**6. Conclusions**

In this book chapter, we propose a constructive methodology for the design of scalable wavelet-based mesh compression systems. Our design strategy differs from conventional designs which simply opt for reusing methods from wavelet-based image coding for the design of mesh coding systems. In particular, our methods are motivated by an informationtheoretic analysis of the statistical dependencies between wavelet coefficients which shows that, intraband dependencies are systematically stronger than interband ones for both normal and non-normal meshes, and that composite models are the best. We also investigate the optimality of successive approximation quantization, commonly used in scalable compression, in the context of wavelet-based mesh compression. Using a Laplacian mixture model, it is shown that successive approximation quantization is an acceptable, but in general not an optimal solution. Anchored in these results, novel intraband and composite coding systems are presented which improve the state-of-the-art in scalable mesh compression, both in terms of scalability and compression efficiency.

## **7. Appendix**

588 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

0.036 bpv, 54.6 dB 0.121 bpv, 62.4 dB 0.179 bpv, 65.5 dB 1.073 bpv, 84.8 dB 1.384 bpv, 90.7 dB

0.036 bpv, 53.9 dB 0.121 bpv, 61.9 dB 0.179 bpv, 65.2 dB 1.073 bpv, 83.4 dB 1.384 bpv, 88.9 dB

Fig. 12. Visual comparison of normal Skull mesh using (top row) the 3xC codec and (bottom row) the PGC codec. The red color intensity reflects the distortion with respect to the uncompressed semi-regular mesh. The rate for the base mesh is not included in the reported

The visual comparisons of the normal mesh Skull at different bpv are shown in Fig. 12. Though, at first glance it may appear that both codecs perform very similar, small differences are noticeable when investigating the meshes more closely. When examining the comparison at 0.036 bpv, we notice that the PGC codec preserves more details in Skull's teeth. The green shade for 3xC at rate 0.179 bpv, however, seems more pure compared to PGC for which it is rather yellowish green. We also observe that no red regions are present

In this book chapter, we propose a constructive methodology for the design of scalable wavelet-based mesh compression systems. Our design strategy differs from conventional designs which simply opt for reusing methods from wavelet-based image coding for the design of mesh coding systems. In particular, our methods are motivated by an informationtheoretic analysis of the statistical dependencies between wavelet coefficients which shows that, intraband dependencies are systematically stronger than interband ones for both normal and non-normal meshes, and that composite models are the best. We also investigate the optimality of successive approximation quantization, commonly used in scalable compression, in the context of wavelet-based mesh compression. Using a Laplacian mixture model, it is shown that successive approximation quantization is an acceptable, but in general not an optimal solution. Anchored in these results, novel intraband and composite coding systems are presented which improve the state-of-the-art in scalable mesh

for 3xC at rate 1.073 bpv, whereas some are visible for PGC at the same rate.

compression, both in terms of scalability and compression efficiency.

rate values.

**6. Conclusions** 

The output distortion *DL* of a Laplacian PDF, quantized using an *n* level EDSQ and reconstructed using midpoint reconstruction, can be written as:

$$D\_{L}(\mathbf{Q}\_{\boldsymbol{\xi}\_{n},\Lambda\_{n}}) = \underbrace{\mathcal{J}\Big\|\_{0}^{(1-\xi\_{n})\Lambda\_{n}}}\_{D\_{\mathrm{DZ}}}\mathbf{x}^{2}e^{-\boldsymbol{\lambda}\cdot\mathbf{x}}\,d\mathbf{x}} + \underbrace{\mathcal{J}\sum\_{k=1}^{\infty}\int\_{(k-\xi\_{n})\Lambda\_{n}}^{(k+1-\xi\_{n})\Lambda\_{n}}\left(\mathbf{x} - (k+0.5-\xi\_{n})\Lambda\_{n}\right)^{2}e^{-\boldsymbol{\lambda}\cdot\mathbf{x}}\,d\mathbf{x}}\_{D\_{\mathrm{RST}}}.$$

where *DDZ* and *DREST* denote the distortion contributions of the deadzone and the other quantization cells, respectively. By proper substitution and letting

$$\sum\_{k=1}^{\infty} e^{-\lambda \Lambda\_n k} = \frac{e^{-\lambda \Lambda\_n}}{1 - e^{-\lambda \Lambda\_n}} \text{, as } e^{-\lambda \Lambda\_n} \le 1 \text{,} \tag{27}$$

the following closed-form expression for the distortion is obtained:

$$D\_{L}(Q\_{\delta\_{n},\Lambda\_{n}}) = \frac{2}{\lambda^{2}} + e^{-\lambda\Lambda\_{n}\delta\_{n}} \left| \left(\frac{1}{4} - \delta\_{n}^{2}\right)\Lambda\_{n}^{2} - \left(2\delta\_{n} + \coth\left(\frac{\lambda\Lambda\_{n}}{2}\right)\right)\frac{\Lambda\_{n}}{\lambda}\right| \tag{28}$$

where 1 *n n* .

Similarly, the output rate *RL* of a Laplacian PDF, quantized using an *n* level EDSQ can be written as:

$$R\_L(Q\_{\boldsymbol{\xi}\_n, \boldsymbol{\Lambda}\_n}) = \underbrace{-2\left(\frac{\lambda}{2} \int\_0^{(1-\xi\_n)\boldsymbol{\Lambda}\_n} e^{-\lambda x} \, d\mathbf{x}\right) \log\_2 2\left(\frac{\lambda}{2} \int\_0^{(1-\xi\_n)\boldsymbol{\Lambda}\_n} e^{-\lambda x} \, d\mathbf{x}\right)}\_{R\_{\boldsymbol{\Omega}\boldsymbol{\Sigma}}}...$$

$$\ldots - \underbrace{2\sum\_{k=1}^\infty \left(\frac{\lambda}{2} \int\_{(k-\xi\_n)\boldsymbol{\Lambda}\_n}^{(k+1-\xi\_n)\boldsymbol{\Lambda}\_n} e^{-\lambda x} \, d\mathbf{x}\right) \log\_2 2\left(\frac{\lambda}{2} \int\_{(k-\xi\_n)\boldsymbol{\Lambda}\_n}^{(k+1-\xi\_n)\boldsymbol{\Lambda}\_n} e^{-\lambda x} \, d\mathbf{x}\right)}\_{R\_{\boldsymbol{\Omega}\boldsymbol{\Sigma}\boldsymbol{\Gamma}}}...$$

Again making use of the summation reduction identity of (27) along with the identity

$$\sum\_{k=1}^{\infty} e^{-\lambda \Lambda\_n k} \log\_2 \left( e^{-\lambda \Lambda\_n k} \right) = \log\_2 \left( e^{-\lambda \Lambda\_n} \right) \sum\_{k=1}^{\infty} k \left( e^{-\lambda \Lambda\_n} \right)^k = \frac{\log\_2 \left( e^{-\lambda \Lambda\_n} \right) e^{-\lambda \Lambda\_n}}{\left( 1 - e^{-\lambda \Lambda\_n} \right)^2}, \quad \lambda \to 0$$

the expression for the rate can be reduced to the following closed-form:

$$R\_{\rm L}(Q\_{\delta\_n, \Lambda\_n}) = c\_{\delta} \log\_2 \left( \frac{2d\_{\delta\_n}}{d\_1 c\_1^{\prime d\_1} e^{\Lambda \Lambda\_n (1 - \delta\_n)} d\_{\delta\_n}^{\prime \epsilon\_{\delta\_n}}} \right) \tag{29}$$

where *n n <sup>n</sup> c e* (hence 1 *<sup>n</sup> c e* ) and 1 *n n d c* (hence 1 1 *d c* 1 ).

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