The mathematics of surface reconstruction

This thesis discusses mathematics engineers use to produce computerized three dimensional im- ages of surfaces. It is self-contained in that all background information is included. As a result, mathematicians who know very little about the technology involved in three dimensional imaging should be able to understand the topics herein, and engineers with no differential geometry background will be able to understand the mathematics.

The purpose of this thesis is to unify and understand the notation commonly used by engineers, understand their terminology, and appreciate the difficulties faced by engineers in their pursuitsIt is also intended to bridge the gap between mathematics and engineering.

This paper proceeds as follows. Chapter one introduces the topic and provides a brief overview of this thesis. Chapter two provides background information on technology and differential geometry. Chapter three discusses various methods by which normal vectors are estimated. In Chapter four, we discuss methods by which curvature is estimatedIn Chapter six, we put it all together to recreate the surface. Finally, in chapter seven, we conclude with a discussion of future research. Each chapter concludes with a comparison of the methods discussed. The study of these reconstruction algorithms originated from various engineering papers on surface reconstruction. The background information was gathered from a thesis and various differential geometry texts.

The challenge arises in the nature of the data with which we work. The surface must be recreated based on a set of discrete points. However, the study of surfaces is one of differential geometry which assumes differentiable functions representing the surface. Since we only have a discrete set of points, methods to overcome this shortcoming must be developed. Two categories of surface reconstruction have been developed to overcome this shortcoming.

The first category estimates the data by data by smooth functionsThe second reconstructs the surface using the discrete data directly. We found that various aspects of surface reconstruction are very reliable, while others are only marginally so. We found that methods recreating the surface from discrete data directly produce very similar results suggesting that some underlying facts about surfaces represented by discrete information may be influencing the results.