Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold

Packaging Big Data Visualization Based on Computational Intelligence Information Design

A method based on a computational intelligence information model is proposed to study the visualization of large data packages. Since the CAIM algorithm only considers the distribution of the largest number of classes in an interval, it offers an optimization method and simultaneously determines the appropriate stopping conditions to avoid overcrowding. The effectiveness of the improved algorithm has been experimentally proven. Methods of character reduction and weight determination are used to reduce the index and weight, establishing a large packaging information system. Experimental results show that the improved algorithm in this article produces more classification rules than the CAIM algorithm, because the discrete intervals created by the CAIM algorithm are relatively simple, but the classification rules are few, but less than the number of CAIM algorithms. Classification rules are generated by entropy-based sampling algorithms. This can make the classification rules simple and universal, and it is clear that the optimal sampling algorithm is more accurate than the CAIM algorithm.

Feature Curves and Surfaces of 3D Asymmetric Tensor Fields

3D asymmetric tensor fields have found many applications in science and engineering domains, such as fluid dynamics and solid mechanics. 3D asymmetric tensors can have complex eigenvalues, which makes their analysis and visualization more challenging than 3D symmetric tensors. Existing research in tensor field visualization focuses on 2D asymmetric tensor fields and 3D symmetric tensor fields. In this paper, we address the analysis and visualization of 3D asymmetric tensor fields. We introduce six topological surfaces and one topological curve, which lead to an eigenvalue space based on the tensor mode that we define. In addition, we identify several non-topological feature surfaces that are nonetheless physically important. Included in our analysis are the realizations that triple degenerate tensors are structurally stable and form curves, unlike the case for 3D symmetric tensors fields. Furthermore, there are two different ways of measuring the relative strengths of rotation and angular deformation in the tensor fields, unlike the case for 2D asymmetric tensor fields. We extract these feature surfaces using the A-patches algorithm. However, since three of our feature surfaces are quadratic, we develop a method to extract quadratic surfaces at any given accuracy. To facilitate the analysis of eigenvector fields, we visualize a hyperstreamline as a tree stem with the other two eigenvectors represented as thorns in the real domain or the dual-eigenvectors as leaves in the complex domain. To demonstrate the effectiveness of our analysis and visualization, we apply our approach to datasets from solid mechanics and fluid dynamics.

Mode Surfaces of Symmetric Tensor Fields: Topological Analysis and Seamless Extraction

Mode surfaces are the generalization of degenerate curves and neutral surfaces, which constitute 3D symmetric tensor field topology. Efficient analysis and visualization of mode surfaces can provide additional insight into not only degenerate curves and neutral surfaces, but also how these features transition into each other. Moreover, the geometry and topology of mode surfaces can help domain scientists better understand the tensor fields in their applications. Existing mode surface extraction methods can miss features in the surfaces. Moreover, the mode surfaces extracted from neighboring cells have gaps, which make their subsequent analysis difficult. In this paper, we provide novel analysis on the topological structures of mode surfaces, including a common parameterization of all mode surfaces of a tensor field using 2D asymmetric tensors. This allows us to not only better understand the structures in mode surfaces and their interactions with degenerate curves and neutral surfaces, but also develop an efficient algorithm to seamlessly extract mode surfaces, including neutral surfaces. The seamless mode surfaces enable efficient analysis of their geometric structures, such as the principal curvature directions. We apply our analysis and visualization to a number of solid mechanics data sets.

Multi-Scale Topological Analysis of Asymmetric Tensor Fields on Surfaces

Asymmetric tensor fields have found applications in many science and engineering domains, such as fluid dynamics. Recent advances in the visualization and analysis of 2D asymmetric tensor fields focus on pointwise analysis of the tensor field and effective visualization metaphors such as colors, glyphs, and hyperstreamlines. In this paper, we provide a novel multi-scale topological analysis framework for asymmetric tensor fields on surfaces. Our multi-scale framework is based on the notions of eigenvalue and eigenvector graphs. At the core of our framework are the identification of atomic operations that modify the graphs and the scale definition that guides the order in which the graphs are simplified to enable clarity and focus for the visualization of topological analysis on data of different sizes. We also provide efficient algorithms to realize these operations. Furthermore, we provide physical interpretation of these graphs. To demonstrate the utility of our system, we apply our multi-scale analysis to data in computational fluid dynamics.

Robust and Fast Extraction of 3D Symmetric Tensor Field Topology

3D symmetric tensor fields appear in many science and engineering fields, and topology-driven analysis is important in many of these application domains, such as solid mechanics and fluid dynamics. Degenerate curves and neutral surfaces are important topological features in 3D symmetric tensor fields. Existing methods to extract degenerate curves and neutral surfaces often miss parts of the curves and surfaces, respectively. Moreover, these methods are computationally expensive due to the lack of knowledge of structures of degenerate curves and neutral surfaces. In this paper, we provide theoretical analysis on the geometric and topological structures of degenerate curves and neutral surfaces of 3D linear tensor fields. These structures lead to parameterizations for degenerate curves and neutral surfaces that can not only provide more robust extraction of these features but also incur less computational cost. We demonstrate the benefits of our approach by applying our degenerate curve and neutral surface detection techniques to solid mechanics simulation data sets.

Tensor Field Visualization using Fiber Surfaces of Invariant Space

Scientific visualization developed successful methods for scalar and vector fields. For tensor fields, however, effective, interactive visualizations are still missing despite progress over the last decades. We present a general approach for the generation of separating surfaces in symmetric, second-order, three-dimensional tensor fields. These surfaces are defined as fiber surfaces of the invariant space, i.e. as pre-images of surfaces in the range of a complete set of invariants. This approach leads to a generalization of the fiber surface algorithm by Klacansky et al. [16] to three dimensions in the range. This is due to the fact that the invariant space is three-dimensional for symmetric second-order tensors over a spatial domain. We present an algorithm for surface construction for simplicial grids in the domain and simplicial surfaces in the invariant space. We demonstrate our approach by applying it to stress fields from component design in mechanical engineering.

DT-MRI Streamsurfaces Revisited

DT-MRI streamsurfaces, defined as surfaces that are everywhere tangential to the major and medium eigenvector fields, have been proposed as a tool for visualizing regions of predominantly planar behavior in diffusion tensor MRI. Even though it has long been known that their construction assumes that the involved eigenvector fields satisfy an integrability condition, it has never been tested systematically whether this condition is met in real-world data. We introduce a suitable and efficiently computable test to the visualization literature, demonstrate that it can be used to distinguish integrable from nonintegrable configurations in simulations, and apply it to whole-brain datasets of 15 healthy subjects. We conclude that streamsurface integrability is approximately satisfied in a substantial part of the brain, but not everywhere, including some regions of planarity. As a consequence, algorithms for streamsurface extraction should explicitly test local integrability. Finally, we propose a novel patch-based approch to streamsurface visualization that reduces visual artifacts, and is shown to more fully sample the extent of streamsurfaces.