This work presents the new Synthesized Cell Texture (SCT) algorithm for visualizing related multiple scalar value fields within the same 3D space. The SCT method is particularly well suited to scalar quantities that could be represented in the physical domain as size fractionated particles, such as in the study of sedimentation, atmospheric aerosols, or precipitation. There are two components to this contribution. First a Scaling and Distribution (SAD) algorithm provides a means of specifying a multi-scalar field in terms of a maximum cell resolution (or density of represented values). This information is used to scale the multi-scalar field values for each 3D cell to the maximum values found throughout the data set, and then randomly distributes those values as particles varying in number, size, color, and opacity within a 2D cell slice. This approach facilitates viewing of closely spaced layers commonly found in sigma-coordinate grids. The SAD algorithm can be applied regardless of how the particles are rendered. The second contribution provides the Synthesized Cell Texture (SCT) algorithm to render the multi-scalar values. In this approach, a texture is synthesized from the location information computed by the SAD algorithm, which is then applied to each cell as a 2D slice within the volume. The SCT method trades off computation time (to synthesize the texture) and texture memory against the number of geometric primitives that must be sent through the graphics pipeline of the host system. Analysis results from a user study prove the effectiveness of the algorithm as a browsing method for multiple related scalar fields. The interactive rendering performance of the SCT method is compared with two common basic particle representations: flat-shaded color-mapped OpenGL points and quadrilaterals. Frame rate statistics show the SCT method to be up to 44 times faster, depending on the volume to be displayed and the host system. The SCT method has been successfully applied to oceanographic sedimentation data, and can be applied to other problem domains as well. Future enhancements include the extension to time-varying data and parallelization of the texture synthesis component to reduce startup time.

Data from present day scientific simulations and observations of physical processes often consist of multiple scalar fields. It is important to study the interactions between the fields to understand the underlying phenomena. A visual representation of these interactions would assist the scientist by providing quick insights into complex relationships that exist between the fields. We describe new techniques for visual analysis of multifield scalar data where the relationships can be quantified by the gradients of the individual scalar fields and their mutual alignment. Empirically, gradients along with their mutual alignment have been shown to be a good indicator of the relationships between the different scalar variables. The Jacobi set, defined as the set of points where the gradients are linearly dependent, describes the relationship between the gradient fields. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a feature-rich dataset. For two dimensional domains, we pose the problem of simplification as the extraction of level sets and offset contours and describe a robust technique to simplify and create a multi-resolution representation of the Jacobi set. Existing isosurface-based techniques for scalar data exploration like Reeb graphs, contour spectra, isosurface statistics, etc., study a scalar field in isolation. We argue that the identification of interesting isovalues in a multifield data set should necessarily be based on the interaction between the different fields. We introduce a variation density function that profiles the relationship between multiple scalar fields over isosurfaces of a given scalar field. This profile serves as a valuable tool for multifield data exploration because it provides the user with cues to identify interesting isovalues of scalar fields. Finally, we introduce a new multifield comparison measure to capture relationships between scalar variables. We also show that our measure is insensitive to noise in the scalar fields and to noise in their gradients. Further, it can be computed robustly and efficiently. The comparison measure can be used to identify regions of interest in the domain where interactions between the scalar fields are significant. Subsequent visualization of the data focuses on these regions of interest leading to effective visual analysis. We demonstrate the effectiveness of our techniques by applying them to real world data from different domains like combustion studies, climate sciences and computer graphics.

Scalar fields are used to represent physical quantities measured over a domain of interest. Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. This thesis proposes three methods to detect symmetry in scalar fields. The first method models symmetry detection as a subtree matching problem in the contour tree, which is a topological graph abstraction of the scalar field. The contour tree induces a hierarchical segmentation of features at different scales and hence this method can detect symmetry at different scales. The second method identifies symmetry by comparing distances between extrema from each symmetric region. The distance is computed robustly using a topological abstraction called the extremum graph. Hence, this method can detect symmetry even in the presence of significant noise. The above methods compare pairs of regions to identify symmetry instead of grouping the entire set of symmetric regions as a cluster. This motivates the third method which uses a clustering analysis for symmetry detection. In this method, the contours of a scalar field are mapped to points in a high-dimensional descriptor space such that points corresponding to similar contours lie in close proximity to each other. Symmetry is identified by clustering the points in the descriptor space. We show through experiments on real world data sets that these methods are robust in the presence of noise and can detect symmetry under different types of transformations. Extraction of symmetry information helps users in visualization and data analysis. We design novel applications that use symmetry information to enhance visualization of scalar field data and to facilitate their exploration.