Discrete LOD Systems

Inspired by the PM representation of Hoppe [17] and the TS single-rate codec of Taubin and Rossignac [13], Taubin et al. [26] developed their progressive forest split (PFS) compression algorithm. The PFS scheme is more coarse-grained compared to the fine-coarse-grained PM scheme. In view of the continuous LOD terminology given above, such coarser-grained LOD systems are termed discrete LOD (dLOD) systems.

An important paradigm in many multimedia domains is the wavelet transform

to obtain multiple resolutions and encode them in an efficient way. A wavelet transform iteratively transforms a higher-resolution signal into a lower-resolution signal and a wavelet subband. However, whereas 3D surfaces are inherently irregularly sampled, wavelet coding conventionally considers regularly sampled data. The regular connectivity of semi-regular meshes has been used in literature for building wavelet transforms. Such meshes are often obtained via the interpolating Butterfly [27] scheme or the approximating Loop [28] subdivision scheme, where each subdivision step results in a resolution increment. After a 1-to-4 subdivision, the Butterfly subdivision perturbs the newly added vertices based on neighborhood information. For a vertex on a given edge, this neighborhood information encompasses the two triangles neighboring this edge, and their four additional neighboring triangles. Figure 2.20a depicts this and illustrates the origin of the name ‘butterfly’ subdivision. Denote byj the amount of performed subdivision steps. Given a tension parameterw, each new vertex v_j+1^o is located at

v_j+1^o =1

2(v^A_j + v^B_j) + 2w(v_j^C+ v_j^D) − w(v^E_j + v^F_j + v^G_j + v^H_j ). (2.31) The Loop subdivision scheme on the other hand transforms both new and existing vertices given the masks shown in Figure 2.20b and 2.20c respectively. Each new vertexv^o_j+1is positioned at

v^o_j+1= 3 8v_j^A+3

8v_j^B+1 8v^C_j +1

8v^D_j , (2.32)

while an exist vertex v^e_j with degree ν(v_j^e) = n is repositioned, again given a weighing factorw, to

v^e_j+1= (1 − nw)v^e_j+ w Xn

i=0

v^e,i_j . (2.33)

Wavelet-based mesh coding was initially proposed for such semi-regular meshes: wavelets are defined on the 2D parametric surface defined by the base mesh, and each higher-resolution wavelet subband is then related to the lower-resolution subband given the subdivision scheme. The stencils depicted in Figure 2.20 and the weights given in Equations 2.31, 2.32 and 2.33 determine the scaling coefficients, describing how a mesh is upscaled from a lower resolution to a higher resolution. Wavelet coefficients then refine the vertices of the mesh, for representing detail information which was not present in the lower-resolution representation. Pioneering work was proposed by Khodakovsky et al. [1], describing their progressive geometry compression (PGC) algorithm which processes meshes by using semi-regular wavelet transforms and zerotree coding [2]. Such a tree exploits the fact that the descendants of a wavelet coefficient which is non-significant given a specific thresholdτ are often non-significant as

v^A v^B Figure 2.20: Butterfly and Loop neighborhoods.

well for the same threshold τ , and can be encoded together with a single zero.

This approach and other approaches such as those proposed by Khodakovsky and Guskov [29] or by Avil´es et al. [30] exploit interband correlations, i.e., correlations between wavelet coefficients in subsequent wavelet subbands: at specific regions, the properties are expected to be similar across resolutions and are encoded together. The main issue with such approaches is that they do not allow for resolution scalability.

While intraband correlations, i.e., the correlations between wavelet coefficients within wavelet subbands, have been investigated for image compression (see, for instance, [31–33]), few intraband mesh codecs have been proposed. Payan and Antonini [34] describe a semi-regular mesh codec which employs the Loop [28] Discrete Wavelet Transform and encodes the quantized wavelet coefficients using independent embedded block coding with optimized truncation (EBCOT)[33] of the embedded bit streams. The statistical dependencies within and across wavelet subbands have been analyzed by Satti et al. in [35] and [36] for semi-regular meshes and normal meshes respectively.

These works concluded that intraband dependencies are stronger than interband dependencies, and that composite codecs which exploit both intraband and interband statistical dependencies perform best.

For directly processing irregular meshes, few codecs have been proposed.

Early work was done by Bonneau [37], which compresses the data contained in an irregular mesh, for instance the color data over an irregular mesh representing the earth. As such, this work considers the changing mesh resolutions as “given” and maps the color data to this domain. Actual compression results for the mesh itself are not taken into consideration. Wavemesh, by Valette and Prost [38], is a state-of-the-art wavelet-based irregular mesh coding system. In Wavemesh, the classical 1-to-4 subdivision is generalized to any subdivision of triangles which adds vertices to one, two or all three of the edges of a triangle. As with any irregular mesh

coding system, irregularity comes at a cost: contrary to semi-regular codecs where the mesh connectivity can be implicitly reconstructed at a decoding side, irregular mesh codecs require additional information to properly reconstruct the connectivity. The Wavemesh connectivity encoding has been reused by Valette et al. in [39] which proposes zerotree encoding [2]. Lee et al. make use of the Wavemesh representation but propose novel connectivity and geometry coding approaches [40].

Roy et al. [41] have reformulated the PM representation, which was already encoded in a batched form by Pajarola and Rossignac in [20], as a multiresolution analysis problem. They additionally take into account multiple attributes per vertex, but do not consider them together to optimize the coding performance.

Finally, Maglo et al. [42] similarly use edge collapses to generate several LODs by decimating an original mesh, grouping edge collapse operations which are mutually independent. At the finer LODs, clustering and their independent compression enables random access, allowing for refining specific clusters more or less than their neighboring clusters.