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Continuous reconstruction, rendering, and editing of point-sampled surfaces

The mathematical representation of surfaces is at the core of most computer graphics problems. In particular, the design of modeling and rendering algorithms for computer graphics applications is largely determined by the specific surface representation that is chosen. A variety of mathematical models has been developed for such applications, ranging from parametric to implicit surfaces. In this thesis, we propose to use nonuniform point samples to continuously reconstruct, render, and interactively edit surfaces. Based on results from classical signal processing theory, we develop core methods for reconstructing, filtering, and resampling nonuniformly point-sampled surfaces. From these concepts, we derive algorithms for high quality, antialiased rendering and interactive surface parameterization. The term point sample denotes a sample of a surface, including information about the surface geometry and the surface appearance. A point-sampled surface consists of a set of point samples without connectivity information, where the points are distributed nonuniformly in space. Point-sampled surfaces are attractive for computer graphics applications because they are easily acquired using range sensing techniques and because they are efficient for modeling and rendering due to their simple structure. This thesis starts by reviewing fundamental signal processing theory, paying particular attention to an analysis of the aliasing phenomenon. We then derive the concept of resampling filters as an efficient antialiasing strategy. Next, we shift our attention to nonuniformly sampled signals and discuss a local filtering approach to nonuniform sampling and reconstruction. Extending these local filtering techniques and combining them with resampling filters, we introduce a parametric representation of point-sampled surfaces. We show how to interpret rendering as resampling a parameterized surface, and we derive an antialiased rendering algorithm for point-sampled surfaces. In addition, we present an adaptation of the algorithm suitable for hardware acceleration, and we present an efficient, hierarchical data structure tailored for point-based rendering. We also develop a resampling filter for volume data and design an efficient splatting algorithm for volume rendering. Finally, we present a framework for interactive editing of point-sampled surfaces. We extend the concept of two-dimensional photo editing to three-dimensional objects by providing two fundamental operations on point-sampled surfaces, namely surface parameterization and surface resampling. Both these operators are based on the parametric representation of point-sampled surfaces. With these building blocks, we show how to implement editing operations such as painting, texturing, carving, displacement mapping, and filtering. Further, we explain a constrained minimum distortion parameterization algorithm that allows the user to perform sophisticated texturing operations. We also provide an efficient strategy for solving the resulting linear equation system using a nested iteration approach.