Erik de Jong & Willem Bouma

 Arithmetic Coding  Octree  Compression  Surface Approximation  Child Cells Configurations  Single Child cell configurations  Results  Questions

 Assign to every symbol a range from the interval [0-1]. The size of the range represents the probability of the symbol occurring. Example: A:60% [ 0.0, 0.6 ) B: 20% [ 0.6, 0.8 ) C: 10% [ 0.8, 0.9 ) D: 10% [ 0.9, 1.0 ) (end of data symbol)

Ranges: A – [0,0.6), B – [0.6,0.8), C – [0.8,0.9), D – [0.9,1) The example shows the decoding of A B C D

 Huffman coding is a specialized case of arithmetic coding  Every symbol is converted to a bit sequence of integer length  Probabilities are rounded to negative powers of two  Advantage: Can decode parts of the input stream  Disadvantage: Arithmetic coding comes much closer to the optimal entropy encoding

SQUEEL

 Estimate/Approximate as much as possible  Store only the differences w.r.t. the estimate  A better estimate ->  smaller numbers  lower entropy  better compression

 What is an Octree?

 Per cell we store only the occupied child cells  Options:  Store a single byte, each bit representing a child cell (for example )  Store the number of occupied cells e and a tupel T with the indices of the occupied cells (for example e=4, T ={0,1,4,5})

 We will approximate/estimate/compress:  The surface  Number of non-empty child cells  Child cell configuration  Index compression  Single child cell configuration

 Every level of the octree will yield a preliminary approximation Q of the complete point cloud P  For a cell that is to be subdivided:  Predict surface F based on Moving Least Squares (MLS) on k nearest points in Q

 Prediction of number of non-empty cells e  Based on estimation of the sampling density ρ  Local sampling density ρ i at point p i in P :  k, nearest points  p i, point in the point cloud P  q i, point on the surface approximation Q

 Sampling density:  Guess the number of child cells e based on:  The area of the plane F  The sampling density ρ

 Quality of prediction Graphs show the difference between te estimated value and the true value. (a)The level 5 octree (b)The level 7 octree (c)The entire octree.

 Given the number of non-empty child cells e there are only a limited number of configurations:

 We have an array with all weighted possible configurations, sorted in ascending order  Each configuration of the subdivision is encoded as an index of the array.  Common configurations get lower weights, means smaller indices, means lower entropy.

 Cell centers tend to be close to F.

 To find the weight of a configuration:  Sum up the (L1) distances from the cell centers to F

 Index of the configuration in the sorted array is encoded using arithmetic coding under two contexts  First context: the octree level of the cell C  Second context: the expressiveness e(F)

 e(F) reflects the angle of the plane to the coordinate directions

 In order to use e(F) as a context for arithmetic coding it has to be quantized.  It has been found that five bins was sufficient and delivered the best results.

 We can exploit an observation for cells that have only one occupied child cell.  Scanning devices often have a regular sampling grid.  It is possible to predict samples on the surface, rather than just close to the surface.  This is relevant for the finer levels in the octree hierarchy.

 For cells with only one child cell we can predict T based on the nearest neighbours of points m, centroid of the k nearest neighbours

 Quite suprisingly, the cell center projections on F that are farthest away are most likely to be occupied.  The area farther away can be seen as undersampled.  So a sample in that area becomes more likely since we expect the surface to be regular and no undersampling should exist.

 The weights for the eight possible configurations are given as c(T), cell center of T prj(F,c(T)), projection of c(T) on F

 Extra attributes can be encoded with an octree  Color  Normals

 Compressing color  Same two-step method as with coordinates  Different prediction functions are needed

ModelNumber of points Raw sizeCompressed (bpp) Compressed size Dragon ,44 MB5,061,66 MB Venus~ KB11,27184 KB Rabbit~ KB11,3793 KB MaleWB~ KB8,87160 KB (bpp) Bits Per Point Raw size is assumed to use 3 times 4 Bytes per point. The octree uses 12 levels. Except for the Dragon for which it is unknown.

(b) uses 1.89 bpp

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