Surface Topological Analysis for Image Synthesis (a Ph.D. thesis)

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Eugene Zhang

Advisor: Greg Turk

Abstract

My thesis work is related to solving two problems on mesh surfaces by performing topological analysis. Many graphics applications, such as high-quality and interactive image synthesis, benefit from the solutions to these problems. The two problems that I address are: surface parameterization and vector field design on surfaces.

Surface parameterization refers to segmenting a mesh surface into a number of patches and unfolding these patches onto a plane without any overlap. Surface parameterization is primarily used for storing surface signals into texture maps, which speeds up the subsequent rendering process. One of the most important quality measurements for surface parameterization is stretch. Stretch causes uneven sampling rate across the surface and needs to be avoided whenever possible. My parameterization technique is based on the idea that a surface can be approximated by a collection of relative simple shapes, such as cylinders, cones, flat disks, and spheres. Unfolding them results in relatively little stretch. I decompose the surface by identifying major features contained in the surface such as handles and large protrusions. This is achieved by performing topological analysis of a distance-based function on the surface and locating its local maxima and saddles. Also, I will describe two techniques to reduce stretch during patch unfolding: the Green-Lagrange tensor, and a virtual boundary.

Vector field design on surfaces allows a user to create a wide variety of vector fields on mesh surfaces and to have control over the topology of these vector fields. A vector field design system is useful for many graphics applications, such as texture synthesis, non-photorealistic rendering, and fluid simulation. Vector field design is a new and challenging problem. First, a vector field design system should provide control over the geometric and topological characteristics of a vector field. Second, vector field design for mesh surfaces requires an e±cient way of representing and analyzing surface vector fields. In this thesis, I present a vector field design system that allows a user to interactively create a wide variety of vector fields with control over its analytical characteristics, such as the curl and divergence, and its topological characteristics, such as the number and location of the singularities. This is achieved through a set of editing operations provided by the system. In addition, I will describe a new piecewise interpolating scheme for representing continuous vector fields defined on 3D mesh surfaces. This is based on the well-known concepts of geodesic polar maps and parallel transport from classical differential geometry. Furthermore, I apply the system to several graphics applications: painterly rendering of still images, pencil-sketches of surfaces, and texture synthesis.