Pattern mapping with quad-pattern-coverable quad-meshes

Abstract

We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texture-mapped with locally translated copies of a single square-texture pattern. This implies, for instance, that every positive-genus surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify sufficient conditions for meshes to be classified as “quad-pattern-coverable”, and we present several methods to construct such meshes. Moreover, we identify some mesh operations that preserve the quad-pattern-coverability property. For instance, since vertex insertion remeshing, which is the remeshing operation behind Catmull–Clark subdivision, preserves quad-pattern-coverability, it is possible to cover any surface of positive genus with iteratively finer versions of the same texture.

Introduction

Texture mapping [1] is very popular in computer graphics applications, since it allows the creation of complicated-looking images without increasing the complexity of the surface geometry. Texturing arbitrary surfaces with repetitive patterns (i.e, pattern mapping [2]) is particularly useful, since repeating a pattern reduces the memory cost, by mapping the same texture image to all faces. Moreover, pattern mapping does not require painting a texture image for each surface or generating a global texture map on each surface. Pattern mapping can potentially provide natural-looking materials, such as stone, wood, or marble, as well as human-made materials, such as wallpapers or repeating tiles.

One of the main challenges arising when mapping patterns to arbitrary polyhedral meshes is to avoid texture discontinuities caused by singularities in the mesh structure. Texture discontinuities differ from shape discontinuities, although the source of both kinds of discontinuities is the same. In quad-meshes, non-4-valent vertices correspond to mesh singularities. For instance, in Catmull–Clark subdivision, non-4-valent vertices cause C² discontinuities, but only at points that correspond to non-4-valent vertices [3]. However, discontinuities can also appear at seams along the edges, which can be visually distracting. Unfortunately, it is not always possible to avoid non-4-valent vertices, since 4-regular quad-meshes exist only for genus-1 surfaces.

In this paper, we show that for any surface of positive genus, there exist quad-meshes that do not cause texture discontinuities. Using such quad-meshes, which we call quad-pattern-coverable meshes (abbr. QPC), it is possible to seamlessly cover a surface of positive genus periodically with any plane symmetric wallpaper pattern. Fig. 1(a) and (b) is examples of a QPC mesh covered periodically by one of the two wallpaper patterns shown in Fig. 2.

QPC meshes can also be covered aperiodically, by using more than one quad pattern. Periodic and aperiodic patterns are contrasted in Section 2. After introducing some definitions, we establish sufficient conditions in Section 3 for a mesh to be classified as QPC.

Our results imply that a quad-mesh is not QPC if the valence of at least one vertex is not divisible by 4. This observation implies, in turn, that there exists no genus-0 QPC mesh, since a genus-0 quad-mesh always has some vertices with valences smaller than 4. For positive genus surfaces, by way of contrast, there exist a wide variety of mesh structures that can satisfy the sufficiency conditions. Theoretical and practical algorithms for construction of QPC meshes are provided in 6 Construction of QPC meshes from regular meshes, 5 Constructing QPC meshes with permutation voltage graphs.

One obvious problem with vertex valences that are large multiples of 4 is that it is hard to avoid texture distortions near such a vertex. It is preferable, therefore, to reduce the large multiples to valence 8, the smallest non-trivial multiple of 4. We introduce an operation in Section 6 that can transform the 4k-valent vertices in a mesh into 8-valent vertices, while preserving the QPC property, regardless of surface genus. Note that 8 valence vertices in saddle regions do not produce significant texture distortions as it can be seen in our examples such is the ones shown in Fig. 1, Fig. 4.

Vertex insertion remeshing that replaces each quad by four smaller ones, as the Catmull–Clark subdivision algorithm being the prime example, preserve quad-pattern-coverability (see Section 3). It is possible, therefore, to cover any surface of positive genus with iteratively finer versions of a given texture. Catmull–Clark subdivision is also useful in creating smooth models. In our examples such as Fig. 1, starting with very coarse QPC meshes, we obtain smooth versions by Catmull–Clark subdivision. Instead of re-texturing the mesh, we bi-linearly interpolate texture coordinates. The result is equivalent to using curved quads, as shown in Fig. 3.

We also show in Section 2 that any wallpaper pattern can be created by translations of one rectangular image, and that such a rectangular pattern can be directly mapped to a toroidal surface, using a (4, 4) mesh. Thus, wallpaper patterns can seamlessly cover any toroidal surface. However, other surfaces do not have such a toroidal parameterization and cannot be unfolded onto the Euclidean plane. QPC meshes provide an alternative parameterization that allow mapping of such patterns to any positive-genus surface.

A significant advantage of using wallpaper patterns is that the seamless texturing does not require any unique solution. Cyclic translations of wallpaper patterns are also wallpaper patterns. Therefore, one can control the results, by cyclic translations of the wallpaper pattern in each quadrilateral. Using this property, it is also possible to create seamless texture animations.

Quad-pattern-coverability does not require using a single pattern. We also show that if the boundaries of the patterns match [4], then it is possible to obtain aperiodic covering. For example, the aperiodic covering in Fig. 4(a) and (b) use the four quad patterns in Fig. 5(a) and (b) respectively.