HighPass Quantization for Mesh Encoding

1
High-Pass Quantization
for Mesh Encoding

• Olga Sorkine, Daniel Cohen-Or, Sivan Toledo
• Eurographics Symposium on Geometry Processing,
  Aachen 2003

2
Overview

• Geometry quantization
• Visual quality
• Connection to spectral properties

3
Geometry quantization introduction

• Each mesh vertex is represented by Cartesian
  coordinates, in floating-point.
• Geometry compression requires quantization,
  normally 10-16 bits/coordinate

(xi, yi, zi)
4
Geometry quantization introduction

• Using smoothness assumptions, the quantized
  coordinates are predicted and the prediction
  errors are entropy-coded Touma and Gotsman 98

5
Quantization error

• Quantization necessarily introduces errors.
• The finer the sampling, the more it suffers.

6
Quantization error

• An example coarsely-sampled sphere

original
Quantized to 8 bits/coordinate
7
Quantization error

• A finely-sampled sphere with the same quantization

original
Same quantization to 8 bits/coordinate
8
Quantization error discussion

• Quantization of the Cartesian coordinates
  introduces high-frequency errors to the surface.
• High-frequency errors alter the visual appearance
  of the surface affect normals and lighting.
• Only conservative quantization (usually 12-16
  bits) avoids these visual artifacts.

9
Quantization our approach

• Transform the Cartesian coordinates to another
  space using the Laplacian matrix of the mesh.
• Quantize the transformed coordinates.
• The quantization error in the regular Cartesian
  space will have low frequency.
• Low-frequency errors are less apparent to a human
  observer.

10
Relative (laplacian) coordinates

• Represent each vertex relatively to its
  neighbours

average of the neighbours
the relative coordinate vector
11
Laplacian matrix

• A? Matn?n(R) the adjacency matrix
• D? Matn?n(R) the degree-diagonal matrix
• Then, the Laplacian matrix L? Matn?n(R) is

12
Laplacian matrix

• The previous form is not symmetric. We will use
  the symmetric Laplacian

13
Properties of L

• Sort the eigenvalues of L in accending order
• We can represent the geometry in Ls eigenbasis

frequencies
eigenvectors
low frequency components
high frequency components
14
Previous usages of Laplacian matrix

• Karni and Gotsman 00 progressive geometry
  compression
• Use the eigenvectors of L as a new basis of Rn
• Transmit the coordinates according to this
  spectral basis
• First transmit the lower-eigenvalue coefficients
  (low frequency components), then gradually add
  finer details by transmitting more coefficients.

15
Previous usages of Laplacian matrix

• Taubin 95 surface smoothing
• Push every vertex towards the centroid of its
  neighbours, i.e.
• v (I ?L)v
• Iterate, with positive and negative values of ?
  (to reduce shrinkage effect)

16
Previous usages of Laplacian matrix

• Ohbuchi et al. 01 mesh watermarking
• Embed a bitstring in the low-frequency
  coefficients
• Changes in low-frequency components are not
  visible
• Alexa 02 morphing using relative coordinates
• Produces locally smoother morphs
• Gotsman et al. 03
• A more general class of Laplacian matrices
• Mesh embedding on a sphere using eigenvectors

17
Quantizing the ? - coordinates

• Transform Cartesian to ?-coordinates
• Quantize ?-coordinates
• To get back Cartesian coordinates

(fixed-point quantization)
18
Discussion of the linear system

• The matrix L is singular, so L?1 doesnt exist.
• Adding one anchor point fixes this problem
  (substitute one vertex (x, y, z) removes
  translation degrees of freedom)

19
Discussion of the linear system

• By quantizing the ?, we put high-frequency error
  into ?.
• L has very small eigenvalues, so L?1 has very
  large eigenvalues (? ?? 1/?)
• Thus, L?1 amplifies small errors and reverses the
  frequencies.

Small quantization error for ?, high frequency
NOT so small !! low frequency
20
Spectrum of quantization error

• Write x as x a1 e1 a2 e2 an en
• Therefore, ? Lx ?1a1 e1 ?2a2 e2
  ?n-1an-1 en-1 ?nan en
• Quantization error for ? (? ? ? q? ) is
  
• q? c1 e1 c2 e2
  cn-1 en-1 cn en
• Resulting error in x
• qx L?1 q? (1/?1)c1 e1 (1/?2)c2 e2
  (1/?n-1)cn-1 en-1 (1/?n)cn en

large ?i high frequencies
Small ?i low frequencies
low frequencies small ci
high frequency error here ci are large
(1/?i) is small attenuates high-frequency
errors
(1/?i) is large amplifies low-frequency errors
Thus, the error in x will contain strong
low-frequency components but weak
high-frequency components.
21
Discussion of the linear system

• Example of low-frequency error
• Find the differences between the horses

22
Discussion of the linear system

• Example of low-frequency error
• This one is the original horse model

23
Discussion of the linear system

• Example of low-frequency error
• This is the model after quantizing ? to 8
  bits/coordinate
• There is one anchor point (front left leg)

24
Making the error lower

• We add more anchor points, whose Cartesian
  coordinates are known, as well as the ? -
  coordinates.
• This nails the geometry in place, reducing the
  low-frequency error

25
Rectangular Laplacian

• We add equations for the anchor points
• By adding anchors the matrix becomes rectangular,
  so we solve the system in least-squares sense

L
constrained anchor points
26
Choosing the anchor points

• A greedy scheme.
• Add one anchor point at a time.
• Each time nail down the vertex that achieved the
  maximal error after reconstruction.
• This process is slow, but it is done only by the
  encoder.
• Only a small number of anchors is needed. We
  experiment with 0.1, which gives very good
  results.

27
The effect of anchors on the error
Positive error vertex moves outside of the
surface
0
Negative error vertex moves inside the surface
?-quantization 7b/c 4 anchors
?-quantization 7b/c 20 anchors
?-quantization 7b/c 2 anchors
Cartesian quantization 8b/c
28
Visual error metric

• Euclidean distance between and does not
  faithfully represent the visual error (Cartesian
  quantization errors are small but the normals
  change a lot...)
• Karni and Gotsman 2000 propose a visual
  metric
• x xvis ?x x2 (1 ?)GL(x)
  GL(x)2
• ? 0.5
• We are not sure that?? should be 0.5

29
Visual error metric

• We measured the two error components separately
• Mq x x2
  
• Sq GL(x) GL(x)2
• Evis ? Mq (1 ?) Sq

30
Rate-distortion curves
31
Some results
We compare to Touma-Gotsman predictive coder that
uses Cartesian quantization
original
?-quantization, entropy 7.62
Cartesian quantization, entropy 7.64
Evis?0.5 2.5 Evis?0.15 2.6
Evis?0.5 5.3 Evis?0.15 2.3
32
Some results
We compare to Touma-Gotsman predictive coder that
uses Cartesian quantization
original
?-quantization, entropy 6.69
Cartesian quantization, entropy 7.17
Evis?0.5 1.8 Evis?0.15 0.9
Evis?0.5 4.8 Evis?0.15 4.9
33
Some results
We compare to Touma-Gotsman predictive coder that
uses Cartesian quantization
original
?-quantization, entropy 10.3
Cartesian quantization, entropy 10.3
Evis?0.5 6.4 Evis?0.15 3.9
Evis?0.5 5.0 Evis?0.15 5.1
34
Time statistics

• The least-squares system was solved using QR
  factorization of the normal equations
• Implementation on 2 GHz P4 using TAUCS library

35
Conclusions

• The spectrum of the quantization error affects
  the visual quality of the quantized mesh
• We have proposed a quantization method that
  concentrates the error in the low-frequencies
• The method requires the computational effort of
  solving a linear least-squares system.
• Much more research is needed to fully understand
  the spectral behavior of meshes

36
Acknowledgements

• Christian Rössl and Jens Vorsatz from
  Max-Planck-Institut für Informatik for the models
• Israel Science Foundation founded by the Israel
  Academy of Sciences and Humanities
• The Israeli Ministry of Science
• IBM Faculty Partnership Award
• German Israel Foundation (GIF)
• EU research project Multiresolution in Geometric
  Modelling (MINGLE), grant HPRN-CT-1999-00117.

37
Thank you!
38
Discussion of the linear system

• Another example

39
Discussion of the linear system

• Original model

40
Discussion of the linear system

• This is the model after quantizing ? to 7
  bits/coordinate, one anchor

41
Invertible square Laplacian

• We could simply eliminate the anchors from the
  system, erasing the rows and the columns of the
  anchor vertices
• Use this reduced Laplacian instead of L and
  remember the anchors (x, y, z) positions
  separately

42
Invertible Laplacian artifacts

• Produces bad results when we quantize ?, because
  no smoothness constraints are posed on the anchors