Spatial and Spectral Methods for Irregular Sampling in Computer Graphics

Most general rasterization algorithms in computer graphics are based on point sampling of the image to be rendered. One peculiarity of computer graphics is that irregular sampling patterns are widely used, mainly to prevent moiré artifacts in rendered images. The best irregular sampling patterns have a blue noise characteristic in the spectral domain: such patterns attain a particularly good tradeoff between moiré prevention and noise-free rendition of low image frequencies. Despite considerable research on blue noise sampling over the last 30 years, several important questions have not been answered completely so far: What is the most desirable irregular sampling pattern? To what extent can such a sampling pattern be realized in practice? What geometric properties of a sampling pattern are especially desirable? Some progress towards answering these questions has been made during the last few years. This thesis continues this line of research and presents new experimental and theoretical results on irregular sampling patterns. Our focus is on the interaction between geometric and spectral properties of sampling patterns and their impact on the sampling process. The main contributions of this thesis fall broadly into three different areas. First, we extend previous results on the spectral analysis of irregular sampling to explain in more detail how the shape of the power spectrum of a sampling pattern affects the visual appearance of aliasing. We then study the limiting case of Poisson disk sampling, which is the prevalent form of irregular sampling used in computer graphics, and demonstrate that it leads to sampling patterns with certain undesirable properties. Finally, we study the mathematical relationship between spatial statistics and spectral measures to make two important contributions to the theory of blue noise sampling. First, we study two realizability conditions, which explain how spatial and spectral characteristics of a point set constrain each other. And second, we show how to derive efficient irregular sampling patterns directly from a specification of their desired spectral properties.