Discrete computational methods for volume data processing in scientific visualization

This dissertation focuses on discretization and its relationship to and substantial impact on computer graphics and scientific visualization algorithms. Discretization has fundamentally influenced the evolution of computer graphics hardware and algorithms, and with the modern graphics processing units (GPUs) implementing the rasterization-based graphics pipeline, an array of new methods are taking advantage of the discretization in the pipeline, processing on discrete elements, and operations on the discretized output domain for hardware acceleration and simplification of the algorithms. This dissertation looks into two fields in scientific visualization that take advantage of discrete computations that not only fit into the GPU framework, but also allow visualizations that have not been explored before.; In the field of scattered data approximation and visualization, a new way of looking at Sibson's scheme for natural neighbor interpolation is presented. By discretizing the input domain and applying reconstruction over a regular grid, the new algorithm shows that Sibson's method reduces to a problem of scattering d-dimensional spheres onto the output grid. Furthermore, the notion of operating on a discretized domain and scattering discrete elements is generalized to other scattered data methods, and a framework is presented. This dissertation also demonstrates how more complex scattered data such as streaming and moving scattered data can be handled efficiently with the new framework.; In the field of flow visualization, this dissertation presents a novel way of visualizing and exploring flow fields by constructing a discrete streamline density field and applying a multi-dimensional transfer function. A density field is constructed by tracing particles in each iteration and bucketing particles on a buffer. The density field enables both feature-accentuating visualization and a dense visualization of a given flow field depending on how transfer functions are applied. The streamline density field is efficiently processed by utilizing the GPU pipeline for both streamline computation and density field accumulation processed on a discrete domain. The presented method is also extended to the time-varying flow fields.