This thesis describes methods for simplifying complex polygonal surfaces and optimizing their approximations. Such densely sampled surfaces are common in many computer graphics applications, and must often be simplified by orders of magnitude to allow interactive display, efficient storage and transmission, and faster processing.

I describe a method for memoryless simplification which uses simple geometric heuristics, based on changes in volume and area, for ordering a sequence of edge collapses. This technique differs from most existing methods, in that the error metric is defined with respect to the partially simplified model, as opposed to the original. This method is fast and memory efficient, and produces models with low mean geometric errors.

In order to handle models that are too complex to fit in main memory, I propose a method for out-of-core simplification based on vertex clustering and quadric error metrics. This method is able to efficiently process an arbitrarily large mesh in a single pass, and outputs simplified meshes of high geometric quality.

Many computer graphics applications require that the original and the simplified model are visually similar--a quality which is not necessarily implied by geometric closeness. I describe an image-driven simplification algorithm in which rendered images of a model are used to guide the coarsening process, allowing changes in visual appearance due to surface properties such as shape, normals, color, texture, and visibility to be measured and accounted for in a more direct manner. Using this framework based on image metrics, I also present a technique for image-driven mesh optimization. This method improves the visual quality of an already simplified mesh by performing multidimensional optimization of geometry and surface attributes while making local connectivity changes to the mesh where appropriate.

Finally, I describe a new image metric that is motivated by human visual perception. This metric is able to exploit some of the limitations in the human visual system and predict where significant perceptible differences occur. When used as the driving criterion in simplification and optimization, this metric occasionally allows a considerable improvement in perceived model quality.