Globally smooth parameterizations with low distortion

Abstract

Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and metric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures and the excellent rate distortion performance of semi-regular remeshes produced with these parameterizations. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.

Mesh Parameterization Methods and Their Applications

“…Morphing and Detail Transfer A map between the surfaces of two objects allows the transfer of detail from one object to another [81,99,121], or the interpolation between the shape and appearance of several objects [2,63,66,71,109]. By varying the interpolation ratios over time, one can produce morphing animations.…”

“…By varying the interpolation ratios over time, one can produce morphing animations. In spatially varying and frequency-varying morphs, the rate of change can be different for different parts of the objects, or different frequency bands (coarseness of the features being transformed) [63,66,71]. Such a map can either be computed directly or, as more commonly done, computed by mapping both object surfaces to a common domain (Sections 5 and 6).…”

“…One common way to remesh surfaces, or to replace one triangulation by another, is to parameterize the surface, then map a desirable, well-understood, and easy to create triangulation of the domain back to the original surface. For example, Gu et al [41] use a regular grid sampling of a planar square domain, while subdivision based methods [49,63,72] use regular subdivision (usually one-to-four triangle splits) on the faces of a simplicial domain. Such locally regular meshes can usually support the creation of smooth surfaces as the limit process of applying subdivision rules.…”

Optimizing Surface Triangulation Via Near Isometry with Reference Meshes

“…Surface parameterization is the procedure to map the points on one surface onto a parameter domain (such as a plane or sphere) under certain constraints such as preservation of areas or angles [15]. Although surface parameterization has been widely used in computer graphics and visualization [16,17,18,19,20,21], in this section we explore an interesting connection between it and mesh optimization.…”

Recent Advances in Compression of 3D Meshes

“…Moreover, recent experiments confirm the importance of the smoothness of the parameterization for semi-regular remeshing and hence for geometry compression [KLS03]. Finally, a model-based bit-allocation technique has been proposed by Payan and Antonini [PA02] to efficiently allocate the bits across wavelet subbands according to their variance.…”