Mesh Regularity

The most important property within this dissertation is mesh regularity. On the one hand, regularity results in predictability which benefits the coding performance.

However, as will be detailed next, on the other hand regularity limits the approximation quality of the representation.

Geometry, Parameter and Connectivity Information In many works, including this dissertation, the information to represent a model is split into the geometry of the model, which determines the positions of the vertices, and the connectivity of the model, determining how faces are formed between vertices.

However, as is illustrated by Khodakovsky et al. [1], the information for the positions of vertices can be further split into geometry and parameter information.

This is based on the idea that any tangential displacement of vertices along the surface does not change the approximated surface geometry. Such parameter information is related to where samples are located on a surface, and as such altering this information affects the sampling noise ǫS. This is illustrated in Figure 2.6 with a 2D example. Figure 2.6a depicts an original curve, and two approximations are given in Figures 2.6b and 2.6c, representing the same geometry information but differing in parameter information. Assuming a perfect acquisition and no quantization noise (i.e.,ǫA = ǫQ = 0), it is clear that the reconstruction error due to sampling depends on the parameter information. For densely-sampled surfaces, the effect of parameter information on the reconstruction quality becomes negligible.

(a) Original curve (b) Regular parameter information

(c) Surface-adapted parameter information Figure 2.6: Parameter information.

Explicit vs Implicit Information A semi-regular mesh has a regular connectivityover large portions of its surface, with irregularity only found around a limited amount of so-called extra-ordinary vertices. In practice, such a semi-regular mesh is obtained by starting with a coarse approximation of a mesh, which is called the base mesh, and by then iteratively subdividing its faces, for instance using1-to-4 subdivision which subdivides each triangle into four new triangles as shown in Figure 2.7a. Figure 2.7b then shows the sphere of Figure 2.2b together with the underlying base mesh. The vertices of the original icosahedron can still be recognized as the extra-ordinary vertices with valence5 whereas the other, regular vertices have valence 6. Once all information on the extra-ordinary vertices is known, i.e. once the base mesh is known, no additional connectivity information is required: any decoder can reconstruct the correct mesh connectivity if it is given the base mesh.

This immediately determines a first of three classes of semi-regular meshes.

Subdivision surfaces only require a base mesh to be encoded. No additional

(a)1-to-4 subdivision (b) Icosahedron after three subdivisions Figure 2.7: Semi-regular meshes.

geometry or parameter information is given, and higher-resolution meshes are obtained through the implicit subdivision rules. Secondly, if vertices created by the subdivision procedure are only moved along the surface normal, only geometry information is added per subdivision step. This actually alters the geometry of the model but leaves parameter information unaltered. Normal meshes [3] are based on this idea. And finally, in general semi-regular meshes, the vertices obtained after subdivision can be perturbed tangentially over the surface of the sphere.

On the other hand, an irregular mesh is no longer implicitly constructed starting from a base mesh. In this case, connectivity information has to be explicitly provided in addition to the geometry and parameter information.

(a) Base mesh (b) Subdivision surface

(c) Semi-regular mesh

(d) Irregular mesh

Figure 2.8: Mesh regularity. As is described in [1], a mesh can in fact be seen as composed of geometry, parameter and connectivity information. All meshes share the same geometry information, yet (b) is represented solely based on the subdivision of (a),

the (c) has additional parameter information, and (d) also has connectivity information which makes it an irregular mesh.

These ideas are illustrated in Figure 2.8. A low-resolution base mesh is shown in Figure 2.8a, and a resulting subdivision surface is depicted in Figure 2.8b.

Parameter information is added to obtain the general semi-regular mesh shown in Figure 2.8c. Observe that the same (implicit) connectivity is seen as in Figure 2.8b;

the vertices are merely displaced within the plane without altering the geometry.

Finally, Figure 2.8d depicts the same plane, but now the connectivity does not show any relationship with the connectivity at the lower resolution (Figure 2.8a).

Table 2.5 summarizes this.

type of mesh Connectivity Parameter Geometry subdivision surface meshes implicit implicit implicit

normal meshes implicit implicit explicit semi-regular meshes implicit explicit explicit irregular meshes explicit explicit explicit

Table 2.5: Implicit vs explicit information. In addition to a possible base mesh, this table depicts possible additional information that can be provided.

Mesh Coding To process irregular meshes, any semi-regular mesh codec converts irregular meshes to semi-regular ones in a preprocessing step [1, 3, 4], referred to as remeshing. This remeshing step finds an appropriate base mesh and approximates the original surface via iterative subdivision. In applications where original sampling and its connectivity information are relevant, such a remeshing step is considered as a lossy step. After remeshing, normal meshes and other semi-regular coding schemes exploit the fact that connectivity is predetermined by the base mesh, while during remeshing they reduce the parameter information as much as possible, aiming for better coding performance by reducing irregularity and increasing predictability.

However, in this dissertation irregular mesh coding is considered, where arbitrary faces can be formed between the vertices as in Figure 2.8d. While it is challenging for an irregular mesh codec to achieve coding performances similar to those of semi-regular ones due to the added cost of storing connectivity information explicitly, its main advantage over semi-regular codecs is that irregular meshes allow for better approximations using fewer vertices and faces, and consequently using less memory for representing the intermediate approximations.

Indeed, it is a well-known fact that regular sampling, and by extension semi-regular meshes with minimal parametric information, require higher sampling densities compared to irregular sampling to preserve all details. A 2D example is shown in Figure 2.9, which shows two approximations of the purple curve which has high-frequency information on the left side and low-frequency information on the right side. The approximations illustrate high-density and low-density regular sampling. The high-density sampling in Figure 2.9a allows for accurately reconstructing all information in the purple curve. However, many samples are used on the right half of the curve which no longer necessarily aid in improving the reconstructed quality. Using fewer samples in the right half, such as depicted in Figure 2.9b suffices for representing these details. However, in this case the high-frequency information on the left side of the curve is lost. This effect is

Original signal Reconstructed signal

(a) High sampling density

Original signal Reconstructed signal

(b) Low sampling density Figure 2.9: Sampling densities.

also found in semi-regular mesh representations. Similar to the 2D example of Figure 2.9a, sufficient vertices need to be defined near high-frequency details to ensure sufficient preservation in these regions. Consequently, this results in denser vertex distributions in regions where this is not required. Irregular mesh representations tackle this, allowing for a better surface-adapted distribution of vertices and reducing the required amount of memory for rendering.