Siggraph Course Notes Mesh Parameterization: Theory and Practice

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications. Prerequisites The audience should have had some prior exposure to mesh representation of geometric models and a working knowledge of vector calculus, elementary linear algebra, and the fundamentals of computer graphics. Optional prerequisites: some lectures may also assume some familiarity with differential geometry, graph theory, harmonic functions, and numerical optimization. Graduate students, researchers, and application developers who seek to understand the concepts and technologies used in mesh parameterization and wish to utilize them. Listeners get an overview of the spectrum of processing applications that benefit from para-meterization and learn how to evaluate different methods in terms of specific application requirements. Sources These notes are largely based on the following survey papers, which can be found on the author webpages: • M. S. Floater and K. Hormann. Surface parameterization: a tutorial and survey. Supplemental material and presentation slides are also available on the presenter web-pages. i Course Overview For any two surfaces with similar topology, there exists a bijective mapping between them. If one of these surfaces is a triangular mesh, the problem of computing such a mapping is referred to as mesh parameterization. The surface that the mesh is mapped to is typically called the parameter domain. Parameterization was introduced to computer graphics for mapping textures onto surfaces. Over the last decade, it has gradually become a ubiquitous tool for many mesh processing applications, including detail-mapping, detail-transfer, morphing, mesh-editing, mesh-completion, remeshing, compression, surface-fitting, and shape-analysis. In parallel to the increased interest in applying parameterization, various methods were developed for different kinds of parameter domains and parameterization properties. The goal of this course is to familiarize the audience with the theoretical and practical aspects of mesh parameterization. We aim to provide the skills needed to implement or improve existing methods, to investigate new approaches, and to critically evaluate the suitability of the techniques for a particular application. The course starts with an introduction to the general concept of parameterization and an overview of its applications. The first half of the course then focuses on pla-nar parameterizations while the second addresses more recent approaches for alternative domains. The course covers the mathematical …