**5. Scalable mesh compression overview**

In this section, we give a brief overview of the scalable mesh compression systems. Based on the design choices established earlier, we designed intraband and composite mesh coding systems which provide state-of-the-art compression performance, together with resolution as well as quality scalability of the compressed mesh.

## **5.1 Progressive Geometry Compression (PGC)**

The first scalable wavelet-based geometry compression technique is the progressive geometry compression (PGC) codec proposed by Khodakovsky et al. in (Khodakovsky et al., 2000). PGC makes use of the well-know zero-tree coding (Shapiro, 1993) of wavelet coefficient's bitplanes in order to encode the decomposed mesh structure. Significant improvements in the compression performance against the contemporary scalable as well as non-scalable mesh coding systems were reported in (Khodakovsky et al., 2000). However, a major drawback of PGC schemes is their inability to provide resolution scalability. This is caused by the zero-tree structure which, for a given bitplane, spans all the wavelet decomposition levels. For a detailed understanding of the PGC system we refer to (Khodakovsky et al., 2000).

### **5.2 Scalable Intraband Mesh Compresion (SIM)**

Despite of the great success of zerotree-based coding techniques in image coding, the choice of an interband codec design is not necessarily the best option in the context of scalable mesh coding. This was illustrated in Section 4 where different types of dependencies among wavelet coefficients were studied. Based on this analysis, we opt for an intraband dependency model in our codec design. As mentioned before, favoring intraband models over 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.

In the designed scalable intraband mesh (SIM) compression system (Denis et al., 2010b) each resolution subband is encoded independently of the others. Similar to (Shapiro, 1993), SAQ is applied to each resolution subband to determine the significance of the wavelet coefficients with respect to a series of monotonically decreasing thresholds. Based on the significance outcome, a tree node is split into eight equal volume nodes. The resulting octree nodes may contain an unequal number of wavelet coefficients. In general, the number of coefficients in all nodes of a same tree-depth is roughly the same. This way, an octree is constructed for each resolution subband, wherein the depth of the tree (number of levels in the octree) is equal to the number of bitplanes of the subband. All magnitude bitplanes are sequentially coded using the non-significance, the significance and the refinement coding passes. For a detailed presentation of the SIM codec the interested reader is referred to (Denis et al., 2010b).

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

In this section, we give a brief overview of the scalable mesh compression systems. Based on the design choices established earlier, we designed intraband and composite mesh coding systems which provide state-of-the-art compression performance, together with resolution

The first scalable wavelet-based geometry compression technique is the progressive geometry compression (PGC) codec proposed by Khodakovsky et al. in (Khodakovsky et al., 2000). PGC makes use of the well-know zero-tree coding (Shapiro, 1993) of wavelet coefficient's bitplanes in order to encode the decomposed mesh structure. Significant improvements in the compression performance against the contemporary scalable as well as non-scalable mesh coding systems were reported in (Khodakovsky et al., 2000). However, a major drawback of PGC schemes is their inability to provide resolution scalability. This is caused by the zero-tree structure which, for a given bitplane, spans all the wavelet decomposition levels. For a detailed understanding of the PGC system we refer to

Despite of the great success of zerotree-based coding techniques in image coding, the choice of an interband codec design is not necessarily the best option in the context of scalable mesh coding. This was illustrated in Section 4 where different types of dependencies among wavelet coefficients were studied. Based on this analysis, we opt for an intraband dependency model in our codec design. As mentioned before, favoring intraband models over 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

In the designed scalable intraband mesh (SIM) compression system (Denis et al., 2010b) each resolution subband is encoded independently of the others. Similar to (Shapiro, 1993), SAQ is applied to each resolution subband to determine the significance of the wavelet coefficients with respect to a series of monotonically decreasing thresholds. Based on the significance outcome, a tree node is split into eight equal volume nodes. The resulting octree nodes may contain an unequal number of wavelet coefficients. In general, the number of coefficients in all nodes of a same tree-depth is roughly the same. This way, an octree is constructed for each resolution subband, wherein the depth of the tree (number of levels in the octree) is equal to the number of bitplanes of the subband. All magnitude bitplanes are sequentially coded using the non-significance, the significance and the refinement coding passes. For a detailed presentation of the SIM codec the interested reader is referred to

next.

**5. Scalable mesh compression overview** 

as well as quality scalability of the compressed mesh.

**5.1 Progressive Geometry Compression (PGC)** 

**5.2 Scalable Intraband Mesh Compresion (SIM)** 

(Khodakovsky et al., 2000).

others can be discarded.

(Denis et al., 2010b).

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 subband.

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 as:

$$PSNR = 20 \cdot \log\_{10} \left( \frac{peak}{RMS} \right) \text{(dBs)},$$

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 ones, respectively.

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.

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.

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).

Optimized Scalable Wavelet-Based Codec Designs for Semi-Regular 3D Meshes 587

In the case of normal meshes (Fig. 10) our coder employs the same transform as PGC. Both codecs perform the same at very low bitrates. However, overall, 3xC yields the best compression performance. 3xC gives approximately equivalent results when compared with the intraband SIM codec for normal meshes. This is because the context-conditioning is only possible for the normal component of vector valued wavelet coefficients. Overall, it is clear that the proposed 3xC codec produces similar, and in almost all cases, superior performance

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

0.050 bpv, 51.3 dB 0.098 bpv, 55.0 dB 0.178 bpv, 58.7 dB 0.314 bpv, 62.5 dB 0.540 bpv, 66.3 dB

0.050 bpv, 47.6 dB 0.098 bpv, 51.4 dB 0.178 bpv, 55.4 dB 0.314 bpv, 59.2 dB 0.540 bpv, 63.1 dB

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

compared to PGC and SIM codecs.

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

much more accurately compared to the PGC system.

2.1.2) is not included in the reported rate values.

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.

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

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

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

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.

In the case of normal meshes (Fig. 10) our coder employs the same transform as PGC. Both codecs perform the same at very low bitrates. However, overall, 3xC yields the best compression performance. 3xC gives approximately equivalent results when compared with the intraband SIM codec for normal meshes. This is because the context-conditioning is only possible for the normal component of vector valued wavelet coefficients. Overall, it is clear that the proposed 3xC codec produces similar, and in almost all cases, superior performance compared to PGC and SIM codecs.
