Topological analysis, visualization, and design of vector fields on surfaces

Analysis, visualization, and design of vector fields on surfaces have a wide variety of major applications in both scientific visualization and computer graphics. On the one hand, analysis and visualization of vector fields provide critical insights to the flow data produced from simulation or experiments of various engineering processes. On the other hand, many graphics applications require vector fields as input to drive certain graphical processes. This thesis addresses vector field analysis and design for both visualization and graphics applications. Topological analysis of vector fields provides the qualitative (or structural) information of the underlying dynamics of the given vector data, which helps the domain experts identify the critical features and behaviors efficiently. In this thesis, I introduce a more complete vector field topology called Entity Connection Graph (ECG) by including periodic orbits , an essential component in vector field topology. An efficient technique for periodic orbit extraction is introduced and incorporated into the algorithm for ECG construction. The analysis results are visualized using the improved evenly-spaced streamline placement with all separation features being highlighted. This is the first time that periodic orbits have been extracted from surface flows. Through applications to engine simulation datasets, I demonstrate how the extracted topology helps engineers interpret the flow data that contains certain desirable behaviors which indicate the ideal engineering process. Accuracy is typically of paramount importance for visualization and analysis tasks. However, the trajectory-based vector field topology approaches are sensitive to small perturbations such as error and noise which are contained in the given data and introduced during data acquisition and processing. This makes rigorous interpretation of vector field topology and flow dynamics difficult. To overcome that, I advocate the use of Morse decomposition to define a more reliable vector field topology called Morse Connection Graph (MCG). In particular, I present the pipeline of Morse decomposition of an input vector field. A technique based on the idea of τ – map is introduced to produce desirably fine Morse decompositions of vector fields. To address the issue of slow performance of the global τ – map framework, I describe a hierarchical MCG refinement framework. It enables the τ – map approach to be conducted within a Morse set of interest which greatly reduces the computation cost and leads to faster analysis. It is my hope that the work on Morse decomposition will invoke the investigation of other similar data analysis problems such as scalar field and tensor field analysis. The techniques of time-independent vector field design have been well-studied. However, there is little attention on the systematic design of time-varying vector fields on surfaces. This dissertation addresses this by developing a design system that allows the creation and modification of time-varying vector fields on surfaces. More specifically, I present a number of novel techniques to enable efficient design over important characteristics in the vector field such as singularity paths, pathlines, and bifurcations. These vector field features are used to generate a vector field by either blending basis vector fields or performing a constrained optimization process. Unwanted singularities and bifurcations can lead to visual artifacts, and I address them through singularity and bifurcation editing. I demonstrate the capabilities of the design system by applying it to the design of two types of vector fields: the orientation field and the advection field for the application of texture synthesis and animation.