A Texture Atlas is an efficient color representation for 3D Paint Systems. The model to be textured is decomposed into charts homeomorphic to discs, each chart is parameterized, and the unfolded charts are packed in texture space. Existing texture atlas methods for triangulated surfaces suffer from several limitations, requiring them to generate a large number of small charts with simple borders. The discontinuities between the charts cause artifacts, and make it difficult to paint large areas with regular patterns.

Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another. 1

Embedding of a graph metric in Euclidean space efficiently and accurately is an important problem in general with applications in topology aggregation, closest mirror selection, and application level routing. We propose a new graph embedding scheme called Big-Bang Simulation (BBS), which simulates an explosion of particles under force field derived from embedding error. BBS is shown to be significantly more accurate, compared to all other embedding methods including GNP. We report an extensive simulation study of BBS compared with several known embedding schemes and show its advantage for distance estimation (as in the IDMaps project), mirror selection and topology aggregation.

We present a novel 3D face recognition approach based on geometric invariants introduced by Elad and Kimmel. The key idea of the proposed algorithm is a representation of the facial surface, invariant to isometric deformations, such as those resulting from different expressions and postures of the face. The obtained geometric invariants allow mapping 2D facial texture images into special images that incorporate the 3D geometry of the face. These signature images are then decomposed into their principal components. The result is an efficient and accurate face recognition algorithm that is robust to facial expressions. We demonstrate the results of our method and compare it to existing 2D and 3D face recognition algorithms.

Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.

Many computer graphics operations, such as texture mapping, 3D painting, remeshing, mesh compression, and digital geometry processing, require finding a low-distortion parameterization for irregular connectivity triangulations of arbitrary genus 2-manifolds. This paper presents a simple and fast method for computing parameterizations with strictly bounded distortion. The new method operates by flattening the mesh onto a region of the 2D plane. To comply with the distortion bound, the mesh is automatically cut and partitioned on-the-fly. The method guarantees avoiding global and local self-intersections, while attempting to minimize the total length of the introduced seams.

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We present an efficient computational framework for isometry-invariant comparison of smooth surfaces. We formulate the Gromov-Hausdorff distance as a multidimensional scaling (MDS)-like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples. For isometry-invariant comparison of surfaces in the case of partially missing data, we present the partial embedding distance, which is computed using a similar scheme. The main idea is finding a minimum-distortion mapping from one surface to another, while considering only relevant geodesic distances. We discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency.

Providing a two-dimensional parameterization of threedimensional tesselated surfaces is beneficial to many applications in computer graphics, finite-element surface meshing, surface reconstruction and other areas. The applicability of the parameterization depends on how well it preserves the surface metric structures (angles, distances, areas). For a general surface there is no mapping which fully preserves these structures. The distortion usually increases with the rise in surface complexity. For highly complicated surfaces the distortion can become so strong as to make the parameterization unusable for application purposes. One possible solution is to subdivide the surface or introduce seams in a way which will reduce the distortion.

We present a survey of recent methods for creating piecewise linear mappings between triangulations in 3D and simpler domains such as planar regions, simplicial complexes, and spheres. We also discuss emerging tools such as global parameterization, inter-surface mapping, and parameterization with constraints. We start by describing the wide range of applications where parameterization tools have been used in recent years. We then briefly review the pertinent mathematical background and terminology, before proceeding to survey the existing parameterization techniques. Our survey summarizes the main ideas of each technique and discusses its main properties, comparing it to other methods available. Thus it aims to provide guidance to researchers and developers when assessing the suitability of different methods for various applications. This survey focuses on the practical aspects of the methods available, such as time complexity and robustness and shows multiple examples of parameterizations generated using different methods, allowing the reader to visually evaluate and compare the results.