**5.4 Visual comparison: PGC vs 3xC**

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

(a) (b) Fig. 10. PSNR versus bitrate for normal mesh models in the quality scalability mode: (a) Skull, (b) Dino. The un-lifted Butterly transform is employed for all three codecs.

3xC SIM PGC

*PSNR* (*dBs*)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>60</sup>

*Bitrate* (*bpv*)

Fig. 10 shows compression performance plots for two normal meshes, Skull and Dino. One notices that at low bitrates, PGC tends to compress better. However, the ability of SIM to capture and code more efficiently the high-frequency components is noticeable at high

The mutual information analysis presented earlier showed that the composite dependencies between the wavelet coefficients are by far the strongest. However, one may notice that, employing composite models may hinder, similar to interband models, the possibility of providing resolution scalability. Thus one must be careful in exploiting the parent-children dependencies within composite models. A careful observation reveals that exploiting parent-children dependencies in a causal fashion (Denis et al., 2010b) does not limit resolution scalable decoding of the compressed mesh. Following this observation, we proposed a scalable composite mesh compression system in (Denis et al., 2009), (Denis et al., 2010b). The bitplane coding modules of the SIM codec and the 3xC codec are identical. The two designs differ at the entropy coding level. In particular, for 3xC, parent coefficient based context-conditioning is employed in the entropy coding module. For context-conditioning, significant, non-significant as well as sign information is entropy coded using the designed context tables. The refinement information is encoded without context-conditioning; this is because including the parental information when entropy coding the refinement symbols does not improve compression performance. For a detailed presentation of the 3xC codec

Fig. 9 also depicts the PSNR curves computed for the non-normal Venus and Bunny meshes using our implementation of the un-lifted butterfly based 3xC mesh compression system. The figure clearly demonstrates that, when dealing with non-normal meshes, 3xC

systematically yields superior performance compared to PGC as well as SIM.

bitrates and leads to an improved performance when compared to PGC.

**5.3 Composite Context-conditioned Compression (3xC)** 

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>60</sup>

*Bitrate* (*bpv*)

3xC SIM PGC

*PSNR* (*dBs*)

the interested reader is referred to (Denis et al., 2009).

Visual comparisons of Bunny and Skull meshes, compressed and reconstructed using 3xC at different bits per vertex (bpv), are presented in Fig. 11 and Fig. 12, respectively. The colored regions highlight the distortions introduced by lossy compression. For low-to-medium bitrates, the pure red color indicates areas where the distance between the original and decoded vertex is larger than 0.1% of the diagonal of the bounding box of the semi-regular mesh. For high bitrates, the distortion is visualized with respect to 0.02% of the diagonal. The mesh is shaded greener as the distortion lowers, with pure green indicating no distortion.

When visually comparing the compressed Bunny and Skull meshes produced by 3xC and PGC, it is very clear that 3xC yields superior performance for all bitrates. Taking the result at 0.050 bpv as an example, we observe that many areas which are shaded red for PGC are green for 3xC. At high rates, the differences between the mesh geometries may not be visually significant, yet the colors reveal that 3xC is able to approximate the original mesh much more accurately compared to the PGC system.

Fig. 11. Visual comparison of non-normal Bunny 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 (i.e., *M* 0 - see section 2.1.2) is not included in the reported rate values.

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

Optimized Scalable Wavelet-Based Codec Designs for Semi-Regular 3D Meshes 589

The output distortion *DL* of a Laplacian PDF, quantized using an *n* level EDSQ and

( ) ( 0.5 )

*<sup>L</sup> n n <sup>k</sup> <sup>k</sup> <sup>D</sup> <sup>D</sup>*

*n n n n*

(1 ) <sup>2</sup> (1 ) <sup>2</sup>

*DZ REST*

where *DDZ* and *DREST* denote the distortion contributions of the deadzone and the other

*<sup>e</sup> <sup>e</sup> <sup>e</sup>* , as

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

(1 ) (1 )

*DZ*

*n n n n*

( )2 log 2 ... 2 2

*k k <sup>k</sup>*

... 2 log 2 2

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

log log log

 

 

*R Q e dx e dx*

<sup>2</sup> ( ) ( ) <sup>1</sup>

*R*

2 1 ( ) 2 coth

*n*

*n*

*D Q x e dx x k e dx* ,

 

*k x x*

 

*n n DQ e <sup>L</sup> nn n* , (28)

4 2

(1 ) (1 )

 

*R*

 *n n*

2 2 2

 

*x x*

 

 

*e dx e dx*

*n n n n n n n n REST*

*k k x x*

 

*d c d*

<sup>1</sup>

 

*dc e d*

2

, 2 <sup>1</sup> <sup>1</sup> (1 ) 1 1

 2 2

 

 

1 *<sup>n</sup> e* , (27)

> 

> >

2

1

*n n n n*

*n n d c* (hence 1 1 *d c* 1 ).

    .

,

, (29)

*e*

*n*

*e e*

 

reconstructed using midpoint reconstruction, can be written as:

, <sup>0</sup> ( ) <sup>1</sup>

quantization cells, respectively. By proper substitution and letting

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

*k*

, 2

*L*

*n n*

*n n*

 

*n n*

, 2 0 0

*nn n n*

 

( ) log *<sup>n</sup>*

*<sup>k</sup> k k*

1 1

*e e e ke*

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

*k k*

*L*

*n n*

*RQ c*

*<sup>n</sup> c e* ) and 1

 <sup>1</sup> 1

*k*

*n*

*n n n n*

**7. Appendix** 

where 1 *n n* .

where

 

 *n n <sup>n</sup> c e* (hence 1

written as:

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 rate values.

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