Erik de Jong & Willem Bouma
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Arithmetic Coding
Octree
Compression
 Surface Approximation
 Child Cells Configurations
 Single Child cell configurations
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Results
Questions
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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:
B:
C:
D:
60%
20%
10%
10%
[ 0.0,
[ 0.6,
[ 0.8,
[ 0.9,
0.6 )
0.8 )
0.9 )
1.0 )
(end of data symbol)
A
B
C
D
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 0.538.
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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
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Estimate/Approximate as much as possible
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Store only the differences w.r.t. the estimate
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A better estimate ->
 smaller numbers
 lower entropy
 better compression
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What is an Octree?
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Per cell we store only the occupied child cells
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Options:
 Store a single byte, each bit representing a child cell
(for example 11001100)
 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})
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We will approximate/estimate/compress:
 The surface
 Number of non-empty child cells
 Child cell configuration
 Index compression
 Single child cell configuration
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Every level of the octree will yield a preliminary
approximation Q of the complete point cloud P
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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 ρi at point pi in P:
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k, nearest points
pi, point in the point cloud P
qi, point on the surface approximation Q
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Sampling density:
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Guess the number of child cells e based on:
 The area of the plane F
 The sampling density ρ
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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.
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Given the number of non-empty child cells e
there are only a limited number of
configurations:
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We have an array with all weighted possible
configurations, sorted in ascending order
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Each configuration of the subdivision is
encoded as an index of the array.
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Common configurations get lower weights,
means smaller indices, means lower entropy.
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Cell centers tend to be close to F.
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To find the weight of a configuration:
 Sum up the (L1) distances from the cell centers to F
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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)
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e(F) reflects the angle of the plane to the
coordinate directions
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In order to use e(F) as a context for arithmetic
coding it has to be quantized.
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It has been found that five bins was sufficient
and delivered the best results.
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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.
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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
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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.
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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
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Extra attributes can be encoded with an
octree
 Color
 Normals
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Compressing color
 Same two-step method as with coordinates
 Different prediction functions are needed
Model
Number of
points
Raw size
Compressed
(bpp)
Compressed
size
Dragon
2.748.318
31,44 MB
5,06
1,66 MB
Venus
~134.000
1569 KB
11,27
184 KB
Rabbit
~67.000
768 KB
11,37
93 KB
MaleWB
~148.000
1734 KB
8,87
160 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
?