Low-discrepancy point sampling of 2D manifolds for visual computing

Point distributions are used to sample surfaces for a wide variety of applications within the fields of graphics and computational geometry, such as point-based graphics, remeshing and area/volume measurement. The quality of such point distributions is important, and quality criteria are often application dependent. Common quality criteria include visual appearance, an even distribution whilst avoiding aliasing and other artifacts, and minimisation of the number of points required to accurately sample a surface. Previous work suggests that discrepancy measures the uniformity of a point distribution and hence a point distribution of minimal discrepancy is expected to be of high quality. We investigate discrepancy as a measure of sampling quality, and present a novel approach for generating low-discrepancy point distributions on parameterised surfaces. Our approach uses the idea of converting the 2D sampling problem into a ID problem by adaptively mapping a space-filling curve onto the surface. A ID sequence is then generated and used to sample the surface along the curve. The sampling process takes into account the parametric mapping, employing a corrective approach similar to histogram equalisation, to ensure that it gives a 2D low-discrepancy point distribution on the surface. The local sampling density can be controlled by a user-defined density function, e.g. to preserve local features, or to achieve desired data reduction rates. Experiments show that our approach efficiently generates low-discrepancy distributions on arbitrary parametric surfaces, demonstrating nearly as good results as popular low-discrepancy sampling methods designed for particular surfaces like planes and spheres. We develop a generalised notion of the standard discrepancy measure, which considers a broader set of sample shapes used to compute the discrepancy. In this more thorough testing, our sampling approach produces results superior to popular distributions. We also demonstrate that the point distributions produced by our approach closely adhere to the blue noise criterion, compared to the popular low-discrepancy methods tested, which show high levels of structure, undesirable for visual representation. Furthermore, we present novel sampling algorithms to generate low-discrepancy distributions on triangle meshes. To sample the mesh, it is cut into a disc topology, and a parameterisation is generated. Our sampling algorithm can then be used to sample the parameterised mesh, using robust methods for computing discrete differential properties of the surface. After these pre-processing steps, the sampling density can be adjusted in real-time. Experiments also show that our sampling approach can accurately resample existing meshes with low discrepancy, demonstrating error rates when reducing the mesh complexity as good as the best results in the literature. We present three applications of our mesh sampling algorithm. We first describe a point- based graphics sampling approach, which includes a global hole-filling algorithm. We investigate the coverage of sample discs for this approach, demonstrating results superior to random sampling and a popular low-discrepancy method. Moreover, we develop levels of detail and view dependent rendering approaches, providing very fine-grained density control with distance and angle, and silhouette enhancement. We further discuss a triangle- based remeshing technique, producing high quality, topologically unaltered meshes. Finally, we describe a complete framework for sampling and painting engineering prototype models. This approach provides density control according to surface texture, and gives full dithering control of the point sample distribution. Results exhibit high quality point distributions for painting that are invariant to surface orientation or complexity. The main contributions of this thesis are novel algorithms to generate high-quality density- controlled point distributions on parametric surfaces and triangular meshes. Qualitative assessment and discrepancy measures and blue noise criteria show their high sampling quality in general. We introduce generalised discrepancy measures which indicate that the sampling quality of our approach is superior to other low-discrepancy sampling techniques. Moreover, we present novel approaches towards remeshing, point-based rendering and robotic painting of prototypes by adapting our sampling algorithms and demonstrate the overall good quality of the results for these specific applications.