Patent Publication Number: US-2018053324-A1

Title: Method for Predictive Coding of Point Cloud Geometries

Description:
FIELD OF THE INVENTION 
     The invention relates generally to a method for compressing point cloud geometries, and more particularly to a method for predictive encoding the point cloud geometries 
     BACKGROUND OF THE INVENTION 
     Point clouds can include a set of points in a 3-D space. For example, a given point may have a specific (x,y,z) coordinate specifying its location or geometry. The point locations can be located anywhere in 3-D space, with a resolution determined by the sensor resolution or by any preprocessing performed to generate the point cloud. For fine resolutions, or for coarse resolutions with point clouds spanning a large 3-D space, integer or floating-point binary representations of point locations can require several bits. Storing or signaling every coordinate with high precision allows the capture system to save the point cloud coordinates with high fidelity; however, such representations can consume massive amounts of storage space when saving or bandwidth when signaling. There is a need, therefore, to reduce the number of bits needed to represent the point cloud locations or geometries for subsequent storage or transmission. Even with compression, the size of the compressed representation can be quite large; therefore, there is also a need for a compressed representation that allows one to quickly or easily decode and reconstruct a coarse representation of the point cloud geometry without having to decode the entire file or bit-stream. 
     SUMMARY OF THE INVENTION 
     Some embodiments of the invention are based on recognition that a point cloud can be effectively encoded by parameterizing a given surface and fitting the parameterized surface on to the point cloud. 
     Accordingly, one embodiment discloses a method for encoding a point cloud of representing a scene using an encoder including a processor in communication with a memory, including steps of fitting a parameterized surface onto the point cloud formed by input points; generating model parameters from the parameterized surface; computing corresponding points on the parameterized surface, wherein the corresponding points correspond to the input points; computing residual data based on the corresponding points and the input points of the point cloud; compressing the model parameters and the residual data to yield coded model parameters and coded residual data; and producing a bit-stream from the coded model parameters of the parameterized surface and the coded residual data. 
     Further, some embodiments of the invention are based on recognition that encoded point cloud data can be effectively decoded by receiving a bit-stream that includes model parameters of a parameterized surface and residual data computed from an original point cloud. 
     Accordingly, one embodiment discloses an encoder system for encoding a point cloud of representing a scene, wherein each point of the point cloud is a location in a three dimensional (3D) space, the encoder system includes a processor in communication with a memory; an encoder module stored in the memory, the encoder module being configured to encode a point cloud of representing a scene by performing steps, wherein the steps comprise: fitting a parameterized surface onto the point cloud formed by input points; generating model parameters from the parameterized surface; computing corresponding points on the parameterized surface, wherein the corresponding points correspond to the input points; computing residual data based on the corresponding points and the input points of the point cloud; compressing the model parameters and residual data to yield coded model parameters and coded residual data, respectively; and producing a bit-stream from coded the model parameters of the parameterized surface and the coded residual data. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of an encoding process according to the embodiments of the invention; 
         FIG. 2  is a block diagram of a process for computing a correspondence between a surface model and point locations for organized point clouds according to the embodiments of the invention; 
         FIG. 3  is a block diagram of a process for computing a correspondence between a surface model and point locations for unorganized point clouds according to the embodiments of the invention; 
         FIG. 4  is a process for computing the residual between point locations and corresponding points of a surface model according to the embodiments of the invention; 
         FIG. 5  is a diagram of a decoding process according to the embodiments of the invention according to the embodiments of the invention; and 
         FIG. 6  is a diagram of a hierarchical partitioning process according to the embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Various embodiments of the present invention are described hereafter with reference to the figures. It would be noted that the figures are not drawn to scale elements of similar structures or functions are represented by like reference numerals throughout the figures. It should be also noted that the figures are only intended to facilitate the description of specific embodiments of the invention. They are not intended as an exhaustive description of the invention or as a limitation on the scope of the invention. In addition, an aspect described in conjunction with a particular embodiment of the invention is not necessarily limited to that embodiment and can be practiced in any other embodiments of the invention. 
     The embodiments of the invention provide a method and system for compressing three-dimensional (3D) points cloud using parametric models of surfaces fitted onto a cloud or set of points that serve as predictors of point locations of the 3D points, and achieving compression by quantizing and signaling the parameters and/or prediction errors 
     Some embodiments disclose a method for encoding a point cloud of representing a scene using an encoder including a processor in communication with a memory, wherein each point of the point cloud is a location in a three-dimensional (3D) space, including steps of fitting a parameterized surface onto the point cloud formed by input points; generating model parameters from the parameterized surface; computing corresponding points on the parameterized surface, wherein the corresponding points correspond to the input points; computing residual data based on the corresponding points and the input points of the point cloud; compressing the model parameters and the residual data to yield coded model parameters and coded residual data; and producing a bit-stream from the coded model parameters of the parameterized surface and the coded residual data. 
     Encoding Process 
       FIG. 1  is a block diagram of an encoding process performed by an encoder  100  according to some embodiments of the invention. The encoder  100  includes a processor (not shown) in communication with a memory (not shown) for performing the encoding process. 
     An input to the encoder  100  is a point cloud which comprises a set of N points p i , i={1, 2, . . . , N}  101 , where each p i  is a point location in 3-D space. A point location can be represented by a coordinate in 3-D space. For example, when using the Cartesian coordinate system, p i  can be represented by a triplet {x i , y i , z i }, where x i , y i , z i  are the coordinates of the point. In another example a spherical coordinate system can be used, in which each point p i  is represented by a triplet {r i , θ i , ω i }, where r i  denotes a radial distance, θ i  denotes a polar angle, and ω i  represents an azimuthal angle for a point. 
     For a set or subset of input points p i    101 , a surface model fitting process  102  computes a surface model which approximates the locations of the input points p i    101  or the surface of the object that is represented by the input points. For example, if the input points are a representation or boundary locations of a spherical object in 3-D space, then the surface model  104  can be the equation of a sphere, having two parameters: A radius and a coordinate for the location of the origin of the sphere. In another example, the surface model  104  can be a Bézier surface or Bézier surface patch. In this case, the surface model  104  can be represented by p(u,v), a parametric representation of a location or coordinate in 3-D space. A Bézier surface patch can be represented by the following equation: 
     
       
         
           
             
               
                 
                   
                     
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     The shape of the Bézier surface patch is determined by model parameters b ij , which are known as “control points”. The control points are indexed by integers i and j, where i is an index from 0 to n along the u dimension and j is an index from 0 to m along the v dimension. For example, if m=3 and n=3, then there are a total of 16 control points. The control points b ij  are included in the model parameters  103 . 
     For an organized point cloud, as described later, the predetermined organizational information  117  specifies how the points are organized, and can be input to the surface model fitting process  102 . This organizational information  117  can be a specification of a mapping between each input point p i    101  and its location in an organizational matrix, grid, vector, or address in the memory. In some cases, the organizational information  117  can be inferred based upon the index, or address or order in the memory of each input point p i    101 . 
     The surface model fitting process  102  can select the model parameters  103  to minimize a distortion. The distortion may be represented by the total or average distortion between each input point p i  and each corresponding point f mj    106  on the surface model  104 . In that case, the surface model  104  serves as a prediction of the set or subset of input points p i    101 . The surface model fitting process  102  can also minimize reconstruction error. The reconstruction error is the total or average error between each input point p i  and corresponding reconstructed point. In this case, the reconstructed point is identical to the reconstructed point in the decoding process, which will be described later. 
     Once the model parameters  103  have been computed, the surface model  104  for a set or subset of points is generated using those model parameters. Alternatively, the surface model  104  may be generated during the surface model fitting process  102  before computing the model parameters  103 . In that case, the surface model  104  is already available and does not have to be regenerated. 
     The surface model  104  can be a continuous model f  105  or a discrete model  106 . The continuous model uses continuous values for (u,v) in a unit square, resulting in a surface model f(x,y,z) (f x ,f y ,f z ), where (f x ,f y ,f z ) represents the location of the surface model  104  in Cartesian space. For the discrete model, the parametric representation p(u,v) of the location of the surface model  104  in 3-D space uses discrete parameters (u i ,v i ) (note: i is used as a general indexing term here and is a different index than the i used in the Bézier surface patch equation described earlier.) In the discrete model, the surface model  104  is represented by a set of points fm j   {fm x   j ,fm y   j ,fm z   j }, j={1, 2, . . . , M}  106 , where each corresponding point fm j    106  represents a discrete point on a continuous surface model f. The intent of this part of the encoding process is to generate a set of M surface model points that are a good representation of the set or subset of N input points. 
     Once the surface model  14  has been obtained, the encoder  100  performs a residual computing process  107  to compute a set of residuals r i    {r x   i , r y   i , r z   i }, i={1, 2, . . . , N}  108  corresponding to each input point p i    101 . Each residual r i  can be used as a prediction error, since the surface model can be considered as being a prediction of the input points. In some cases, the prediction error can be computed based on the residual; for example, if the residual r i  is represented by a triplet of differences between each coordinate, which in this case would be r i ={x i −f x   i , y i −f y   i , z i −f z   i }, i={1, 2, . . . , N}, then the prediction error can be computed using a squared-error distortion by summing the squares of each component of the residual. 
     In order to compute a residual r i    108 , each input point p i    101  must have a corresponding point on the surface model fm j    106 . The N input points p i    101  and the M surface model points fm j    106  are input to a process  109  that computes corresponding points f i    110  on the surface model  104 . In another embodiment, the points of the continuous surface f  105  may be used in the correspondence computation process  109 . The process  109  outputs a set of N corresponding surface model points f i   {f x   i  f y   i ,f z   i }, i{1, 2, . . . , N}  110 , in which for a given i in {1, 2, . . . , N}, f i    110  denotes the point on the surface model  104  that corresponds to the input points p i    101 . 
     Once the corresponding point f i    110  on the surface model  104  has been computed, the residuals r i    108  between the corresponding points f i    110  and the input points p i    101  is computed in the residual computing process  107 , as described earlier. 
     The residuals r i , i={1, 2, . . . , N}  108  can be input to a transform process  111  to produce a set of transform coefficients  112 . For example, the transform process  111  can apply a Discrete Cosine Transform (DCT) or other spatial domain to frequency domain transform to the residuals r i    108 , to output a set of transform coefficients  112 . In some cases, the transform process  111  may be a pass-through transform process, in which no operations alter the data, making the transform coefficients  112  identical to the residuals r i    108 . 
     After the transform process  111 , the transform coefficients  112  are input to a quantization process  113 . The quantization process quantizes the transform coefficients  112  and outputs a set of quantized transform coefficients  114 . The purpose of the quantization process  113  is to represent a set of input data for which each element of the data can have a value from a space that contains a large number of different values, by a set of output data for which each element of the data has a value from a space that contains fewer different values. For example, the quantization process  113  may quantize floating-point input data to integer output data. Lossless compression can be achieved if the quantization process  113  is reversible. This indicates that each possible output element from the quantization process  113  can be mapped back to the corresponding unquantized input element. 
     Successively, the quantized transform coefficients  114  are input to an entropy coder  115 . The entropy coder  115  outputs a binary representation of the transform coefficients  115  to an output bit-stream  116 . The output bit-stream  116  can be subsequently transmitted or stored in the memory or in a computer file. The model parameters  103  are also input to the entropy coder  115  for output to a bit-stream  116 . 
     In another embodiment, the entropy coder  115  may include a fixed-length coder. 
     Process to Compute Corresponding Points on a Surface Model 
       FIGS. 2 and 3  show block diagrams of corresponding point computing processes  200  and  300 . There are two types of corresponding point computing processes. The corresponding point computing process  200  is applied to organized point clouds, and the corresponding point computing process  300  is applied to unorganized point clouds. 
     The purpose of the process to compute corresponding points on a surface model  109  is to determine, for each input point p i    101 , a corresponding point f i    110  on the surface model  104 , so that later a residual r i    108  can be computed between each input point p i    101  and each corresponding point f i    110  on the surface model  104 . This process for computing corresponding points on the surface model  104  in a process  109  can operate on the organized or unorganized point clouds. The processes will be described in detail after the explanation of the organized and unorganized point clouds below. 
     In an organized point cloud, each point p i    101  is associated with a position or address in a memory or in a matrix or vector containing one or more elements. For example, a 3×4 matrix contains 12 elements. Each point p i    101  in the point cloud can be associated with, or assigned to, an element or position in the matrix. This association creates an organization among the points that can be independent of the actual coordinates of the point p i    101  in 3-D space. Operations can then be applied to points based upon their organization in the matrix or memory, for example, points can be operated upon based upon their adjacency in the organizational matrix or grid. If the number of points p i , i={1, 2, . . . , N}  101  in the point cloud is less than the total number of memory addresses or matrix elements, then some of the memory addresses or matrix elements will not have input points p i    101  associated with them. These elements can be called “null points”, empty spaces, or holes in the organizational grid. 
     In an unorganized point cloud, there is no organization matrix or grid as described above associated with the input points p i    101 . The location of each point in 3-D space is known, but there is no relation among the points defined by an association with element locations in a matrix or grid or with address locations in a memory. 
     Computing Corresponding Points for an Organized Point Cloud 
       FIG. 2  shows a block diagram of the corresponding point computing process  200 . Given a set of N input points p i , i={1, 2, . . . , N}  101 , the input points p i  are arranged in a memory, or in a matrix according to a predetermined organization or mapping. 
     In the encoding process  100 , the predetermined organization or mapping can be represented by organizational information  117  as described earlier. The organizational information  117  can be a specified mapping or can be inferred based upon the order in which each input point p i    101  is input or stored in the memory. The N input points p i    101  are arranged and stored  201  into the memory or in the matrix or vector having M elements, locations, or addresses. If M≧N, then “null point” placeholders are stored  202  into the memory, matrix, or vector locations. The null point place holders are not associated with any input points p i    101 . In this case, the arranged input points p i    101  and null point placeholders are stored combined  210  in M locations comprising N input points p i    101  and M−N null point placeholders. 
     In the organized point cloud, each input point p i    101  is associated with a position or address in a memory or in a matrix or vector containing one or more elements as indicated in step S 1 . Each input point p i    101  in the organized point cloud can be associated with an element assigned to a predetermined position in the matrix. In this case, an association is made between input points p i    101  and elements in the matrix. The association may be performed independently from the actual coordinates of the input point p i    101  in 3-D space. In step S 2 , arranging operations can be applied to the elements in the matrix based upon the association. In step S 2 , the elements in the matrix can be arranged to M memory locations based upon their adjacency in the matrix. 
     If the number of the input points p i , i={1, 2, . . . , N}  101  in the point cloud is less than the total number of M memory addresses or matrix elements, then some of the memory addresses or matrix elements will not have input points p i    101 . These elements having no input points p i  can be called “null points,” empty spaces, or holes in the organizational matrix. After the steps S 1  and S 2  are performed, the matrix elements are stored into the M memory addresses or the matrix elements in the memory. For example, a matrix having 12 elements stored in the memory is indicated in  FIG. 2 . Note that the null points are indicates by “Ø.” That is, if M≧N, then “null point” placeholders are stored  202  into the memory, matrix, or vector locations that are not associated with any input points p i    101 . In this case, the arranged input points p i    101  and null point placeholders are stored combined  210  in M locations comprising N input points p i    101  and M−N null point placeholders. 
     Once the N input points p i    101  are organized into a memory in step S 4 , matrix or vector having M addresses or elements, surface model points  104 , for example, discrete surface model points fm j , j={1, 2, . . . , M}  106  are generated in a process  203  using the surface model fitting process  102  in step S 4 . 
     In step S 5 , the surface model points fm j  are arranged  204  into memory containing the same number and arrangement of addresses or elements as contained in the combined  210  locations, resulting in an ordered or organized arrangement of surface model points  211 . 
     In some cases, the surface model points may be generated during the fitting process  102 . 
     Because the process to compute corresponding points on the surface model in the process  109  outputs N corresponding surface model points f i , i={1, 2, . . . , N}  110 , the last step (Step  6 ) of the process for computing corresponding points between the model and the point locations for organized point clouds  200  is a method of selecting  205  N points from the M surface model points fm j , j={1, 2, . . . , M}  106 . This selection can be achieved by inspecting the contents of the combined  210  locations in order, for example, from the first element to the last element. The position in this order can be indexed by j, and an index that keeps track of the non-null points can be denoted as i. If the first element, i.e. j=1 of combined  210  locations contains a non-null point, this non-null point will be p 1 , so the corresponding point f 1  is set equal to surface model point fm 1  and i is incremented. If this first element is null, then there is no corresponding point for this location, so no assignment is made. Next, j is incremented and these steps are repeated for the next element in the combined  210  locations. The end result is the output of N corresponding points f i , i={1, 2, . . . , N}  110  which are selected or mapped  212  from the elements of surface model points fm j , j={1, 2, . . . , M} whose locations correspond to non-null entries in the memory storing the combined  210  locations. 
     As discussed above, in the encoding process, the point cloud can be an organized point cloud. In another case, the point cloud can be an unorganized point cloud. 
     Computing Corresponding Points for an Unorganized Point Cloud 
       FIG. 3  shows a block diagram of a corresponding point computing process  300 . The process for computing corresponding points on a surface model for unorganized point clouds  300  is shown in  FIG. 3 . Given a set of N input points p i , i={1, 2, . . . , N}  101 , the process  300  operates on unorganized point clouds, so the input points p i , i={1, 2, . . . , N}  101  are not required to be arranged in the memory, or in a matrix according to a predetermined organization or mapping. 
     Earlier, it was described that the surface model can be represented by p(u,v), a parametric representation of a location or coordinate in 3-D space. For an unorganized point cloud and optionally for an organized point cloud, the surface model fitting process  102  can compute the parameters (u,v) used to generate a location on the surface model patch that corresponds to a given input point. Thus, for each of the input points p i , i={1, 2, . . . , N}  101  there is a pair of parameters (u i ,v i )  301 . A set of parameter pairs (u i ,v i )  301  can additionally be signaled as model parameters  103 . The choice of each parameter pair (u i ,v i )  301  to be associated with each input point p i    101  can be computed during the surface model fitting process  102 , for example, by minimizing the difference or distortion between each input point p i    101  and the location on the surface patch p(u i ,v i ). 
     Given the input points p i    101  and computed parameters (u i ,v i )  301 , the corresponding points ( 110 ) on the surface model fm i , i={1, 2, . . . , N}  311  are computed  303  by setting the corresponding points on the surface model fm i    311  to the locations on the surface patch p(u i ,v i ) for the given (u i ,v i )  301  parameter pairs. 
     The N input points p i , i={1, 2, . . . , N}  101  are arranged  310  in order with the surface model points fm i , i={1, 2, . . . , N}  311 , whose order matches that of an ordered list  320  of the parameter pairs (u i ,v i )  301 . Since the number of input points  101  and the number of surface model points fm i  are both N, there is a one-to-one correspondence between the input points p i  and the model points fm i , so the corresponding points f i , i={1, 2, . . . , N}  110  of the surface model  104  are set in a process  305  with a one-to-one mapping  312  to the surface model points fm i , i={1, 2, . . . , N}  311 . 
     Process to Compute the Residual Between Point Locations and Corresponding Points of the Surface Model 
     A process to compute the residual between point locations of an organized point cloud and corresponding points of the surface model  104  is shown in  FIG. 4 . Given the set of combined  210  locations, which contain the input points p i , i={1, 2, . . . , N}  101  and null point placeholders as output from the arranging  201  and storing  202  processes of  FIG. 2 , and given the corresponding points f i , i={1, 2, . . . , N}  110  of the surface model computed in the process  109  during the encoding process  100  of  FIG. 1 , the residuals r i , i={1, 2, . . . , N}  108  are computed  440  as the difference between each input point p i    101  and the corresponding point f i  of the surface model  104 . This difference can be computed for each coordinate of the point, in which r i ={x i  f x   i , y i −f y   i , z i −f z   i }, i={1, 2, . . . , N}. When processing an unorganized point cloud, there are no null point placeholders in the arranged points  210 , so the list of residuals has a one-to-one correspondence to the input points  101  and corresponding points  110  on the surface model  104 . When processing an organized point cloud, there is still a one-to-one correspondence in computing the residuals, but the residuals are arranged in the same way as the arranged points  210 . In  FIG. 4 , the list of corresponding points  430  are arranged in a larger memory so that locations occupied by null points in the list of arranged points  210 , have no entries corresponding to the null point locations in the list of arranged points  210 . By subtracting this list of corresponding points  430  from the list of arranged points  210 , the memory storing the residuals  441  can contain null point entries in the same locations as in the memory storing the arranged points  210 . Thus, for organized point clouds, the residuals  108  output from the residual computation process  107  also include information, either explicitly or implicitly based on storage location, about the location of null point entries. For an organized point cloud, there are no null point entries, so the output residuals  108  contain a list of differences as computed by the residual computation process  107 . 
     In an embodiment, the residual data represent distances between the corresponding points and the input points. 
     Further, in some cases, the input points p i    101  may include attributes. In this case, the corresponding points may include attributes. For example, the attributes may include color information. 
     Further, some embodiments disclose an encoder system for encoding a point cloud of representing a scene, wherein each point of the point cloud is a location in a three dimensional (3D) space, the encoder system including a processor in communication with a memory; an encoder module stored in the memory, the encoder module being configured to encode a point cloud of representing a scene by performing steps, wherein the steps comprise: fitting a parameterized surface onto the point cloud formed by input points; generating model parameters from the parameterized surface; computing corresponding points on the parameterized surface, wherein the corresponding points correspond to the input points; computing residual data based on the corresponding points and the input points of the point cloud; compressing the model parameters and residual data to yield coded model parameters and coded residual data, respectively; and producing a bit-stream from coded the model parameters of the parameterized surface and the coded residual data. 
     Decoding Process 
     Some embodiments of the invention are based on recognition that encoded point cloud data can be effectively decoded by receiving a bit-stream that includes model parameters of a parameterized surface and residual data computed from an original point cloud. Some embodiments disclose a method for decoding a point cloud representing a scene using a decoder including a processor in communication with a memory, including steps of receiving model parameters for a parameterized surface; receiving residual data; determining the parameterized surface using the model parameters; computing corresponding points from the parameterized surface according to a predetermined arrangement; and computing reconstructed input points by combining the residual data and the corresponding points. 
       FIG. 5  is a diagram of the decoder or decoding process for an embodiment of the invention. The input to the decoder is a bit-stream  116 . The bit-stream  116  is decoded by an entropy decoder  515 , which produces a set of model parameters  103  and quantized transform coefficients  114 . The quantized transform coefficients are inverse quantized  513  to produce in inverse quantized transform coefficients  512 . The inverse quantized transform coefficients  512  are input to an inverse transform  511 , which produces a set of reconstructed residuals {circumflex over (r)} i ={1, 2, . . . , N}  508 , which can be, for example, {circumflex over (r)} i   {{circumflex over (r)} x   i ,{circumflex over (r)} y   i ,{circumflex over (r)} z   i } when using a Cartesian coordinate system. 
     The model parameters  103  decoded from the entropy decoder  515  are input to a surface model generation process  515 , which generates a surface model  104  in the same manner as in the surface model fitting process  102  of the encoding process  100 , but without the fitting part of that process because the model parameters are provided by decoding them from the bit-stream  116 . The surface model  104  can be a continuous f(x,y,z) (f x ,f y ,f z )  105  or discrete fm j   {fm x   j ,fm y   j , fm z   j }, j={1, 2, . . . , M}  106 . If the surface model  104  is a Bézier surface patch, then the model parameters  103  include control points b ij . The surface model  104  is input to a process to compute corresponding points on a surface model in the process  109 , which computes corresponding points f i , i={1, 2, . . . , N}  110  of the surface model  104  according to a predetermined arrangement. For example, the predetermined arrangement may be performed by an adjacency of parameters in the models. In this case, the corresponding points from the parameterized surface can be computed according to an arrangement, and the model parameters include a specification of the arrangement. 
     Given the reconstructed residuals {circumflex over (r)} i  and corresponding points f i  on the surface model  104  as input, a combiner  507  computes the reconstructed point cloud or reconstructed input points {circumflex over (p)} i , i={1, 2, . . . , N}  501 . The process performed in the combiner  507  can be, for example, addition, in which {circumflex over (p)} i ={{circumflex over (r)} x   i +f x   i ,{circumflex over (r)} y   i +f y   i {circumflex over (r)} z   i +f z   i }, i={1, 2, . . . , N}. 
     In another example, the process by the combiner  507  can be performed by adding the residual data and the corresponding points. 
     Analogous to input points p i    101  in the encoding process  100 , the reconstructed point cloud {circumflex over (p)} i    501  can be unorganized or organized. 
     For the unorganized case, the model parameters entropy decoded  515  from the bit-stream  116  can include a list  320  of (u i ,v i )  301  parameter pairs, which are used as parameters for generating a surface model  104 , such as a surface patch p(u i ,v i ) of a Bézier surface model  104 . The corresponding points on the surface model  104  are computed  109  similarly to how they are computed in the encoder, yielding a set of N corresponding points  110  on the surface model  104 . 
     For the organized case, the reconstructed residuals {circumflex over (r)} i    508  include the locations of the null points, i.e. they are arranged into memory in the same way as the residuals  441  generated inside the encoding process  100 . The process to compute  109  corresponding points  110  on the surface model  104  in the decoding process  500  can operate in the same way as is done inside the encoding process  100 . 
     When operating on reconstructed residuals  508  having null points, the combiner  507  copies input null points to its output. 
     In another embodiment, the combiner can copy the corresponding point  110  on the surface model  104  to its output, which in that case would replace all the null point placeholders in the reconstructed point cloud  501  with corresponding points  110  from the surface model  104 . This embodiment can be used to replace “holes” or points missing from a point cloud with points from the surface model  104 . 
     Hierarchical Partitioning Process 
     When more than one surface model  104  is needed, a hierarchical partitioning process can be used to break the point cloud into smaller point clouds, each having their own model. 
     Hierarchical Partitioning Process for Encoding 
     For organized point clouds, the surface model  104  can be a surface patch p(u i ,v i ), where a pair of parameters (u i ,v i )  301  are used to compute a point location in 3-D space for the surface model  104 . For a continuous surface model  104 , the parameters (u,v) of a surface patch p(u,v) span a unit square, i.e. 0≦u, v≦1. For a discrete surface model, the parameter space (u,v) can be discretized as an J×K grid of parameter values (u j ,v k ), j={1, 2, . . . , J}, k={1, 2, . . . , K}. In this case, the surface patch p(u j ,v k ) comprises M×N points in 3-D space. 
       FIG. 6  is a diagram of a hierarchical partitioning process according to the embodiments of the invention. The initial parameter space, which in this case is a 2-D grid (u j ,v k ), j={1, 2, . . . , J}, k={1, 2, . . . , K}  601  can be represented by an initial rectangle  602  having width w=J and height h=K. 
     In some cases, an adjacency may be defined by neighboring positions on the 2-D grid. Further, the neighboring positions can be in the horizontal, vertical and diagonal directions. 
     Given the initial parameter space (u j ,v k )  601  represented by a rectangle  602  having size w×h, a surface model patch p(u i ,v i )  603  is generated by the surface model fitting process  102 . The encoding process  100  and decoding process  500  represent a discrete surface model as fm l , l={1, 2, . . . , L}  106 , so a patch to surface model mapping  604  maps p(u j ,v k ) j={1, 2, . . . , J}, k={1, 2, . . . , K} to fm l , l={1, 2, . . . , L}. Therefore, L=J×K. 
     A fitting error e  603  between the input points associated with the current rectangle representation, which initially can represent all the input points, and the discrete surface model  104   106  is computed  605 . The fitting error e can be measured as, for example, the total or mean-square error between each component (e.g. the x component, y component, and z component) of the input point and its corresponding surface model point. In another embodiment, the fitting error e can be measured as the total or average deviation among surface normal of a surface model  104  and of a previously-computed surface model  104 . 
     If the fitting error e  606  is less than a predetermined fitting error threshold T  607 , then the hierarchical partitioning process  600  is considered as being successful, so a partition flag  610  indicating that the rectangle will no longer be split is output  608 , and the discrete surface model fm l    106  for the points associated with the current rectangle along with the model parameters  103  are also output  608  and this process ends. For the case when the surface model  104  is a Bézier surface, the model parameters  103  can include the control points b ij  shown in equation (1). They can also include the width w and height h of the current rectangle representation. 
     If the fitting error e  606  is not less than a predetermined fitting error threshold T  607 , then the current surface model  104  is considered as not being a good fit to the input points associated with the current rectangle. In this case, the rectangle is partitioned into two or more rectangles by a rectangle partitioning process  609 , and a partition flag  610  is output  611  to indicate that the rectangle will be partitioned. 
     The rectangle partitioning process  609  takes the current rectangle representation of the parameter space, and divides the rectangle which has width w and height h into partitioned rectangles  612  comprising two or more rectangles. For example, a binary partitioning can divide the rectangle into two rectangles, each having width w/2 and height h. In another example, after dividing, each rectangle can have width w and height h/2. If w/2 or h/2 are not integers, then rounding can be done, e.g. one rectangle can have width floor(w/2) and the other can have width floor(w/2)+1, where floor( ) rounds down to the nearest integer. For a binary partitioning, whether to divide along the width or along the height can be either decided by a predetermined process, or the decision as to which dimension to divide can be explicitly signaled in the bit-stream as a flag. In the former case, an example of such a predetermined process is to divide the rectangle across its longer dimension. For example, if w=10 and h=4, then the rectangle can be divided across the width, so partitioned rectangles  612  will each have width w=5 and height h=4. 
     In another embodiment, the rectangle partitioning process  609  can divide a rectangle into more than two rectangles, for example, four rectangles each representing one quadrant of the rectangle. 
     In another embodiment, the rectangle partitioning process  609  can partition the rectangle based upon the density of the input points  101  associated with the rectangle. For example, if the input points for that rectangle represent two distinct objects in 3-D space, then the rectangle partitioning process  609  can divide the rectangle into two parts, each containing one object. To determine where to best divide the rectangle, for example, the density of input points for the rectangle can be measured, and the partitioning can be done along a line that maximizes the density in each partitioned rectangle; or, in another example, the partitioning can be done along a line that minimizes the sum of distances between that line and each point in the partitioned rectangles, e.g. by fitting a line using a least-square approximation. Parameters indicating the location of the partitioning can be coded signaled in the bit-stream as model parameters. 
     Once the partitioned rectangles  612  are obtained, the hierarchical partitioning process  600  is repeated for each of the partitioned rectangles  612  by inputting them to the surface model fitting process  102  in the hierarchical partitioning process  600 . 
     In addition to terminating the partitioning process  609  for a rectangle or partitioned rectangle when its fitting error  606  is less than a threshold  607 , the partitioning process  609  for a rectangle can be terminated when the width w or height h, or the area, i.e. product of the width and height, are less than a predetermined value. For example, if a Bézier surface patch having 16 control points is used as a surface model  104 , and if the area of a rectangle  10 , then it may be more efficient to directly signal the 10 input points  101  associated with the rectangle instead of fitting it with a surface model  104  that requires the signaling of 16 control points. 
     In another embodiment, the number of control points to use for the surface model  104  can depend upon the width and height of a rectangle. For example, if the area of a rectangle is less than 16, then a surface model  104  with fewer control points can be used. A value or index to a look-up table can be signaled in the bit-stream to indicate how many control points are used in the surface model  104  for a given rectangle. 
     In another embodiment, the rank, i.e. the number of linearly independent rows or columns for a matrix of input points  101  or a decomposition of the matrix of input points associated with a rectangle can be measured, and the number of control points to use for generating the surface model  104  can be set to a value less than or equal to the rank. 
     When the hierarchical partitioning process  600  is complete, a sequence of partition flags  610  will have been output. If the partitioning process is predetermined, in that it can be entirely specified by the partitioning flags and the width and height of the initial rectangle representation, for example, by knowing that a binary division always occurs across the longer dimension, then sufficient information will be available to the decoder for recovering the locations of all partitioned rectangles. Thus, the width and height of the initial rectangle representation  602 , the sequence of partition flags  610 , and the model parameters  103  such as control points for each rectangle can be used by the decoder to generate  515  surface models  104  for each rectangle. 
     Processing Partitioned Rectangles in the Decoder 
     In one embodiment, the decoder can decode from the bit-stream the width and height of an initial rectangle representation  602 , i.e. a 2-D organizational grid. The decoder next decodes from the bit-stream a partition flag  610 , where a partition flag of 1 or true indicates that the rectangle is to be partitioned, for example, into two rectangles (a first and second rectangle), and, for example, wherein that split can occur across the longer dimension of the rectangle. If the rectangle is split, then the next partition flag  610  decoded from the bit-stream is the partition flag  610  for the first rectangle. 
     If that partition flag is 0 or false, then the first rectangle will not subsequently be split, and a payload flag can be decoded from the bit-stream to indicate what kind of data will be decoded for the first rectangle. If the payload flag is 1 or true, then control points and data representing the residuals  108 , such as quantized transform coefficients  114  for the first rectangle are entropy decoded from the bit-stream. After decoding the data representing the residuals, model parameters  103 , such as control points for the surface model  104  for this rectangle, can be entropy decoded from the bit-stream. If the payload flag is 0 or false, then no residual is available for the first rectangle, which can happen, for example, if no surface model  104  was used and the encoder directly signaled the input points  101  into the bit-stream. In this case, the decoder will next entropy decode from the bit-stream input points, quantized input points, or quantized transform coefficients of transformed input points for the first rectangle. In another embodiment, no payload flag is used. In that case, a surface model  104  is always used, so data representing residuals will be decoded from the bit-stream. 
     If that partition flag is 1 or true, then the first rectangle will be further partitioned. 
     A data structure representing a hierarchy or tree can be traversed breadth first or depth-first. For the breadth-first case, the next data that is decoded from the bit-stream is the partition flag for the second rectangle. For the depth-first case, the next data that is decoded from the bit-stream is the partition flag for the current rectangle, which in this case indicates whether the first rectangle will be further split. 
     This processing of partitioned rectangles in the decoder is performed on each rectangle until all rectangles have been processed. 
     As is the case during the encoding process, additional criteria can be used to decide whether a block is split or not split. For example, if the dimensions (width and/or height, or area) of a rectangle are below predetermined thresholds, then the partitioning process for that rectangle can be terminated without having to decode a split flag from the bit-stream. The dimensions of each rectangle can be inferred during the splitting process from the height and width of the initial rectangle representation  602 . 
     Additional Embodiments 
     In another embodiment, after a selected set of rectangles or partitioned rectangles  612  have been processed and the reconstructed point cloud  501  is output by the decoding process  600 , additional rectangles can be processed to generate additional points for the reconstructed point cloud  501 , for example, in a scalable decoding system. 
     In another embodiment, the fitting error  606  is computed as the difference between the discrete surface model  106  and the reconstructed point cloud  501  that would result if that surface model were used by the encoding process  100  and decoding process  600 . 
     In another embodiment, instead of computing a difference-based fitting error  606 , e.g. the mean-squared error between the input points  101  and the discrete surface model  106 , an error metric can be the sum of the mean-squared error plus a scaled number of bits occupied in the bit-stream for representing all data associated with the rectangle being processed. 
     In another embodiment, the partitioning of a rectangle can occur across the shorter dimension of the rectangle. 
     In another embodiment, all the partition flags are decoded from the bit-stream  610  before any data associated with each rectangle, such as control points, other model parameters  103 , and data associated with the residuals  108 . This embodiment allows the full partitioning hierarchy to be known by the decoder before the remaining data is decoded from the bit-stream. 
     In another embodiment, some or all of the partition flags are decoded from the bit-stream before the payload flags and associated data are decoded, and then payload flags and data from a selected subset of rectangles can be decoded from the bit-stream. The desired rectangles can be selected by specifying a region of interest on the initial rectangle representation  602 , for example, by drawing an outline a representation of the organizational grid on a computer display, and then only the data in the bit-stream associated with the rectangles contained or partially contained in the region of interest can be decoded from the bit-stream. 
     In another embodiment, a region of interest is selected from 3-D space, e.g. a 3-D bounding box encompasses the input points  101  of interest, then for the point locations in the 3-D region of interest, their corresponding locations in the 2-D rectangle representation  602  are identified, i.e. looked-up or inverse mapped, and a 2-D bounding box is computed over the 2-D rectangle representation that contains all of these selected corresponding locations. This bounding box comprises a new initial sub-rectangle that is populated by the selected corresponding input points  101 . For any locations inside this new sub-rectangle that do not have corresponding 3-D point locations inside the 3-D bounding box, they are populated with null points, wherein a null point is a value indicating that there is no corresponding point in 3-D space, e.g. a “hole”. 
     In another embodiment, the rectangle partitioning process  609  partitions a rectangle so that the number of input points  101  associated with each partitioned rectangle  612  are equal or approximately equal. 
     In another embodiment, each input point  101  has an associated attribute. An attribute is additional data including but not limited to color values, reflectance, or temperature. 
     In another embodiment, the control points for a surface model patch are quantized and optionally transformed during the encoding process  100 , and are inverse quantized and optionally inverse transformed during the decoding process  500 . 
     In another embodiment, for an unorganized point cloud, in which the input points are not arranged in a memory along with null point placeholders in the memory locations that do not contain input points, the positions of the corresponding input point location on the manifold associated with the parameterized model is signaled in the bit-stream. For example, with Bézier surfaces or Bézier patches, a unit square, represented by parameters u and v, where both u and v are between 0.0 and 1.0 inclusive, is mapped to (x,y,z) coordinates in 3-D space. For an organized point cloud, which is the case in the preferred embodiment, the unit square can be sampled with a uniform grid so that each (x,y,z) point in 3-D corresponds to a sample position on the (u,v) unit square plane, and sample positions on the (u,v) unit square plane that do not have a corresponding point in 3-D space can be populated by a null point. The sample positions in the (u,v) unit square plane can be populated by, i.e. associated with, input points  101 , using a predetermined order. For example, a sequence of input points  101  can populate the 2-D sample positions in a raster-scan order, where each row in the 2-D sample position is filled from the first to the last element in the row, and then the next input points go into the next row. 
     In another embodiment, the parameter space  601  is a 2-D grid or manifold with uniform sampling. 
     In another embodiment, the parameter space  601  is a 2-D grid or manifold with non-uniform sampling. For example, the center of the grid can have a higher density of points than the edges. As a further example, a uniform grid can be sampled at every integer position, and then a centered square having one half the width and height of the whole parameter space can be additionally sampled at every half-integer or quarter-integer position. 
     In another embodiment, a 2D  111  transform is applied to the components of the residual data  108  according to their corresponding locations on the 2D organizational grid  117 . 
     In another embodiment, the residual data  108  corresponding to partitioned rectangles  112  are signaled to the bit-stream  116  in an order based upon the dimensions of the partitioned rectangles  612 , for example, from largest to smallest area, or in another example, from smallest to largest area, where area is the area or width times height of a partitioned rectangle  612 . 
     In another embodiment, the reconstructed input points comprise a three-dimensional map (3D map). A vehicle can determine its position in 3D space by capturing a point cloud from sensors located on the vehicle, and then comparing the captured point cloud with the reconstructed input points or reconstructed point cloud. By registering, i.e. aligning, points on the captured point cloud with reconstructed input points, the position of objects or points in the captured point cloud can be associated with objects or points in the reconstructed point cloud. Given that the position of objects captured by the vehicle&#39;s sensors is known relative to the position of the vehicle or vehicle&#39;s sensors, and given that after the registration process the positions of objects captured by the vehicle&#39;s sensors are known relative to the reconstructed point cloud, the position of the vehicle in the reconstructed point cloud can be inferred, and therefore, the position of the vehicle in the 3D map can be inferred and thus known. 
     EFFECT OF THE INVENTION 
     According to some embodiments of the invention, a point cloud can be effectively encoded and decoded, and the embodiments can be useful for compressing three dimensional representations of objects in the point cloud. Further, the methods of encoding and decoding according to some embodiments of the invention can generate a compressed representation that allows one to quickly or easily decode and reconstruct a coarse representation of the point cloud geometry without having to decode the entire file or bit-stream. 
     Although several preferred embodiments have been shown and described, it would be apparent to those skilled in the art that many changes and modifications may be made thereunto without the departing from the scope of the invention, which is defined by the following claims and their equivalents.