**4. Analysis of wavelet coefficient dependencies**

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

ratio (*SNR*) difference which is utilized to measure the performance gap of different

*SNR SNR R SNR R*

which is computed over a rate range for *N* rate points, where *SNR R*( ) denotes the

total distortion in the transform domain. The difference in *SNR* is computed relative to the uniform embedded quantizer (UEQ), i.e., *z* 1 . *SNR* for five embedded deadzone

We determined experimentally that at lower standard deviation ratios, *SNR* is positive

performs better compared to all other quantizers. Similarly, *z* 2 (i.e. the SAQ) performs the

results show that one cannot determine a single embedded quantizer that provides the best performance for *all* 3D meshes. However, an optimal quantizer per wavelet coordinate can

Given the fact that SAQ is closely linked to bit-plane coding and that it can be implemented using simple binary arithmetic, one concludes that SAQ is not an optimal, but an acceptable

 2 1 

2 1 120 . For

quantizers is plotted in Fig. 7. over a wide range of standard deviation ratios

 

general, small standard deviation ratios correspond to

 

Fig. 7. *SNR* difference for five EDSQs with respect to UEQ.

solution in scalable coding of meshes.

 

normal meshes, while higher ratios correspond to

be determined based on the corresponding

2

1 1

 

extracted from the model.

2 1 120 , SAQ is not always the optimum, but lies not far

2 1 290 600 , while *z* 2.5 performs the best for 600

2 1 120 , the difference between SAQ and the UEQ is significant, and hence

<sup>10</sup> *SNR* 10log ( ) *D* is computed in dBs, where *D* is the

0.9 is considered, as mentioned in Section 3.2.

2 1 120 290 , the quantizer with *z* 1.5

1 , observed in normal meshes. These

 2 1 

close to 1 , observed in non-

 2 1 . In

. In Fig.

<sup>1</sup> () () 

*z z*

*<sup>N</sup>* ,

embedded quantizers is defined as:

discrete *SNR*-rate function. The

7., the commonly observed proportion

and the UEQ is optimal for

 

from the optimum.

UEQ is the optimal choice. For

Overall, for

best in the range

Similar to images, parent-children and neighboring wavelet coefficient dependencies exist in wavelet decomposed mesh structure. In Fig. 8 (middle, right), the positions of the wavelet coefficients at different levels of the transform are shown with the help of white and dark circles. In particular, wavelet coefficients have a one-to-one correspondence with the edges of the coarser mesh. For each wavelet coefficient there are rings of neighboring coefficients which lie in the same wavelet subband – see Fig. 8 (right). Also, a set of four wavelet coefficients have a parent coefficient at the next coarser resolution – see Fig. 8 (middle, right).

Fig. 8. Parent-children and neighboring wavelet coefficients: actual mesh (left); coarser meshes after one (middle), and after two wavelet decomposition levels (right).

Statistical intraband dependencies exist between neighboring coefficients of each resolution level. The main reason for the existence of these dependencies is the smoothness of the surface. Wavelet coding paradigms that exploit the intraband dependencies between the wavelet coefficients are known as intraband wavelet codecs such as block-based coding techniques (Munteanu et al., 1999a), quadtree coding approaches (Munteanu et al., 1999b), and the EBCOT codec employed in the JPEG-2000 scalable image coding standard (Taubman, 2000).

Statistical dependencies also exist between the parent and descendants (children) due to the natural decay of the coefficients' magnitude for increasing frequencies. In other words, if a parent coefficient magnitude is below a certain threshold, then there is a high probability that the magnitude of its descendants will be also below this threshold. This corresponds to the so-called zerotree-model, firstly introduced by Shapiro in (Shapiro, 1993). The wavelet coding paradigms that exploit the parent-children dependencies are known as interband wavelet codecs.

Finally, there is a third category of coding paradigms, exploiting both the interband and intraband statistical dependencies between the wavelet coefficients. They are generally known as composite codecs, EZBC (Hsiang & Woods, 2000) and the ECECOW approach of (Wu, 1997) are typical examples of codecs in this category.

In the following, an information theoretical analysis of the aforementioned coefficient dependencies is presented. Our aim is to single out the type of dependencies which can ensure best compression performance in the context of wavelet-based mesh compression.

#### **4.1 Mutual information analysis**

The mutual information is the reduction in the entropy of one random variable due to the knowledge of the other random variable.

$$I(X;Y) = h(X) - h(X/Y) \ , \tag{22}$$

Optimized Scalable Wavelet-Based Codec Designs for Semi-Regular 3D Meshes 583

hand, ensures that there are always sufficient numbers of observations in each bin, and

Mesh Type MESH INTRABAND INTERBAND COMPOSITE INTRABAND INTERBAND COMPOSITE

Table 3. Average mutual information in bits for several non-normal and normal meshes.

Table 3 shows the average mutual information results for interband, intraband and composite dependencies for various mesh models. Since in mesh coding three different components need to be coded for each vertex position in space, the average mutual information ( )/3 *avg X Y Z I III* is reported instead of the mutual information for the three components individually. It is observed from Table 3 that for both normal and non-normal meshes mutual information of interband models is the least, independent of the wavelet transform employed. On the other hand mutual information for intraband models is significantly higher than for the interband models. Finally, composite models which gather the characteristics of both interband and intraband models exhibit even higher mutual information than interband or intraband models alone. Mathematically we can summarize

Experimental results for the mutual information based estimation of interband, intraband and composite dependencies seem to indicate that exploiting the composite dependencies should be preferred. Additionally, it is important to point out that favoring intraband over zerotree-based interband models brings along the additional benefit of resolution scalability. Specifically, by following an intraband codec design, only those wavelet subbands that are needed in order to reconstruct a target mesh resolution-level need to be encoded, while the others can be discarded. This does not hold in case of interband and composite codec designs, due to the tree-structures that span all the wavelet decomposition levels. Since composite models cannot be discarded altogether due to their highest mutual information property, a careful implementation of a composite mesh coding system needs to be carried out into order to get the benefit of both the higher compression efficiency and the resolution

Finally, it is important to point out that the differences in terms of mutual information do not give any indication about the final performance differences between interband, intraband and composite coding systems. Hence, an actual development and comparison of such coding systems is needed in order to experimentally validate the conclusions of this

**Butterfly Loop** 

*IXP IX IXP* ; ; ;; *<sup>X</sup>* **n n** *X XX* . (26)

Venus 0.3727 0.1902 0.6886 0.8320 0.5591 1.5847 Bunny 0.3960 0.1992 0.6844 0.8033 0.5628 1.5427 Horse 0.5615 0.2869 0.9873 1.0482 0.6943 1.9684 Rabbit 0.4048 0.2017 0.7089 0.8996 0.6450 1.7425 Feline 0.8277 0.2134 1.0696 1.1471 0.6285 2.0287

Venus 0.3052 0.2130 0.5741 - - - Skull 0.3381 0.2922 0.7001 - - - Dino 0.3043 0.2804 0.6672 - - -

provides reliable estimates of the mutual information.

**Non-Normal** 

**Normal** 

our numerical findings as:

scalable decoding at the same time.

It is known that *IXY IYX* (,) (, ) . In the wavelet domain, we define the following mutual information quantities:

(; ) *<sup>X</sup> IXP* : denotes the mutual information between a wavelet coefficient *X* and its parent coefficient *PX* .

(; ) **n***<sup>X</sup> I X* : denotes the mutual information between a wavelet coefficient *X* and its neighboring wavelet coefficients 1, 2, , **n** [ , ,.... ] *X X X NX nn n* .

*IXP* (; ; ) *X X* **n** : denotes the composite mutual information.

From the basics of information theory (Cover & Thomas, 1991), we know that:

$$I(X; P\_X; \mathbf{n}\_X) \ge I(X; \mathbf{n}\_X) \quad \text{and} \quad I(X; P\_X; \mathbf{n}\_X) \ge I(X; P\_X). \tag{23}$$

For the estimation of (; ) **n***<sup>X</sup> I X* , we need to estimate the joint PDF (, ) *<sup>x</sup> p x n* which can have high dimensionality depending on the number of considered neighbors. Since the amount of data needed to accurately estimate a PDF increases exponentially with its dimensionality, it is difficult to reliably estimate a high-dimensional PDF. To alleviate this problem, the reduction in dimensionality as proposed in (Liu & Moulin, 2000) is used here. We summarize the neighborhood of *X* through a so-called summarizing function *T g* ( ) **n***<sup>X</sup>* . This function maps the neighboring wavelet coefficients to a single value. We note that such a many-to-one summarizing function cannot increase the mutual information, i.e.,

$$I(X; \mathbf{n}\_X) \ge I(X; T) \,. \tag{24}$$

Equality in the above equation holds if **n***<sup>X</sup> T X* forms a Markov chain. The summarizing function used in our analysis is:

$$T = f(\mathbf{n}\_X) = \sum\_{i=1}^{N} \left(n\_{i,X}\right)^2. \tag{25}$$

Due to this summarizing function, it is sufficient to compute the joint PDF *p*(,) *x t* , *t* is a realization of the random variable *T* , instead of (, ) *<sup>x</sup> p x n* , for the estimation of the intraband mutual information (; ) **n***<sup>X</sup> I X* .

In our analysis, the mutual information for the defined quantities is estimated using the adaptive partitioning method (Darbellay & Vajda, 1999) instead of the traditional histogram method. This is because the histogram method highly depends on the bin size and for a small bin size there may not be sufficient number of observations in some bins to make a correct estimate. The adaptive partitioning method (Darbellay & Vajda, 1999) on the other

The mutual information is the reduction in the entropy of one random variable due to the

It is known that *IXY IYX* (,) (, ) . In the wavelet domain, we define the following mutual

(; ) *<sup>X</sup> IXP* : denotes the mutual information between a wavelet coefficient *X* and its parent

(; ) **n***<sup>X</sup> I X* : denotes the mutual information between a wavelet coefficient *X* and its

For the estimation of (; ) **n***<sup>X</sup> I X* , we need to estimate the joint PDF (, ) *<sup>x</sup> p x n* which can have high dimensionality depending on the number of considered neighbors. Since the amount of data needed to accurately estimate a PDF increases exponentially with its dimensionality, it is difficult to reliably estimate a high-dimensional PDF. To alleviate this problem, the reduction in dimensionality as proposed in (Liu & Moulin, 2000) is used here. We summarize the neighborhood of *X* through a so-called summarizing function *T g* ( ) **n***<sup>X</sup>* . This function maps the neighboring wavelet coefficients to a single value. We note that such

Equality in the above equation holds if **n***<sup>X</sup> T X* forms a Markov chain. The

Due to this summarizing function, it is sufficient to compute the joint PDF *p*(,) *x t* , *t* is a realization of the random variable *T* , instead of (, ) *<sup>x</sup> p x n* , for the estimation of the intraband

In our analysis, the mutual information for the defined quantities is estimated using the adaptive partitioning method (Darbellay & Vajda, 1999) instead of the traditional histogram method. This is because the histogram method highly depends on the bin size and for a small bin size there may not be sufficient number of observations in some bins to make a correct estimate. The adaptive partitioning method (Darbellay & Vajda, 1999) on the other

 **<sup>n</sup>** <sup>2</sup>

, 1 () . *N X iX i Tf n*

( ; ; ) ( ; ) ( ; ; ) ( ; ). *XX X* **nn n** and *XX X IXP IX IXP IXP* (23)

(; ) (;) *<sup>X</sup> IX IXT* **n** . (24)

(25)

From the basics of information theory (Cover & Thomas, 1991), we know that:

a many-to-one summarizing function cannot increase the mutual information, i.e.,

*IXY hX hX Y* (;) () ( /) , (22)

**4.1 Mutual information analysis** 

information quantities:

coefficient *PX* .

knowledge of the other random variable.

neighboring wavelet coefficients 1, 2, , **n** [ , ,.... ] *X X X NX nn n* . *IXP* (; ; ) *X X* **n** : denotes the composite mutual information.

summarizing function used in our analysis is:

mutual information (; ) **n***<sup>X</sup> I X* .


hand, ensures that there are always sufficient numbers of observations in each bin, and provides reliable estimates of the mutual information.

Table 3. Average mutual information in bits for several non-normal and normal meshes.

Table 3 shows the average mutual information results for interband, intraband and composite dependencies for various mesh models. Since in mesh coding three different components need to be coded for each vertex position in space, the average mutual information ( )/3 *avg X Y Z I III* is reported instead of the mutual information for the three components individually. It is observed from Table 3 that for both normal and non-normal meshes mutual information of interband models is the least, independent of the wavelet transform employed. On the other hand mutual information for intraband models is significantly higher than for the interband models. Finally, composite models which gather the characteristics of both interband and intraband models exhibit even higher mutual information than interband or intraband models alone. Mathematically we can summarize our numerical findings as:

$$I(X; P\_X) \ll I(X; \mathbf{n}\_X) \ll I(X; P\_X; \mathbf{n}\_X) \cdot \tag{26}$$

Experimental results for the mutual information based estimation of interband, intraband and composite dependencies seem to indicate that exploiting the composite dependencies should be preferred. Additionally, it is important to point out that favoring intraband over zerotree-based interband models brings along the additional benefit of resolution scalability. Specifically, by following an intraband codec design, only those wavelet subbands that are needed in order to reconstruct a target mesh resolution-level need to be encoded, while the others can be discarded. This does not hold in case of interband and composite codec designs, due to the tree-structures that span all the wavelet decomposition levels. Since composite models cannot be discarded altogether due to their highest mutual information property, a careful implementation of a composite mesh coding system needs to be carried out into order to get the benefit of both the higher compression efficiency and the resolution scalable decoding at the same time.

Finally, it is important to point out that the differences in terms of mutual information do not give any indication about the final performance differences between interband, intraband and composite coding systems. Hence, an actual development and comparison of such coding systems is needed in order to experimentally validate the conclusions of this

Optimized Scalable Wavelet-Based Codec Designs for Semi-Regular 3D Meshes 585

Using the octree-based bitplane coding, separate symbol streams are first generated for all bitplanes of each resolution subband. Depending on the type of scalability, i.e., resolution or quality scalability, the encoded symbol streams are entropy coded using a predefined progression order of bitplanes. For quality scalability, bitplanes of certain significance, from all resolution subbands, are first encoded before encoding the bitplanes of lower significance. However, in resolution scalability mode, all bitplanes of a lower resolution subband are progressively encoded before encoding the next higher resolution

We compared the SIM codec with the PGC codec for both normal and non-normal 3D meshes. The decoded meshes are compared against the original semi-regular input meshes using the peak signal-to-noise ratio (PSNR) as the distortion metric, which is defined

where *peak* and *RMS* denote the size of the bounding box and the root mean squared error calculated on the distances between the decoded vertex positions with respect to the original

Fig. 9 depicts PSNR versus bitrate (bits per semi-regular vertex) plots, evaluated for the semiregular non-normal Venus and Bunny meshes using the Butterly transform. The results demonstrate that for both meshes, SIM yields superior performance when compared to PGC.

(a) (b) Fig. 9. PSNR versus bitrate for non-normal mesh models in the quality scalability mode: (a) Venus, (b) Bunny . The lifted Butterly transform is employed for all three codecs.

40

60

80

*PSNR* (*dBs*) 100

120

140

3xC SIM PGC

The averaged gain in PSNR when compressing the Venus and Bunny meshes goes up to 2.22 dB and 2.35 dB, respectively. One may also notice the increasing performance difference with increasing bitrates; this indicates that the SIM coder tends to code the high frequency information more efficiently. For the spatially adaptive wavelet transform (SAWT) the

compression results are reported in (Denis et al., 2010a).

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>40</sup>

*Bitrate* (*bpv*)

*peak PSNR*

dBs <sup>10</sup> 20 log

*RMS* ,

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>20</sup>

*Bitrate* (*bpv*)

subband.

ones, respectively.

3xC SIM PGC

*PSNR* (*dBs*)

as:

information-theoretic analysis of wavelet-based mesh coding designs, which is presented next.
