Almost Isometric Mesh Parameterization through Abstract Domains

Abstract

Abstract-In this paper we propose a robust, automatic technique to build a global hi-quality parameterization of a two-manifold triangular mesh. An adaptively chosen 2D domain of the parameterization is built as part of the process. The produced parameterization exhibits very low isometric distortion, because it is globally optimized to preserve both areas and angles. The domain is a collection of equilateral triangular 2D regions enriched with explicit adjacency relationships (it is abstract in the sense that no 3D embedding is necessary). It is tailored to minimize isometric distortion, resulting in excellent parameterization qualities, even when meshes with complex shape and topology are mapped into domains composed of a small number of large continuous regions. Moreover, this domain is in turn remapped into a collection of 2D square regions, unlocking many advantages found in quad-based domains (e.g. ease of packing). The technique is tested on a variety of cases, including challenging ones, and compares very favorably with known approaches. An open source implementation is made available.

Isotropic 2D Quadrangle Meshing with Size and Orientation Control

“…Because the main steps of the algorithm only deal with labeling the triangles of a background triangle mesh, our implementation is simple and reliable, and is less prone to numerical robustness issues. Resorting to local parameterizations on trivial convex domains such as [23] provides us with scalability and robustness. For these reasons our approach could be extended to reliable quadrangle surface remeshing (including anisotropic quadrangle surface remeshing).…”

“…Including methods devised to generate quadrangle surface tilings, the rich literature on this topic contains approaches which proceed by quadrangulation [9,3], square packing [28], advancing front [22], conversion [8], decimation [18], Morse-Smale complexes [14,33,17], clustering [4,20], local and global operators [23,19], whisker weaving [32], medial axis [24], streamlining [1,21] and parameterization [25,29,7].…”

“…Regularity may be improved a posteriori through, e.g., grid-preserving operators so as to generate simple base complexes [6]. Strict local conformance to both sizing and cross fields is notoriously delicate for most approaches which involve a global variational formulation, and almost none of the fine-to-coarse approaches based on decimation [18,23] leads to meshes that conform to both sizing and cross fields.…”

“…We call butterfly a set of four tiles incident to a short meta-edge connecting two degree-3 meta-vertices (see Figure 8). Inspired by [23] we remove many of these butterfly configurations through local parameterizations on square domains, which merge 2 degree-3 meta-vertices into one degree-4 meta-vertex.…”

A Multi-Resolution Photogrammetric Framework for Digital Geometric Recording of Large Archeological Sites: Ajloun Castle-Jordan

“…The.ply files were imported into MeshLab, where they processed and re-sampled using the so-called iso-parameterization algorithm. This algorithm calculated an almost-isometric surface triangulation, while preserving the main shape of the volume [42]. The derived model is not satisfactory even it has huge number of faces.…”