1. Frederic Payan, Marc Antonini
MSE Approximation for model-based compression of multiresolution semiregular meshes
Frederic Payan, Marc Antonini
I3S laboratory - CReATIVe Research Group
Universite de Nice Sophia Antipolis - FRANCE
13th European Conference on Signal Processing, Antalya, Turkey, 2OO5

2. Motivations
Design an efficient wavelet-based lossy compression method for the geometry of semiregular meshes Q
Entropy Coding
DWT
1010…
Wavelet coefficients
Semiregular mesh
The objective of these works is to design an efficient wavelet-based coder for the geometry of semiregular meshes.
Here is the global scheme of such a coder.
First a wavelet transform is applied on the input data. allows to obtain several subbands of wavelet coefficients. In our case, the input data is a semiregular mesh, that can be created with any remeshing method. In these works, we focus on the MAPS method of Lee, and the Normal remesher of Guskov, and we choose the butterfly scheme as WT.
These subbands are then quantized. Here, we choose the scalar quantization method.
Finally, an entropy coder is applied on the quantized coefficients to obtain the bitstream.
DWT: discrete wavelet transform
Q: quantization

3. Summary
Background
Let’s introduce now the background and the problem statement of my works.

4. One-level decomposition
I. Background
Wavelet transform …
Details
Details
Details
Details
One-level decomposition
N-level multiresolution decomposition:
low frequency (LF) mesh: connectivity + geometry
N sets of wavelet coefficients (3D vectors): geometry
Let’s first present the WT.
The principle of a WT is to project the signal in a space where the data compression will be more efficient
During the analysis step, a WT applied on a SR Mesh produces
- One Low Freq mesh that is a coarse mesh of the input data.
- and a set of 3D vectors which are the HF details (also named the wavelet coefficients) lost between the original data and the LF one.
By applying this analysis step several times, we finally obtain a multiresolution representation of the mesh,
composed by the LF mesh and N sets of WC.

5. Compression: principle
I. Background
Compression: principle D
Compression  Optimization of the rate-distortion (RD) tradeoff R
Multiresolution data  how dispatching pertinently the bits across the subbands in order to obtain the highest quality for the reconstructed mesh?
=> Solution: bit allocation process
From a general point of view, the objective of a compression method is to find the highest quality for the smallest file size. In other words, a compression method aims to optimize the rate-distortion tradoff.
Since we use a multiresolution representation of the original data, the problem of the RD optimization becomes: “how will we dispatch the bit budget across the wavelet subbands in order to obtain the highest quality for the mesh once reconstructed after decompression ?” One relevant solution is to use a bit allocation process whose the objective is to minimize the total D in function of the total R.

6. Proposed bit allocation
I. Background
Proposed bit allocation
find the set of optimal quantization steps that minimizes the total distortion at one user-given target bitrate
Distortion criterion: Mean Square Error
More precisely, the objective of the bit allocation we propose today is the following.
Determine the optimal quantization steps (used to encode the different coefficient subbands) that will minimize the total distortion at one user-given target bitrate.
As distortion criterion we use the MSE between the original mesh geometry and the reconstructed one after compression/decompression.
semiregular vertices
Quantized vertices
Number of vertices

7. Problem statement In order to speed the allocation process up, how
I. Background
Problem statement
The distortion is measured on the vertices (Euclidean Space)
The quantization is done on the coefficient subbands (Transformed space)
Now, I van present the problem statement of our works.
Note that during the bit allocation the distortion will be measured on the vertices, whereas the allocation will optimize the quantization of the wavelet coefficients.
So, each time we need to evaluate the distortion during the BAP, we have to apply the inverse wavelet transform before computing the MSE on the vertices => slow down the process.
Therefore, In order to speed the allocation process up, we would like to evaluate the MSE directly in the wavelet space,
The problem is finally, how can we express the MSE directly from the quantization errors of each coefficient subband?
In order to speed the allocation process up, how
expressing MSEsr directly from the quantization errors of each coefficient subband?

8. Summary
Background
MSE approximation for semiregular meshes

9. II. MSE approximation for semiregular meshes
Previous works
The MSE of data quantized by a wavelet coder can be approximated by a weighted sum of the MSE of each subband
The weights depend on the coefficients of the synthesis filters
But… shown only for data sampled on square grids and not for the mesh geometry!
In previous works, it has been already shown that biorthogonal filters weights the amount of quantization error which appears on a reconstructed data.
the MSE can be indeed approximated by a weighted sum of the MSE of each coefficient subband. the weights depending on the coefficients of the synthesis filters
Unfortunately, it has been shown only for data sampled on square grids like image for instance and not for for a mesh geometry. Our challenge is thus to develop an MSE approximation for data sampled on triangular grids
Challenge: develop an MSE approximation for a data sampled on a triangular grid

10. MSE approximation for meshes
II. MSE approximation for semiregular meshes
MSE approximation for meshes
Triangular sampling:
Principle of a wavelet coder/decoder for meshes n1 n2
LF coset (0) 2 3
HF coset 1 1
HF coset 2
HF coset 3
As I said previously, a mesh geometry is a data sampling on a triangular grid. So After a one decomposition of the mesh geometry, we obtain 4 cosets of details.
The first is the set of white dots which are the LF coefficients, and the third other cosets composed by the orange, the red and the green dots, represent the HF details. So a wavelet transform for the mesh geometry corresponds to a 4 channel filter bank and the corresponding coder is represented by this scheme.
We can see that 4 analysis filters followed by an subsampling are needed to obtain the 4 cosets. These 4 cosets are then quantized. During decompression, an upsampling followed by the application of 4 synthesis filters allows to obtain the reconstructed mesh geometry. M D Q + ^ h3 h0 g3 g0 s0 s3

11. Method: global steps We follow a deterministic approach
II. MSE approximation for semiregular meshes
Method: global steps
We follow a deterministic approach
quantization error  additive noise
We exploit the polyphase notations
with
During this presentation I can’t develop the entire method, because it s a little complex. So I just give the main keys.
What you have to know is that, contrary to previous works, we choose to use a deterministic approach and that we consider that the quantization error of a coefficient subband is modeled as an additive noise.
Moreover, We exploit the polyphase notation of the synthesis filters.
The polyphase notation is a matrix-based representation of a wavelet transform.
This allows to introduce the polyphase components in the formulation of the MSE approximation, which will be finally very useful, as you will see later.
Polyphase notation of the synthesis filters
the polyphase components

12. Solution For a one-level decomposition
II. MSE approximation for semiregular meshes
Solution
For a one-level decomposition
MSE approximation for a N-level decomposition
with
MSE of the coset i
Now I show you the solution of out problem.
We find also that for a one level decomposition, the MSE can be approximated by a weighted sum of MSE of the subband. But in our case, the weights depends on the polyphase components, contrary to previous works where the weights depend on the coefficients of the synthesis filter bank.
For A multilevel decomposition, we also find a weighted sum of MSE of the subband. In this case the weights logically depend on the weights previsously found.
with

13. Model-based algorithm
II. MSE approximation for semiregular meshes
Model-based algorithm
Probability density Function of the coordinate sets: Generalized Gaussian Distribution (GGD)
=> Model-based algorithm
Complexity : 12 operations / semiregular vertex
Example : 0.4 second (PIII 512 Mb Ram)
=> Fast allocation process
As said before, using this approximation as distortion criterion is a relevant solution to speep the allocation process up.
Moreover, in parallel I have shown that the PDF of the coordinate sets can be modeled by a Generalized Gaussian Distribution (GGD).
Hence, we can use a model-based algorithm initially proposed by Parisot, whichs exploit some theoretical models for the distortion and the rate of each subband. The complexity of this algorithm is very low, since it requires 12 operation / SR vertex to compute the set of optimal steps.

14. Summary Background MSE approximation for semiregular meshes
Experimental results
Let me show you experimental results

15. Simulations Two versions of our algorithm are proposed:
for MAPS meshes + Lifted butterfly scheme
for Normal meshes + Unlifted butterfly scheme
Comparison with the zerotree coders PGC (for MAPS meshes) and NMC (for Normal meshes)
Comparison criterion: PSNR based on the Hausdorff distance (computed with MESH)
To show that the weighted MSE can be exploited for any semiregular mesh independently of the remeshing method and for any WT, we deal with two
versions of this coder. The first version, named the MAPS coder, exploits the remesher MAPS of Lee and the lifted Butterfly-based transform.
The second version, named the NORMAL coder, exploits the Normal Remesher of Guskov and the unlifted Butterfly-based transform.
We compare our coders with the state-of-the-art zerotree coders. Our maps coder is compared to the coder PGC and our normal coder with NMC.
As comparison criterion we use the frequently used PSNR based on the Hausdorf distance.

16. Curves PSNR-Bitrate for our MAPS Coder
On this slide we can see the curves of the psnr in function of the total bitrate for the model bunny,
obtained with maps, according different coders.
IN dark blue the proposed coder with the weighted mse as distortion criterion during the bit allocation,
In green the pgc coder,
And in light blue and dotted line our coder but with an unweighted mse as criterion (in other words, we dont take into account the nonorthogonalty of the filters for this simulation)

17. Curves PSNR-Bitrate for the Normal Coder

18. Summary Background MSE approximation for semiregular meshes
Experimental results
Conclusion

19. V. Conclusions and perspectives
Contribution:
derivation of an MSE approximation for the geometry of semiregular meshes
Interest:
fast model-based bit allocation optimizing the quality of the quantized mesh
An efficient compression method for semiregular meshes outperforming the state of the art zerotree methods (up to 3.5 dB)

20. This is the end….
My homepage:

21. MSE approximation for meshes
II. MSE approximation for semiregular meshes
MSE approximation for meshes
Proposed MSE approximation is well-adapted for the lifting schemes because the polyphase components of such transforms depend on only the prediction and update operators
Notice that the MSE approximation that we propose is well-adapted for the lifting schemes because, as we can see below, the polyphase components of such transforms depend on only the prediction and update operators
Only the knowledge of these operators is sufficient to comptute the weights, contrary to previous works which need to develop the synthesis filter bank notation.

22. Geometrical comparison
IV. Experimental results
Geometrical comparison
NMC (62.86 dB)
Proposed algorithm (65.35 dB)
Bitrate = 0.71 bits/iv

23. MSE of one subband i MSE relative to the tangential components
III.Optimization of the Rate-Distorsion trade-off
MSE of one subband i
MSE relative to the tangential components
MSE relative to the normal components

24. Optimization of the Rate-Distorsion trade-off
III.Optimization of the Rate-Distorsion trade-off
Optimization of the Rate-Distorsion trade-off
Objective :
find the quantization steps that maximize the quality of the reconstructed mesh
Scalar quantization (less complex than VQ)
3D Coefficients => data structuring?

25. How solving the problem?
III.Optimization of the Rate-Distorsion trade-off
How solving the problem?
Find the quantization steps and lambda that minimize the following lagrangian criterion:
Method:
=> first order conditions
Distortion
Constraint relative to the bitrate

26. Solution Need to solve (2N + 4) equations with (2N + 4) unknowns
III.Optimization of the Rate-Distorsion trade-off
Solution
Need to solve (2N + 4) equations with (2N + 4) unknowns
PDF of the component sets: Generalized Gaussian Distribution (GGD) => model-based algorithm (C. Parisot, 2003)

27. Model-based algorithm
III.Optimization of the Rate-Distorsion trade-off
Model-based algorithm
compute the variance and α for each subband
compute the bitrates for each subband λ
Target bitrate reached?
Look-up tables
new λ
compute the quantization step of each subband