoded in a CTU3D header so that all CU3D in this CTU3D share the same reorder indices. A flag is defined to indicate if a reorder operation is allowed. This flag is encoded in either a model header or a layer header. 
     A zdep_array[RS] is defined to store a reordered index of an RS number of 2D planes. zdep_array[n] indicates that an index n of a reordered 2D plane is from the index zdep_array[n] of an original 2D plane. Because a first 2D plane does not need to be reordered to another location, zdep_arrya[0] is always 0. 
     Because a reorder operation is always performed in a closed form, a zdep_array can be scanned starting from next_idx=first_idx, and next_idx=zdep_array[next_idx] is iteratively assigned. A loop will be found in which zdep_array[next_idx]=first_idx. If not all indices of zdep_array are included in this loop, the zdep_array can be continuously scanned starting from a first index that is not included in all previous loops, and next_idx=zdep_array[next_idx] is iteratively assigned. 
     A queue is defined to store a sequence of next_idx, using the aforementioned zdep_array scanning method. A pseudo code for queue generation is listed below: 
     
       
         
           
               
               
             
               
                   
                   
               
             
            
               
                   
                 if (zdep_arry[++idx] != −1) { 
               
               
                   
                  first_idx = idx; 
               
               
                   
                  while (zdep_arry[idx] != first_idx) { 
               
               
                   
                   next_idx = zdep_arry[idx]; 
               
               
                   
                   queue.push_back(next_idx); 
               
               
                   
                    zdep_arry[idx] = −1; 
               
               
                   
                   idx = next_idx; 
               
               
                   
                  } 
               
               
                   
                  queue.push_back(−1); 
               
               
                   
                  zdep_arry[idx] = −1; 
               
               
                   
                  idx = first_idx; 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     Based on the reordering method, content of first and last locations of this queue can be inferred, and the content of the last location of any given loop can be inferred as well. 
     An example of a corresponding syntax table is listed below in Tables 9-11: 
     
       
         
           
               
               
             
               
                   
                 TABLE 9 
               
               
                   
                   
               
             
            
               
                   
                 nnr_header( ) { 
               
               
                   
                  ...... 
               
               
                   
                  enable_zdep_reorder 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     enable_zdep_reorder being 0 indicates that reorder of zdep_array is not allowed, and enable_zdep_reorder being 1 indicates that reorder of the zdep_array is allowed. 
     
       
         
           
               
               
             
               
                   
                 TABLE 10 
               
               
                   
                   
               
             
            
               
                   
                 ctu3d_header( ) { 
               
               
                   
                  ...... 
               
               
                   
                  zdep_array( ) 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 11 
               
               
                   
                   
               
             
            
               
                   
                 zdep_array( ) { 
               
               
                   
                  ...... 
               
               
                   
                  reorder_flag=false 
               
               
                   
                  if(enable_zdep_reorder &amp;&amp; zdep_size&gt;2) 
               
               
                   
                   reorder_flag 
               
               
                   
                  if(!reorder_flag){ 
               
               
                   
                   for(n=0;n&lt;zdep_size;++n) 
               
               
                   
                    zdep_array[n]=n; 
               
               
                   
                   return; 
               
               
                   
                  } 
               
               
                   
                  queue[0]=−1; 
               
               
                   
                  for(n=1;n&lt;zdep_size−1;++n){ 
               
               
                   
                   signalled_flag 
               
               
                   
                   queue[n]=(signalled_flag)?1:(queue[n−1]&gt;0)?−2:−1; 
               
               
                   
                  } 
               
               
                   
                  queue[zdep_size−1]=(queue[zdep_size−2]&gt;0)?−2:−1; 
               
               
                   
                  for(n=0;n&lt;zdep_size;++n){ 
               
               
                   
                   zdep_array[n]=−1; 
               
               
                   
                   if(queue[n]==1){ 
               
               
                   
                    qval_minus_one 
               
               
                   
                    queue[n]=qval_minus_one+1; 
               
               
                   
                   } 
               
               
                   
                  } 
               
               
                   
                  qidx=0, zidx=−1; 
               
               
                   
                  do{ 
               
               
                   
                   while(zdep_array[++zidx]!=−1); 
               
               
                   
                   if(queue[qidx]==−1){ 
               
               
                   
                    zdep_array[zidx]=zidx; 
               
               
                   
                   }else{ 
               
               
                   
                    first_zidx=zidx; 
               
               
                   
                    while(queue[qidx]!=−2){ 
               
               
                   
                     zdep_array[zidx]=queue[qidx]; 
               
               
                   
                     zidx=queue[qidx]; 
               
               
                   
                     ++qidx; 
               
               
                   
                    } 
               
               
                   
                    zdep_array[zidx]= first_zidx; 
               
               
                   
                    zidx= first_zidx; 
               
               
                   
                   } 
               
               
                   
                   ++qidx; 
               
               
                   
                  }while(qidx&lt;zdep_size); 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 reorder_flag being 0 indicates that zdep_array is not reordered, and reorder_flag being 1 indicates that the zdep_array is reordered. 
               
               
                   
                 signalled_flag being 0 indicates that content of location n of a queue is inferred, and signalled_flag being 1 indicates that the content of the location n of the queue is signaled. 
               
               
                   
                 qval_minus_one indicates that the content of the location n of the queue = qval_minus_one + 1. 
               
            
           
         
       
     
     In further embodiments, a 3D-Octree structure, a 3D-Tagtree structure, a 3D-Unitree structure and corresponding encoding methods may be used to encode a CU3D. 
     3D-Octree Coding 
     As shown in  FIG.  6   , an Octree  605  is a tree data structure in which each internal node  610  has exactly eight child nodes  615 . A 3D-Octree is used to partition a three-dimensional tensor  620  by recursively subdividing it along z, y, x axes into eight octants  625 . 
     In embodiments, a 3D-Octree structure may be used to represent significant (non-zero) states of coefficients in a CU3D. 
     A 3D-Octree for a CU3D is constructed as follows. A node value of 1 for a 3D-Octree location at a last depth indicates that a codebook index (if a codebook coding method is used) or a coefficient (if a direct quantization coding method is used) in a corresponding CU3D is other than zero. A node value of 0 for a 3D-Octree location at a bottom depth indicates that a codebook index or a coefficient in a corresponding CU3D is zero. A node value for a 3D-Octree location at another depth is defined as a maximum or minimum value of its eight child nodes. 
     After the 3D-Octree is constructed, all nodes are scanned using a predefined scan order to encode respective node values. 
     In one embodiment, starting from a top node, a depth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     In another embodiment, starting from a top node, a breadth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     Several node skipping methods are introduced to obtain a compact representation of a bitstream. 
     If a node value for a 3D-Octree location at another depth is defined as a maximum value of its eight child nodes, and if a value of a parent node is 0, scanning and encoding of its child nodes (and their child nodes) is skipped as their values should always be 0. If the value of the parent node is 1 and the values of all but last child nodes are all 0s, a last child node is still scanned, but the encoding of its value is skipped as it should always be 1. 
     If a node value for a 3D-Octree location at another depth is defined as a minimum value of its eight child nodes, and if a value of a parent node is 1, scanning and encoding of its child nodes (and their child nodes) is skipped as their values should always be 1. If the value of the parent node is 0 and the values of all but last child nodes are all ls, a last child node is still scanned, but the encoding of its value is skipped as it should always be 0. 
     An encoding_start_depth is defined to indicate a first depth that participates in an encoding process. When all nodes are scanned using a predefined scan order, encoding of a current node value is skipped if a depth of this node is above encoding_start_depth. 
     If a value of a node at a bottom depth is 1, its corresponding non-zero codebook index is encoded if a codebook coding method is used. If a direct quantization coding method is used, a sign bit of its corresponding non-zero coefficient is encoded, followed by an absolute value of the nonzero coefficient. 
     For some CU3Ds with different depths/heights/widths, there are not enough coefficients to construct a complete 3D-Octree in which all parent nodes have all eight child nodes available. Scanning and encoding of these non-exist child nodes are skipped if a parent node does not have all eight child nodes. 
     An example of a corresponding syntax table is listed below in Table 12: 
     
       
         
           
               
             
               
                 TABLE 12 
               
               
                   
               
             
            
               
                 octree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  proceed=nzflag=bmp[depth][z_idx][y_idx][x_idx]=1 
               
               
                  if(depth&gt;=start_depth){ 
               
               
                   if(!skip){ 
               
               
                    nzflag 
               
               
                    bmp[depth][z_idx][y_idx][x_idx]=nzflag 
               
               
                   } 
               
               
                   proceed=nzflag 
               
               
                  } 
               
               
                  if(proceed){ 
               
               
                   if(depth&lt;total_depth−1){ 
               
               
                    skip=false 
               
               
                    next_z_idx=(z_idx&lt;&lt;1) 
               
               
                    next_y_idx=(y_idx&lt;&lt;1) 
               
               
                    next_x_idx=(x_idx&lt;&lt;1) 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                  octree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in 
               
               
                 next depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in 
               
               
                 next depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and bit value of all 
               
               
                 other child nodes are zero 
               
               
                  octree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    return 
               
               
                   } 
               
               
                   if(codebook_size){ 
               
               
                    index 
               
               
                    map[z_idx][y_idx][x_idx]=index 
               
               
                   }else{ 
               
               
                    sign 
               
               
                    abs_q 
               
               
                    map[z_idx][y_idx][x_idx]=(sign?-int(abs_q):abs_q) 
               
               
                   } 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag non-zero flag of the index 
               
               
                 index index value 
               
               
                 sign sign bit of the quantized coefficient 
               
               
                 abs_q absolute value of the quantized coefficient 
               
            
           
         
       
     
     3D-Tagtree Coding 
     In embodiments, a 3D-Tagtree structure may be used to represent absolute values of coefficients in a CU3D. 
     A 3D-Tagtree for a CU3D is constructed as follows. A node value for a 3D-Tagtree location at a last depth indicates an absolute value of a codebook index (if a codebook coding method is used) or an absolute coefficient (if a direct quantization coding method is used) in a corresponding CU3D is other than zero. A node value for a 3D-Tagtree location at another depth is defined as a maximum or minimum value of its eight child nodes. 
     After the 3D-Tagtree is constructed, all nodes are scanned using a predefined scan order to encode respective node values. 
     In one embodiment, starting from a top node, a depth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     In another embodiment, starting from a top node, a breadth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     A node value is encoded if a corresponding node is a top node that does not have a parent node. For any child node, a difference between a parent node and the child node is encoded. Several node skipping methods are introduced to obtain a compact representation of a bitstream. 
     If a node value for a 3D-Tagtree location at another depth is defined as a maximum value of its eight child nodes, and if a value of a parent node is 0, scanning and encoding of its child nodes (and their child nodes) is skipped as their values should always be 0. If the value of a parent node is X and the values of all but last child nodes are smaller than X, a last child node is still scanned, but the encoding of its value is skipped as it should always be X. 
     If a node value for a 3D-Tagtree location at another depth is defined as a minimum value of its eight child nodes, and if a value of a parent node is X and values of all but last child nodes are bigger than X, a last child node is still scanned, but encoding of its value is skipped as it should always be X. 
     An encoding_start_depth is defined to indicate a first depth that participates in an encoding process. When all nodes are scanned using a predefined scan order, encoding of a current node value is skipped if a depth of this node is above encoding_start_depth. An actual value instead of a difference between a parent node and this current node is encoded if the depth of this node equals to encoding_start_depth. 
     If a value of a node at a bottom depth is not zero and a direct quantization coding method is used, a sign bit of its corresponding non-zero coefficient is encoded. 
     For some CU3Ds with different depths/heights/widths, there are not enough coefficients to construct a complete 3D-Tagtree in which all parent node have all eight child nodes available. Scanning and encoding of these non-exist child nodes are skipped if a parent node does not have all eight child nodes. 
     An example of a corresponding syntax table is listed below in Table 13: 
     
       
         
           
               
             
               
                 TABLE 13 
               
               
                   
               
             
            
               
                 tagtree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  proceed=nzflag=1 
               
               
                  if(depth) 
               
               
                   rmap[depth][z_idx][y_idx][x_idx]=rmap[depth−1] [z_idx&gt;&gt;1] 
               
               
                 [y_idx&gt;&gt;1][x_idx&gt;&gt;1] 
               
               
                  if(depth&gt;=start_depth){ 
               
               
                   if(!skip){ 
               
               
                    if(codebook_size){ 
               
               
                     if(depth==start_depth){ 
               
               
                      index=0 
               
               
                      nzflag_index 
               
               
                      if(nzflag_index) 
               
               
                       index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=index 
               
               
                     }else{ 
               
               
                      delta_index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]−delta_index 
               
               
                     } 
               
               
                    }else{ 
               
               
                     if(depth==start_depth){ 
               
               
                      abs_q=0 
               
               
                      nzflag_q 
               
               
                      if(nzflag_q) 
               
               
                       abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=abs_q 
               
               
                     }else{ 
               
               
                      delta_abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]−delta_abs_q 
               
               
                     } 
               
               
                    } 
               
               
                    nzflag=(rmap[depth][z_idx][y_idx][x_idx]!=0) 
               
               
                   } 
               
               
                   if(depth==total_depth−1&amp;nzflag&amp;&amp;codebook_size==0){ 
               
               
                    sign_q 
               
               
                    rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                     (sign?- 
               
               
                 int(rmap[depth][z_idx][y_idx][x_idx]):rmap[depth][z_idx][y_idx][x_idx]) 
               
               
                   } 
               
               
                   proceed=nzflag 
               
               
                  } 
               
               
                  if(proceed){ 
               
               
                   if(depth&lt;total_depth−1){ 
               
               
                    skip=false 
               
               
                    next_z_idx=(z_idx&lt;&lt;1) 
               
               
                    next_y_idx=(y_idx&lt;&lt;l) 
               
               
                    next_x_idx=(x_idx&lt;&lt;l) 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,skip) 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in 
               
               
                 next depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    return 
               
               
                   } 
               
               
                   map[z_idx][y_idx][x_idx]=rmap[depth][z_idx][y_idx][x_idx] 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag_index non-zero flag of the index 
               
               
                 index index value 
               
               
                 delta_index index value = parent node index value + delta_index 
               
               
                 nzflag_q non-zefo flag of the quantized coefficient 
               
               
                 abs_q absolute value of quantized coefficient 
               
               
                 delta_abs_q absolute value of quantized coefficient = parent node value + delta_abs_q 
               
               
                 sign_q sign bit of the quantized coefficient 
               
            
           
         
       
     
     Escape Coding 
     In a codebook coding method, an escape index is a special index in which a coefficient that is represented by the escape index can have different quantized coefficient values. The quantized coefficient values for all escape indices need to be encoded in a bitstream explicitly. 
     After 3D-Octree or 3D-Tagtree coding is completed, an escape coding procedure is launched if a codebook coding method is used. All codebook indices are scanned, and if an escape index is found, a non-zero flag of a corresponding quantized coefficient value is encoded. If the coefficient value is not zero, a sign bit followed by the absolute value of the quantized coefficient value are encoded. 
     An example of a corresponding syntax table is listed below in Table 14: 
     
       
         
           
               
               
             
               
                   
                 TABLE 14 
               
               
                   
                   
               
             
            
               
                   
                 escape( ){ 
               
               
                   
                  ...... 
               
               
                   
                  if(codebook_size) 
               
               
                   
                   for(z=0;z&lt;cu_cdepth;++z) 
               
               
                   
                    for(y=0;y&lt;cu_height;++y) 
               
               
                   
                     for(x=0;x&lt;cu_width:++x) 
               
               
                   
                      if(map[z][y][x]==codebook_size){ 
               
               
                   
                       q=0 
               
               
                   
                       nzflag 
               
               
                   
                       if(nzflag){ 
               
               
                   
                        sign 
               
               
                   
                        abs_q 
               
               
                   
                        q=(sign?-int(abs_q):abs_q) 
               
               
                   
                       } 
               
               
                   
                      } 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 nzflag non-zero flag 
               
               
                   
                 sign sign bit 
               
               
                   
                 abs_q quantized coefficient q=(sign?-int(abs_q):abs_q) 
               
            
           
         
       
     
     CTU3D Header 
     An encoding algorithm is applied to use both a 3D-Octree method and a 3D-tagteee method, and one mode with a better RD is chosen as the winner. A mode decision is recorded, and a ctu3d_map_mode_flag is defined to indicate if all CU3Ds in this CTU3D share the same map_mode. An enable_start_depth is also defined to indicate if CU3D encoding can start from a depth other than a bottom depth. 
     An example of a corresponding syntax table is listed below in Tables 15 and 16: 
     
       
         
           
               
               
             
               
                   
                 TABLE 15 
               
               
                   
                   
               
             
            
               
                   
                 ctu3d( ){ 
               
               
                   
                  ...... 
               
               
                   
                  ctu3d_header( ) 
               
               
                   
                  cu3d(0,0,0) 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
                 TABLE 16 
               
               
                   
                   
               
             
            
               
                   
                 ctu3d_header( ){ 
               
               
                   
                  ...... 
               
               
                   
                  ctu3d_map_mode_flag 
               
               
                   
                  if(!ctu3d_map_mode_flag) 
               
               
                   
                   map_mode 
               
               
                   
                  enable_start_depth 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
            
           
         
       
     
     ctu3d_map_mode_flag being 0 indicates that each cu3d use its own map_mode, and ctu3d_map_mode_flag being 1 indicates that all cu3d share one map_mode. 
     map_mode being 0 indicates that a 3D-Octree coding method is selected, and map_mode being 1 indicates that a 3D-Tagtree coding method is selected. 
     enable_start_depth being 0 indicates cu3d encoding always start at a bottom depth, and enable_start_depth being 1 indicates that cu3d encoding can start from a depth other than the bottom depth. 
     Mode Decision 
     An encoding algorithm is applied to use both a 3D-Octree method and a 3D-tagteee method, and one mode with a better RD is chosen as the winner. A mode decision is recorded in a CU3D syntax section. An encoding_start_depth is also recorded in the CU3D syntax section. 
     An example of a corresponding syntax table is listed below in Table 17: 
     
       
         
           
               
               
             
               
                   
                 TABLE 17 
               
               
                   
                   
               
             
            
               
                   
                 cu3d(depth,y_idx,x_idx){ 
               
               
                   
                  ...... 
               
               
                   
                  if(ctu3d_map_mode_flag) 
               
               
                   
                   map_mode 
               
               
                   
                  start_depth_delta=0 
               
               
                   
                  if(enable_start_depth) 
               
               
                   
                   start_depth_delta 
               
               
                   
                  start_depth=total_depth−1−start_depth_delta 
               
               
                   
                  if(map_mode==0) 
               
               
                   
                   octree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  elseif(map_mode==1) 
               
               
                   
                   tagtree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  escape( ) 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 map_mode being 0 indicates that a 3D-Octree method is selected, and map_mode being 1 indicates that a 3D-Tagtree method is selected. 
               
               
                   
                 start_depth_delta start_depth=total_depth−1−start_depth_delta 
               
            
           
         
       
     
     3D-Unitree Coding 
     If a node value for a 3D-Octree location at another depth is defined as a maximum value of its eight child nodes, the node value that equals to zero indicates that all its child nodes (and their child nodes, including nodes at a last depth) have identical values of zero. A value of zero of a node at the last depth indicates that a corresponding codebook index or coefficient is zero as well. 
     In embodiments, the above requirement is relaxed so that the aforementioned identical value can be any value, not just zero. 
     A 3D-Unitree for a CU3D is constructed as follows. A node value of 1 for a 3D-Unitree location at a depth other than a last depth indicates that its child nodes (and their child nodes, including nodes at the last depth) have non-unified (different) values. A node value of 0 for a 3D-Unitree location at a depth other than the last depth indicates that all its child nodes (and their child nodes, including nodes at the last depth) have unified (identical) values. 
     For nodes at a last depth that have the same parent node, a value of 1 is assigned to these nodes, if codebook indices (if a codebook coding method is used) or an absolute value of coefficients (if a direct quantization coding method is used) in a corresponding CU3D have non-unified values. A value of 0 is assigned to these nodes, if the codebook indices (if the codebook coding method is used) or the absolute value of the coefficients (if the direct quantization coding method is used) in the corresponding CU3D have unified values of any values including zero. 
     After the 3D-Unitree is constructed, all nodes are scanned using a predefined scan order to encode respective node values. 
     In one embodiment, starting from a top node, a depth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     In another embodiment, starting from a top node, a breadth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     To have a compact representation of a bitstream, after a value of a given node is encoded, its corresponding unified value is also encoded if the node value is zero, but scanning and encoding of its child nodes (and their child nodes) is skipped as their values should always equal to a unified value. 
     An encoding_start_depth is defined to indicate a first depth that participates in encoding process. When all nodes are scanned using a predefined scan order, encoding of a current node value is skipped if a depth of this node is above encoding_start_depth. 
     If a node at a last depth is reached, if encoding_start_depth is the last depth or its parent node value is not zero, its corresponding codebook index or coefficient is encoded into a bitstream. Otherwise, if a direct quantization coding method is used and its corresponding unified value is not zero, a sign bit of the coefficient is encoded into the bitstream. 
     For some CU3Ds with different depths/heights/widths, there are not enough coefficients to construct a complete 3D-Octree in which all parent nodes have all eight child nodes available. Scanning and encoding of these non-exist child nodes are skipped if a parent node does not have all eight child nodes. 
     An example of a corresponding syntax table is listed below in Table 18: 
     
       
         
           
               
             
               
                 TABLE 18 
               
               
                   
               
             
            
               
                 unitree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  if(depth&lt;total_depth−1){ 
               
               
                   nzflag=utree[depth][z_idx][y_idx][x_idx]=0 
               
               
                   if(depth&gt;=start_depth){ 
               
               
                    if(!skip){ 
               
               
                     nzflag 
               
               
                     utree[depth][z_idx][y_idx][x_idx]=nzflag 
               
               
                     if(!nzflag){ 
               
               
                      map_nzflag 
               
               
                      if(map_nzflag){ 
               
               
                       if(codebook_size){ 
               
               
                        cmap_val 
               
               
                        map_val=cmap_val 
               
               
                       }else{ 
               
               
                        qmap_val 
               
               
                        map_val=qmap_val 
               
               
                       } 
               
               
                      } 
               
               
                     } 
               
               
                    } 
               
               
                   } 
               
               
                   next_z_idx=(z_idx&lt;&lt;1) 
               
               
                   next_y_idx=(y_idx&lt;&lt;1) 
               
               
                   next_x_idx=(x_idx&lt;&lt;1) 
               
               
                   bskip=(depth&gt;=start_depth)?!nzflag:false; 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   return 
               
               
                  } 
               
               
                  if(start_depth=total_depth−1 || utree[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]){ 
               
               
                   map_nzflag 
               
               
                   if(map_nzflag){ 
               
               
                    if(codebook_size){ 
               
               
                     index 
               
               
                     map[z_idx][y_idx][x_idx]=index 
               
               
                    }else{ 
               
               
                     sign 
               
               
                     abs_q 
               
               
                     map[z_idx][y_idx][x_idx]=(sign?-int(abs_q):abs_q) 
               
               
                    } 
               
               
                   } 
               
               
                  }else{ 
               
               
                   sign=0 
               
               
                   if(!codebook_size &amp;&amp; map_val) 
               
               
                    map_sign 
               
               
                   map[z_idx][y_idx][x_idx]=(sign?-int(map_val):map_val) 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag utree node value 
               
               
                 map_nzflag non-zero flag of the unified value 
               
               
                 cmap_val index value of the unified value 
               
               
                 qmap_val absolute value of the unified quantized coefficient 
               
               
                 index index value 
               
               
                 sign sign bit of the quantized coefficient 
               
               
                 abs_q absolute value of the quantized coefficient 
               
               
                 map_sign sign bit of the unified quantized coefficient 
               
            
           
         
       
     
     CTU3D Header 
     An encoding algorithm is applied to use both a 3D-Octree method and a 3D-tagteee method, and a 3D-Unitree method is also applied when using the 3D-Octree method. One mode with a better RD is chosen as the winner. A mode decision is recorded, and a ctu3d_map_mode_flag is defined to indicate if all CU3Ds in this CTU3D share the same map_mode. An enable_start_depth is also defined to indicate if CU3D encoding can start from a depth other than a bottom depth. 
     Mode Decision 
     An encoding algorithm is applied to use both a 3D-Octree method and a 3D-tagteee method, and a 3D-Unitree method is also applied when using the 3D-Octree method. One mode with a better RD is chosen as the winner. A mode decision is recorded in a CU3D syntax section. 
     An example of a corresponding syntax table is listed below in Table 19: 
     
       
         
           
               
               
             
               
                   
                 TABLE 19 
               
               
                   
                   
               
             
            
               
                   
                 cu3d(depth,y_idx,x_idx){ 
               
               
                   
                  ...... 
               
               
                   
                  if(ctu3d_map_mode_flag) 
               
               
                   
                   map_mode 
               
               
                   
                  start_depth_delta=0 
               
               
                   
                  if(enable_start_depth) 
               
               
                   
                   start_depth_delta 
               
               
                   
                  start_depth=total_depth−1−start_depth_delta 
               
               
                   
                  if(map_mode==0){ 
               
               
                   
                   uni_mode 
               
               
                   
                   if(uni_mode) 
               
               
                   
                    unitree3d(start_depth,0,0,0,0,false) 
               
               
                   
                   else 
               
               
                   
                    octree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  }else if(map_mode==1) 
               
               
                   
                   tagtree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 map_mode being 0 indicates that a 3D-Octree method is selected, and map_mode being 1 indicates that a 3D-Tagtree method is selected. 
               
               
                   
                 start_depth_delta start_depth=total_depth−1−start_depth_delta 
               
               
                   
                 uni_mode being 0 indicates that a 3D-Octree method is selected, and uni_mode being 1 indicates that a 3D-Unitree method is selected. 
               
            
           
         
       
     
     Modification in 3D-Tree Coding Method 
     In an original disclosure, tree[z][y][x] is used to access location [z][y][x] at a last depth of a 3D-Tree (3D-Octree, 3D-Unitree or 3D-Tagtree), and map[z][y][x] is used to access an index map of a CU3D. 
     Because zdep_array[n] represents an original n th  2D plane before a kernel plane reorder operation, the kernel plane reorder operation requires that a z-axis is indexed by the zdep_array[n] instead of n. 
     In embodiments, a z-axis access for an encoding method is modified so that tree[zdep_array[z]][y][x] is used to access location [z][y][x] at a last depth of a 3D-Tree (3D-Octree, 3D-Unitree or 3D-Tagtree), and map[zdep_array[z]][y][x] is used to access an index map of a CU3D. 
     An example of a corresponding syntax table modification is listed below in Tables 20 and 21: 
     
       
         
           
               
             
               
                 TABLE 20 
               
               
                   
               
               
                 Original syntax table 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 unitree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  if(depth&lt;total_depth−1){ 
               
               
                   nzflag=utree[depth][z_idx][y_idx][x_idx]=0 
               
               
                   if(depth&gt;=start_depth){ 
               
               
                    if(!skip){ 
               
               
                     nzflag 
               
               
                     utree[depth][z_idx][y_idx][x_idx]=nzflag 
               
               
                     if(!nzflag){ 
               
               
                      map_nzflag 
               
               
                      if(map_nzflag){ 
               
               
                       if(codebook_size){ 
               
               
                        cmap_val 
               
               
                        map_val=cmap_val 
               
               
                       }else{ 
               
               
                        qmap_val 
               
               
                        map_val=qmap_val 
               
               
                       } 
               
               
                      } 
               
               
                     } 
               
               
                    } 
               
               
                   } 
               
               
                   next_z_idx=(z_idx&lt;&lt;1) 
               
               
                   next_y_idx=(y_idx&lt;&lt;1) 
               
               
                   next_x_idx=(x_idx&lt;&lt;1) 
               
               
                   bskip=(depth&gt;=start_depth)?!nzflag:false; 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   return 
               
               
                  } 
               
               
                  if(start_depth=total_depth−1 || utree[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]){ 
               
               
                   map_nzflag 
               
               
                   if(map_nzflag){ 
               
               
                    if(codebook_size){ 
               
               
                     index 
               
               
                     map[z_idx][y_idx][x_idx]=index 
               
               
                    }else{ 
               
               
                     sign 
               
               
                     abs_q 
               
               
                     map[z_idx][y_idx][x_idx]=(sign?-int(abs_q):abs_q) 
               
               
                    } 
               
               
                   } 
               
               
                  }else{ 
               
               
                   sign=0 
               
               
                   if(!codebook_size &amp;&amp; map_val) 
               
               
                    map_sign 
               
               
                   map[z_idx][y_idx][x_idx]=(sign?-int(map_val):map_val) 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     To address a 2D plane at a last depth, a variable zs_idx is defined as new z_idx. 
     
       
         
           
               
             
               
                 TABLE 21 
               
               
                   
               
               
                 Modified syntax table 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 unitree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  zs_idx=(depth==total_depth−1)?zdep_array[z_idx]:z_idx 
               
               
                  if(depth&lt;total_depth−1){ 
               
               
                   nzflag=utree[depth][z_idx][y_idx][x_idx]=0 
               
               
                   if(depth&gt;=start_depth){ 
               
               
                    if(!skip){ 
               
               
                     nzflag 
               
               
                     utree[depth][zs_idx][y_idx][x_idx]=nzflag 
               
               
                     if(!nzflag){ 
               
               
                      map_nzflag 
               
               
                      if(map_nzflag){ 
               
               
                       if(codebook_size){ 
               
               
                        cmap_val 
               
               
                        map_val=cmap_val 
               
               
                       }else{ 
               
               
                        qmap_val 
               
               
                        map_val=qmap_val 
               
               
                       } 
               
               
                      } 
               
               
                     } 
               
               
                    } 
               
               
                   } 
               
               
                   next_z_idx=(z_idx&lt;&lt;1) 
               
               
                   next_y_idx=(y_idx&lt;&lt;1) 
               
               
                   next_x_idx=(x_idx&lt;&lt;1) 
               
               
                   bskip=(depth&gt;=start_depth)?!nzflag:false; 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,bskip) 
               
               
                   if(location [next_z_idx+1][next_y_idx+1][next_x_idx] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,bskip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth) 
               
               
                  unitree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,bskip) 
               
               
                   return 
               
               
                  } 
               
               
                  if(start_depth=total_depth−1 || utree[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]){ 
               
               
                   map_nzflag 
               
               
                   if(map_nzflag){ 
               
               
                    if(codebook_size){ 
               
               
                     index 
               
               
                     map[zs_idx][y_idx][x_idx]=index 
               
               
                    }else{ 
               
               
                     sign 
               
               
                     abs_q 
               
               
                     map[zs_idx][y_idx][x_idx]=(sign?-int(abs_q):abs_q) 
               
               
                    } 
               
               
                   } 
               
               
                  }else{ 
               
               
                   sign=0 
               
               
                   if(!codebook_size &amp;&amp; map_val) 
               
               
                    map_sign 
               
               
                   map[zs_idx][y_idx][x_idx]=(sign?-int(map_val):map_val) 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     In further embodiments, an additional 3D-Tagtree mode, a corresponding flag and a corresponding encoding method are used to encode a CU3D. 
     3D-Tagtree Mode Decision 
     As described above, one version of a 3D-Tagtree is that a node value for a 3D-Tagtree location is defined as a maximum value of its eight child nodes (tagtree3d). Another version of a 3D-Tagtree is that a node value for a 3D-Tagtree location is defined as a minimum value of its eight child nodes (tagtree3dm). One of the two versions of a 3D-Tagteee is used to encode a CU3D. 
     In alternative embodiments, both versions of a 3D-Tagtree is used to encode the CU3D, and a candidate with a best RD is chosen as the winner. 
     In detail, an encoding algorithm may use both a 3D-Octree method and a 3D-tagteee method. A 3D-Unitree method is applied when the 3D-Octree method is used, and both versions of the 3D-Tagteee method is applied when the 3D-Tagtree method is used. A mode with a better RD is chosen as the winner, and a mode decision is recorded in a cu3d syntax section. 
     An example of a corresponding syntax table for a tagtree3d mode decision is listed below in Table 22: 
     
       
         
           
               
               
             
               
                   
                 TABLE 22 
               
               
                   
                   
               
             
            
               
                   
                 cu3d(depth,y_idx,x_idx){ 
               
               
                   
                  ...... 
               
               
                   
                  if(ctu3d_map_mode_flag) 
               
               
                   
                   map_mode 
               
               
                   
                  start_depth_delta=0 
               
               
                   
                  if(enable_start_depth) 
               
               
                   
                   start_depth_delta 
               
               
                   
                  start_depth=total_depth−1−start_depth_delta 
               
               
                   
                  if(map_mode==0){ 
               
               
                   
                   uni_mode 
               
               
                   
                   if(uni_mode) 
               
               
                   
                    unitree3d(start_depth,0,0,0,0,false) 
               
               
                   
                   else 
               
               
                   
                    octree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  }else if(map_mode==1){ 
               
               
                   
                   tgt_mode 
               
               
                   
                   if(tgt_mode) 
               
               
                   
                    tagtree3dm(start_depth,0,0,0,0,false) 
               
               
                   
                   else 
               
               
                   
                    tagtree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  } 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 map_mode being 0 indicates that an Octree method is selected, and map-mode being 1 indicates that a Tagtree3d method is selected. 
               
               
                   
                 start_depth_delta start_depth=total_depth−1−start_depth_delta 
               
               
                   
                 uni_mode being 0 indicates that the Octree method is selected, and uni_mode being 1 indicates that an Unitree3d method is selected. 
               
               
                   
                 tgt_mode being 0 indicates that a Tagtree3d method version 1 is selected, and tgt_mode being 1 indicates that a Tagtree3d method version 2 is selected. 
               
            
           
         
       
     
     An example of a corresponding syntax table for a tagtree3d coding method 1 is listed below in Table 23: 
     
       
         
           
               
             
               
                 TABLE 23 
               
               
                   
               
             
            
               
                 tagtree3d(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  proceed=nzflag=1 
               
               
                  if(depth) 
               
               
                   rmap[depth][z_idx][y_idx][x_idx]=rmap[depth−1] [z_idx&gt;&gt;1] 
               
               
                 [y_idx&gt;&gt;1][x_idx&gt;&gt;1] 
               
               
                  if(depth&gt;=start_depth){ 
               
               
                   if(!skip){ 
               
               
                    if(codebook_size){ 
               
               
                     if(depth==start_depth){ 
               
               
                      index=0 
               
               
                      nzflag_index 
               
               
                      if(nzflag_index) 
               
               
                       index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=index 
               
               
                     }else{ 
               
               
                      delta_index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]−delta_index 
               
               
                     } 
               
               
                    }else{ 
               
               
                     if(depth==start_depth){ 
               
               
                      abs_q=0 
               
               
                      nzflag_q 
               
               
                      if(nzflag_q) 
               
               
                       abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=abs_q 
               
               
                     }else{ 
               
               
                      delta_abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]−delta_abs_q 
               
               
                     } 
               
               
                    } 
               
               
                    nzflag=(rmap[depth][z_idx][y_idx][x_idx]!=0) 
               
               
                   } 
               
               
                   if(depth==total_depth−1&amp;nzflag&amp;&amp;codebook_size==0){ 
               
               
                    sign_q 
               
               
                    rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                     (sign?- 
               
               
                 int(rmap[depth][z_idx][y_idx][x_idx]):rmap[depth][z_idx][y_idx][x_idx]) 
               
               
                   } 
               
               
                   proceed=nzflag 
               
               
                  } 
               
               
                  if(proceed){ 
               
               
                   if(depth&lt;total_depth−1){ 
               
               
                    skip=false 
               
               
                    next_z_idx=(z_idx&lt;&lt;1) 
               
               
                    next_y_idx=(y_idx&lt;&lt;1) 
               
               
                    next_x_idx=(x_idx&lt;&lt;1) 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next 
               
               
                 depth) 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,skip) 
               
               
                    if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next 
               
               
                 depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,skip) 
               
               
                    } 
               
               
                    if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in 
               
               
                 next depth){ 
               
               
                     if(depth&gt;=start_depth) 
               
               
                      skip=this is the last child node and value of all other 
               
               
                 child nodes are smaller than value of parent node 
               
               
                  tagtree3d(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,skip) 
               
               
                    } 
               
               
                    return 
               
               
                   } 
               
               
                   map[z_idx][y_idx][x_idx]=rmap[depth][z_idx][y_idx][x_idx] 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag_index non-zefo flag of the index 
               
               
                 index index value 
               
               
                 delta_index index value = parent node index value − delta_index 
               
               
                 nzflag_q non-zefo flag of the quantized coefficient 
               
               
                 abs_q absolute value of quantized coefficient 
               
               
                 delta_abs_q absolute value of quantized coefficient = parent node value − delta_abs_q 
               
               
                 sign_q sign bit of the quantized coefficient 
               
            
           
         
       
     
     An example of a corresponding syntax table for a tagtree3d coding method 2 is listed below in Table 24: 
     
       
         
           
               
             
               
                 TABLE 24 
               
               
                   
               
             
            
               
                 tagtree3dm(start_depth,depth,z_idx,y_idx,x_idx,skip){ 
               
               
                  ...... 
               
               
                  if(depth) 
               
               
                   rmap[depth][z_idx][y_idx][x_idx]=rmap[depth−1] [z_idx&gt;&gt;1] 
               
               
                 [y_idx&gt;&gt;1][x_idx&gt;&gt;1] 
               
               
                  if(depth&gt;=start_depth){ 
               
               
                   if(!skip){ 
               
               
                    if(codebook_size){ 
               
               
                     if(depth==start_depth){ 
               
               
                      index=0 
               
               
                      nzflag_index 
               
               
                      if(nzflag_index) 
               
               
                       index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=index 
               
               
                     }else{ 
               
               
                      delta_index 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]+delta_index 
               
               
                     } 
               
               
                    }else{ 
               
               
                     if(depth==start_depth){ 
               
               
                      abs_q=0 
               
               
                      nzflag_q 
               
               
                      if(nzflag_q) 
               
               
                       abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]=abs_q 
               
               
                     }else{ 
               
               
                      delta_abs_q 
               
               
                      rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                       rmap[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1]+delta_abs_q 
               
               
                     } 
               
               
                    } 
               
               
                   } 
               
               
                   nzflag=(rmap[depth][z_idx][y_idx][x_idx]!=0) 
               
               
                   if(depth==total_depth−1&amp;nzflag&amp;&amp;codebook_size==0){ 
               
               
                    sign_q 
               
               
                    rmap[depth][z_idx][y_idx][x_idx]= 
               
               
                     (sign?- 
               
               
                 int(rmap[depth][z_idx][y_idx][x_idx]):rmap[depth][z_idx][y_idx][x_idx]) 
               
               
                   } 
               
               
                  } 
               
               
                  if(depth&lt;total_depth−1){ 
               
               
                   next_z_idx=(z_idx&lt;&lt;1) 
               
               
                   next_y_idx=(y_idx&lt;&lt;1) 
               
               
                   next_x_idx=(x_idx&lt;&lt;1) 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx] exist in next depth){ 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx][ next_y_idx][ next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx,next_y_idx,next_x_idx+1,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx][ next_y_idx+1][ next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx,next_y_idx+1,next_x_idx+1,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx] exist in next depth) 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx,skip) 
               
               
                   if(location [next_z_idx+1][ next_y_idx][ next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx+1,next_y_idx,next_x_idx+1,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx,skip) 
               
               
                   } 
               
               
                   if(location [next_z_idx+1][ next_y_idx+1][ next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     skip=this is the last child node and value of all other child 
               
               
                 nodes are bigger than  value of parent node 
               
               
                  tagtree3dm(start_depth,depth+1,next_z_idx+1,next_y_idx+1,next_x_idx+1,skip) 
               
               
                   } 
               
               
                   return 
               
               
                  } 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag_index non-zefo flag of the index 
               
               
                 index index value 
               
               
                 delta_index index value = parent node index value + delta_index 
               
               
                 nzflag_q non-zefo flag of the quantized coefficient 
               
               
                 abs_q absolute value of quantized coefficient 
               
               
                 delta_abs_q absolute value of quantized coefficient = parent node value + delta_abs_q 
               
               
                 sign_q sign bit of the quantized coefficient 
               
            
           
         
       
     
     In further embodiments, an additional 3D-Tagtree mode, a corresponding flag and a corresponding encoding method are used to encode a CU3D. 
     Combined 3D Unitree and Tagtree Coding 
     Unified coefficients and non-unified coefficients can co-exist in one CU3D, so a combination of a 3D-Unitree coding method and a 3D-Tagtree coding method can be used to encode such a CU3D. 
     Both a 3D-Tagtree and a 3D-Unitree are constructed using 3D-Tagtree and 3D-Unitree construction methods. After the 3D-Tagtree and 3D-Unitree are constructed, all nodes are scanned using a predefined scan order to encode respective node values. 
     In one embodiment, starting from a top node, a depth-first-search is used to scan all child nodes. A scan order for the child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     In another embodiment, starting from a top node, a breadth-first-search is used to scan all child nodes. A scan order for the child nodes that share the same parent node can be defined arbitrarily, such as (0,0,0)-&gt;(0,0,1)-&gt;(0,1,0)-&gt;(0,1,1)-&gt;(1,0,0)-&gt;(1,0,1)-&gt;(1,1,0)-&gt;(1,1,1). 
     To have a compact representation of a bitstream, a value of a given node from a 3D-Unitree is encoded first. If the 3D-Unitree node value is zero, its corresponding unified value is encoded, and scanning and encoding of its child nodes (and their child nodes) are skipped as their values should always be equal to a unified value. If the 3D-Unitree node value is not zero, a 3D-Tagtree coding method is used to encode either a 3D-Tagtree value if a node is a top node that does not have a parent node or a difference of 3D-Tagtree values between a parent node and this child node. Node skipping methods introduced in the above Tagtree coding section are adopted as well. 
     An encoding_start_depth is defined to indicate a first depth that participate in an encoding process. When all nodes are scanned using a predefined scan order, encoding of a current node value is skipped if a depth of this node is above encoding_start_depth. 
     If a node at a last depth is reached, and if a value of a node at a bottom depth is not zero and a direct quantization coding method is used, a sign bit of its corresponding non-zero coefficient is encoded. 
     For some CU3Ds with different depths/heights/widths, there are not enough coefficients to construct a complete 3D-Octree in which all parent node have all eight child nodes available. Scanning and encoding of these non-exist child nodes are skipped if a parent node does not have all eight child nodes. 
     Combined 2D Unitree and Tagtree Coding 
     In embodiments, the aforementioned combined 3D Unitree and Tagtree coding is used to encode a CU3D. In further embodiments, instead of using a 3D-Octree structure, a sequence of 2D planes is encoded independently, and for each independent 2D plane, a Quadtree structure is used to construct a 2D-Unitree and a 2D-Tagtree. 
     An Octree structure is used to represent a 3D-Tagtree and a 3D-Unitree. In these embodiments, a Quadtree structure is used to represent a 2D-Tagtree and a 2D-Unitree. The 2D-Tagtree and 2D-Unitree are constructed using the same principle as a 3D-Tagtree and a 3D-Unitree. 
     After the 2D-Tagtree and 2D-Unitree are constructed, all nodes are scanned using a predefined scan order to encode respective node values. 
     In one embodiment, starting from a top node, a depth-first-search is used to walk through all child nodes. A scan order for the child nodes that share the same parent node can be defined arbitrarily, such as (0,0)-&gt;(0,1)-&gt;(1,0)-&gt;(1,1). 
     In another embodiment, starting from a top node, a breadth-first-search is used to scan all child nodes. A scan order for child nodes that share the same parent node can be defined arbitrarily, such as (0,0)-&gt;(0,1)-&gt;(1,0)-&gt;(1,1). 
     To have a compact representation of a bitstream, a value of a given node from Unitree first is encoded first. If the Unitree node value is zero, its corresponding unified value is encoded, and scanning and encoding of its child nodes (and their child nodes) are skipped as their value should always equal to a unified value. If the Unitree node value is not zero, a Tagtree coding method is used to encode either a Tagtree value if a node is top node that does not have a parent node or a difference of Tagtree values between a parent node and this child node. Node skipping methods introduced in the above Tagtree coding section are adopted as well. 
     An encoding_start_depth is defined to indicate a first depth that participate in encoding process. When all nodes are scanned using a predefined scan order, encoding of a current node value is skipped if a depth of this node is above encoding_start_depth. 
     If a node at a last depth is reached, and if a value of a node at a bottom depth is not zero and a direct quantization coding method is used, a sign bit of its corresponding non-zero coefficient is encoded. 
     After encoding of one 2D plane is completed, a next 2D plane starts to be encoded. The process is repeated until a last 2D plane is encoded. 
     For some CU3Ds with different heights/widths, there are not enough coefficients to construct a complete 2D-Quadtree in which all parent node have all four child nodes available. Scanning and encoding of these non-exist child nodes are skipped if a parent node does not have all four child nodes. 
     Mode Decision 
     Along with other coding methods, a combined Unitree and Tagtree coding method may be used to encode a CU3D, and a candidate with the best RD among all of the coding methods is chosen as the winner. 
     In detail, an encoding algorithm uses both a 3D-Octree method and a 3D-Tagtree method. A 3D-Unitree method is applied when the 3D-Octree method is used, and a Combined-Unitree-Tagtree method is applied when the 3D-Tagtree method is used. A mode with a better RD among the 3D-Octree method and the 3D-Tagtree method is chosen as the winner, and a mode decision is recorded in a cu3d syntax section. 
     An example of a corresponding syntax table for a tagtree3d mode decision is listed below in Table 25: 
     
       
         
           
               
               
             
               
                   
                 TABLE 25 
               
               
                   
                   
               
             
            
               
                   
                 cu3d(depth,y_idx,x_idx){ 
               
               
                   
                  ...... 
               
               
                   
                  if(ctu3d_map_mode_flag) 
               
               
                   
                   map_mode 
               
               
                   
                  start_depth_delta=0 
               
               
                   
                  if(enable_start_depth) 
               
               
                   
                   start_depth_delta 
               
               
                   
                  start_depth=total_depth−1−start_depth_delta 
               
               
                   
                  cbook_esc_mode=0 
               
               
                   
                  if(enable_escape_reorder) 
               
               
                   
                   cbook_esc_mode 
               
               
                   
                  if(map_mode==0){ 
               
               
                   
                   uni_mode 
               
               
                   
                   if(uni_mode) 
               
               
                   
                    unitree3d(start_depth,0,0,0,0,false) 
               
               
                   
                   else 
               
               
                   
                    octree3d(start_depth,0,0,0,0,false) 
               
               
                   
                  }else if(map_mode==1){ 
               
               
                   
                   tgt_mode 
               
               
                   
                   if(tgt_mode) 
               
               
                   
                    tagtree3d(start_depth,0,0,0,0,false) 
               
               
                   
                   else 
               
               
                   
                    for(z=0;z&lt;zdep_size;++z) 
               
               
                   
                     uni_tagtree3d(start_depth,0,z,0,0,false,false) 
               
               
                   
                  } 
               
               
                   
                  ...... 
               
               
                   
                 } 
               
               
                   
                   
               
               
                   
                 map_mode being 0 indicates that an Octree method is selected, and map_mode being 1 indicates that a Tagtree3d method is selected. 
               
               
                   
                 start_depth_delta start_depth=total_depth−1−start_depth_delta 
               
               
                   
                 cbook_esc_mode being 0 indicates that an escape is not reordered, and cbook_esc_mode being 1 indicates that the escape is reordered. 
               
               
                   
                 uni_mode being 0 indicates that the Octree method is selected, and uni_mode being 1 indicates that a Unitree3d method is selected. 
               
               
                   
                 tgt_mode being 0 indicates that a uni_tgtree3d method is selected, and tgt_mode being 1 indicates that the Tagtree3d method is selected. 
               
            
           
         
       
     
     An example of a corresponding syntax table for a uni_tagtree3d coding method is listed below in Table 26: 
     
       
         
           
               
             
               
                 TABLE 26 
               
               
                   
               
             
            
               
                 uni_tagtree3d(start_depth,depth,z_idx,y_idx,x_idx,uni_skip,tgt_skip){ 
               
               
                  ...... 
               
               
                  zs_idx=(depth==total_depth−1)?zdep_array[z_idx]:z_idx 
               
               
                  uniflag=tgt[depth][zs_idx][y_idx][x_idx][0]=0 
               
               
                  if(depth) 
               
               
                   tgt[depth][zs_idx][y_idx][x_idx][1]=tgt[depth− 
               
               
                 1][z_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1][1] 
               
               
                  if(depth&gt;=start_depth){ 
               
               
                   if(!uni_skip){ 
               
               
                    if(start_depth&lt;total_depth−1 || ((y_idx&amp;1)==0 &amp;&amp; (x_idx&amp;1)==0)){ 
               
               
                     nzflag_uni 
               
               
                    }else{ 
               
               
                     nzflag_uni=tgt[depth][zs_idx][y_idx&amp;(~1)][x_idx&amp;(~1)][0] 
               
               
                  tgt[depth][zs_idx][y_idx][x_idx][1]=abs(tgt[depth][zs_idx][y_idx&amp;(~1)][x_idx&amp;(~1)][1]) 
               
               
                    } 
               
               
                    uniflag=tgt[depth][zs_idx][y_idx][x_idx]=nzflag_uni 
               
               
                    if(!tgt_skip &amp;&amp; 
               
               
                     (start_depth&lt;total_depth−1 || (((y_idx&amp;1)==0 &amp;&amp; (x_idx&amp;1)==0) || 
               
               
                 uniflag))) { 
               
               
                     if(codebook_size){ 
               
               
                      if(depth==start_depth){ 
               
               
                       index=0 
               
               
                       nzflag_index 
               
               
                       if(nzflag_index) 
               
               
                        index 
               
               
                       tgt[depth][zs_idx][y_idx][x_idx][1]=index 
               
               
                      }else{ 
               
               
                       delta_index 
               
               
                       tgt[depth][zs_idx][y_idx][x_idx][1]= 
               
               
                        tgt[depth− 
               
               
                  1][zs_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1][1]+delta_index 
               
               
                      } 
               
               
                     }else{ 
               
               
                      if(depth==start_depth){ 
               
               
                       abs_q=0 
               
               
                       nzflag_q 
               
               
                       if(nzflag_q) 
               
               
                        abs_q 
               
               
                       tgt[depth][zs_idx][y_idx][x_idx][1]=abs_q 
               
               
                      }else{ 
               
               
                       delta_abs_q 
               
               
                       tgt[depth][zs_idx][y_idx][x_idx][1]= 
               
               
                        tgt[depth− 
               
               
                  1][zs_idx&gt;&gt;1][y_idx&gt;&gt;1][x_idx&gt;&gt;1][1]+delta_abs_q 
               
               
                      } 
               
               
                     } 
               
               
                    } 
               
               
                   } 
               
               
                   nzflag=(tgt[depth][zs_idx][y_idx][x_idx][1]!=0) 
               
               
                   if(depth==total_depth−1&amp;nzflag&amp;&amp;codebook_size==0){ 
               
               
                    sign_q 
               
               
                    tgt[depth][zs_idx][y_idx][x_idx][1]= 
               
               
                     (sign?- 
               
               
                 int(tgt[depth][zs_idx][y_idx][x_idx][1]):tgt[depth][zs_idx][y_idx][x_idx][1]) 
               
               
                   } 
               
               
                  } 
               
               
                  if(depth&lt;total_depth−1){ 
               
               
                   next_y_idx=(y_idx&lt;&lt;1) 
               
               
                   next_x_idx=(x_idx&lt;&lt;1) 
               
               
                   uskip=(depth&gt;=start_depth)?!uniflag:false 
               
               
                   if(location [z_idx][next_y_idx][next_x_idx] exist in next depth){ 
               
               
                  uni_tagtree3d(start_depth,depth+1,z_idx,next_y_idx,next_x_idx,uskip,false) 
               
               
                   } 
               
               
                   if(location [z_idx][next_y_idx][next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     tskip=this is the last child node and value of all other child nodes 
               
               
                 are bigger than value of parent node 
               
               
                  uni_tagtree3d(start_depth,depth+1,z_idx,next_y_idx,next_x_idx+1,uskip,tskip) 
               
               
                   } 
               
               
                   if(location [z_idx][next_y_idx+1][next_x_idx] exist in next depth){ 
               
               
                    if(depth&gt;=start_depth) 
               
               
                     tskip=this is the last child node and value of all other child nodes 
               
               
                 are bigger than value of parent node 
               
               
                  uni_tagtree3d(start_depth,depth+1,z_idx,next_y_idx+1,next_x_idx,uskip,tskip) 
               
               
                   } 
               
               
                   if(location [z_idx][next_y_idx+1][next_x_idx+1] exist in next depth){ 
               
               
                    if(depth&gt;=start depth) 
               
               
                     tskip=this is the last child node and value of all other child nodes 
               
               
                 are bigger than value of parent node 
               
               
                  uni_tagtree3d(start_depth,depth+1,z_idx,next_y_idx+1,next_x_idx+1,uskip,tskip) 
               
               
                   } 
               
               
                   return 
               
               
                  } 
               
               
                  map[zs_idx][y_idx][x_idx]=tgt[depth][zs_idx][y_idx][x_idx][1] 
               
               
                  ...... 
               
               
                 } 
               
               
                   
               
               
                 nzflag_uni utree node value 
               
               
                 nzflag_index non-zero flag of index 
               
               
                 index index value 
               
               
                 delta_index index value = parent node index value + delta_index 
               
               
                 nzflag_q non-zero flag of quantized coefficient 
               
               
                 abs_q absolute value of quantized coefficient 
               
               
                 delta_abs_q absolute value of quantized coefficient = parent node value + delta_abs_q 
               
               
                 sign_q sign bit of quantized coefficient 
               
            
           
         
       
     
       FIG.  7    is a flowchart of a method  700  of adaptive block partitioning for neural network model compression, according to embodiments. In some implementations, one or more process blocks of  FIG.  7    may be performed by the platform  220 . In some implementations, one or more process blocks of  FIG.  7    may be performed by another device or a group of devices separate from or including the platform  220 , such as the user device  210 . 
     As shown in  FIG.  7   , in operation  710 , the method  700  includes reshaping a four-dimensional (4D) parameter tensor of a neural network into 3D parameter tensor of the neural network, the 3D parameter tensor including a convolution kernel size, an input feature size and an output feature size 
     In operation  720 , the method  700  includes partitioning the 3D parameter tensor along a plane that is formed by the input feature size and the output feature size, into 3D coding tree units (CTU3Ds). 
     In operation  730 , the method  700  includes partitioning each of the CTU3Ds into a plurality of 3D coding units (CU3Ds) recursively until a maximum depth, using a quad-tree. Note that the maximum depth can be zero, which indicates that no partition of CTU3D to CU3Ds is needed. 
     In operation  740 , the method  700  includes constructing a 3D-Tree for each of the plurality of CU3Ds. 
     In operation  750 , the method  700  includes entropy encoding each of a plurality of values of a plurality of nodes of the 3D-Tree. 
     The 3D-Tree may be a 3D-Octree. Based on a codebook index or coefficient of a respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the 3D-Octree may be 1. Based on the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Octree may be 0. A value of a parent node at another depth of the 3D-Octree may be a maximum or minimum value of child nodes of the parent node. 
     The 3D-Tree may be a 3D-Tagtree. Based on an absolute value of a codebook index or coefficient of a respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Tagtree may be 0. A value of a parent node at another depth of the 3D-Tagtree may be a maximum value of child nodes of the parent node. 
     The method  700  may further include constructing another 3D-Tagtree for each of the plurality of CU3Ds. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the other 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the other 3D-Tagtree may be 0. A value of a parent node at another depth of the other 3D-Tagtree may be a minimum value of child nodes of the parent node of the other 3D-Tagtree. The method  700  may further include entropy encoding each of a plurality of values of a plurality of nodes of the other 3D-Tagtree, and selecting the entropy-encoded plurality of values of the plurality of nodes of one of the 3D-Tagtree and the other 3D-Tagtree that has a higher rate distortion. 
     The 3D-Tree may be a 3D-Unitree. Based on child nodes of a parent node included in a depth other than a last depth of the 3D-Unitree having different values, a value of the parent node may be 1. Based on the child nodes of the parent node having identical values, the value of the parent node may be 0. Based on codebook indices or absolute values of coefficients of a respective one of the plurality of CU3Ds having different values, a value of a node included in the last depth may be 1. Based on the codebook indices or the absolute values of the coefficients of the respective one of the plurality of CU3Ds having identical values, the value of the node included in the last depth may be 0. 
     The method  700  may further include constructing a 3D-Tagtree for each of the plurality of CU3Ds. Based on an absolute value of a codebook index or coefficient of the respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Tagtree may be 0. A value of a parent node at another depth of the 3D-Tagtree may be a minimum value of child nodes of the parent node of the 3D-Tagtree. The entropy encoding each of the plurality of values of the plurality of nodes of the 3D-Tree may include entropy encoding a value of a current node of the 3D-Unitree, based on the entropy-encoded value of the current node of the 3D-Unitree being zero, skipping entropy encoding of values of child nodes of the current node of the 3D-Unitree, based on the entropy-encoded value of the current node of the 3D-Unitree being other than zero and the current node of the 3D-Unitree being a top node, entropy encoding a value of a current node of the 3D-Tagtree corresponding to the current node of the 3D-Unitree, and based on the entropy-encoded value of the current node of the 3D-Unitree being other than zero and the current node of the 3D-Unitree not being the top node, entropy encoding a difference between the value of the current node of the 3D-Tagtree and a value of a parent node of the current node of the 3D-Tagtree. 
     The method  700  may further include reordering two-dimensional (2D) planes in at least one of the CTU3Ds and the plurality of CU3Ds, along an axis that is formed by the convolutional kernel size, to generate reorder indices respectively corresponding to orders of the 2D planes, and encoding the reorder indices in a header of the at least one of the CTU3Ds and the plurality of CU3Ds. 
     Although  FIG.  7    shows example blocks of the method  700 , in some implementations, the method  700  may include additional blocks, fewer blocks, different blocks, or differently arranged blocks than those depicted in  FIG.  7   . Additionally, or alternatively, two or more of the blocks of the method  700  may be performed in parallel. 
       FIG.  8    is a diagram of an apparatus  800  for adaptive block partitioning for neural network model compression, according to embodiments. As shown in  FIG.  8   , the apparatus  800  includes reshaping code  810 , first partitioning code  820 , second partitioning code  830 , first constructing code  840  and first entropy encoding code  850 . 
     The reshaping code  810  is configured to cause at least one processor of the apparatus  800  to reshape a four-dimensional (4D) parameter tensor of a neural network into 3D parameter tensor of the neural network, the 3D parameter tensor including a convolution kernel size, an input feature size and an output feature size. 
     The first partitioning code  820  is configured to cause the at least one processor to partition the 3D parameter tensor along a plane that is formed by the input feature size and the output feature size, into 3D coding tree units (CTU3Ds). 
     The second partitioning code  830  is configured to cause the at least one processor to partition each of the CTU3Ds into a plurality of 3D coding units (CU3Ds) recursively until a maximum depth, using a quad-tree. 
     The first constructing code  840  is configured to cause the at least one processor to construct a 3D-Tree for each of the plurality of CU3Ds. 
     The first entropy encoding code  850  is configured to cause the at least one processor to entropy encode each of a plurality of values of a plurality of nodes of the 3D-Tree. 
     The 3D-Tree may be a 3D-Octree. Based on a codebook index or coefficient of a respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the 3D-Octree may be 1. Based on the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Octree may be 0. A value of a parent node at another depth of the 3D-Octree may be a maximum or minimum value of child nodes of the parent node. 
     The 3D-Tree may be a 3D-Tagtree. Based on an absolute value of a codebook index or coefficient of a respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Tagtree may be 0. A value of a parent node at another depth of the 3D-Tagtree may be a maximum value of child nodes of the parent node. 
     The apparatus  800  may further include second constructing code  860  configured to cause the at least one processor to construct another 3D-Tagtree for each of the plurality of CU3Ds. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being other than zero, a value of a node included in a last depth of the other 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the other 3D-Tagtree may be 0. A value of a parent node at another depth of the other 3D-Tagtree may be a minimum value of child nodes of the parent node of the other 3D-Tagtree. The apparatus  800  may further include second entropy encoding code  870  configured to cause the at least one processor to entropy encode each of a plurality of values of a plurality of nodes of the other 3D-Tagtree, and selecting code  880  configured to cause the at least one processor to select the entropy-encoded plurality of values of the plurality of nodes of one of the 3D-Tagtree and the other 3D-Tagtree that has a higher rate distortion. 
     The 3D tree may be a 3D-Unitree. Based on child nodes of a parent node included in a depth other than a last depth of the 3D-Unitree having different values, a value of the parent node may be 1. Based on the child nodes of the parent node having identical values, the value of the parent node may be 0. Based on codebook indices or absolute values of coefficients of a respective one of the plurality of CU3Ds having different values, a value of a node included in the last depth may be 1. Based on the codebook indices or the absolute values of the coefficients of the respective one of the plurality of CU3Ds having identical values, the value of the node included in the last depth may be 0. 
     The second constructing code  860  may be further configured to cause the at least one processor to construct a 3D-Tagtree for each of the plurality of CU3Ds. Based on an absolute value of a codebook index or coefficient of the respective one of the plurality of CU3Ds being non-zero, a value of a node included in a last depth of the 3D-Tagtree may be 1. Based on the absolute value of the codebook index or coefficient of the respective one of the plurality of CU3Ds being zero, the value of the node included in the last depth of the 3D-Tagtree may be 0. A value of a parent node at another depth of the 3D-Tagtree may be a minimum value of child nodes of the parent node of the 3D-Tagtree. The first entropy encoding  850  may be further configured to cause the at least one processor to entropy encode a value of a current node of the 3D-Unitree, based on the entropy-encoded value of the current node of the 3D-Unitree being zero, skip entropy encoding of values of child nodes of the current node of the 3D-Unitree, based on the entropy-encoded value of the current node of the 3D-Unitree being non-zero and the current node of the 3D-Unitree being a top node, entropy encode a value of a current node of the 3D-Tagtree corresponding to the current node of the 3D-Unitree, and based on the entropy-encoded value of the current node of the 3D-Unitree being non-zero and the current node of the 3D-Unitree not being the top node, entropy encode a difference between the value of the current node of the 3D-Tagtree and a value of a parent node of the current node of the 3D-Tagtree. 
     The apparatus  800  may further include reordering code  890  configured to cause the at least one processor to reorder two-dimensional (2D) planes in at least one of the CTU3Ds and the plurality of CU3Ds, along an axis that is formed by the convolutional kernel size, to generate reorder indices respectively corresponding to orders of the 2D planes, and encoding code  900  configured to cause the at least one processor to encode the reorder indices in a header of the at least one of the CTU3Ds and the plurality of CU3Ds. 
     The foregoing disclosure provides illustration and description, but is not intended to be exhaustive or to limit the implementations to the precise form disclosed. Modifications and variations are possible in light of the above disclosure or may be acquired from practice of the implementations. 
     As used herein, the term component is intended to be broadly construed as hardware, firmware, or a combination of hardware and software. 
     It will be apparent that systems and/or methods, described herein, may be implemented in different forms of hardware, firmware, or a combination of hardware and software. The actual specialized control hardware or software code used to implement these systems and/or methods is not limiting of the implementations. Thus, the operation and behavior of the systems and/or methods were described herein without reference to specific software code—it being understood that software and hardware may be designed to implement the systems and/or methods based on the description herein. 
     Even though combinations of features are recited in the claims and/or disclosed in the specification, these combinations are not intended to limit the disclosure of possible implementations. In fact, many of these features may be combined in ways not specifically recited in the claims and/or disclosed in the specification. Although each dependent claim listed below may directly depend on only one claim, the disclosure of possible implementations includes each dependent claim in combination with every other claim in the claim set. 
     No element, act, or instruction used herein may be construed as critical or essential unless explicitly described as such. Also, as used herein, the articles “a” and “an” are intended to include one or more items, and may be used interchangeably with “one or more.” Furthermore, as used herein, the term “set” is intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items, etc.), and may be used interchangeably with “one or more.” Where only one item is intended, the term “one” or similar language is used. Also, as used herein, the terms “has,” “have,” “having,” or the like are intended to be open-ended terms. Further, the phrase “based on” is intended to mean “based, at least in part, on” unless explicitly stated otherwise.