Abstract-Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures.Index Terms-Vector Fields, Error Quantification, Edge Maps. ! MOTIVATIONSV ECTOR fields are a common form of simulation data appearing in a wide variety of applications ranging from computational fluid dynamics (CFD) and weather prediction to engineering design. Visualizing and analyzing the flow behavior of these fields can help provide critical insights into simulated physical processes. However, achieving a consistent and rigorous interpretation of vector fields is difficult, in part because traditional numerical techniques for integration do not preserve the expected invariants of vector fields.To better understand this challenge inherent in traditional numerical techniques, we reconsider the most common way to store vector fields. Both a discretization of the domain of the field (often in the form of a triangulated mesh) as well as a set of sample vectors (defined at the vertices of the mesh) are required. The vector field on the interior of a triangle is approximated by interpolating vector values from the samples at the triangle's corners. Subsequently, computing properties that require integrating these vector values presents a significant computational challenge. For example, consider computing the flow paths (streamlines) of massless particles that travel using the instantaneous velocity defined by the field. Naive integration techniques may violate the property that every two of these paths are expected to be pairwise disjoint
Vector fields, represented as vector values sampled on the vertices of a triangulation, are commonly used to model physical phenomena. To analyze and understand vector fields, practitioners use derived properties such as the paths of massless particles advected by the flow, called streamlines. However, currently available numerical methods for computing streamlines do not guarantee preservation of fundamental invariants such as the fact that streamlines cannot cross. The resulting inconsistencies can cause errors in the analysis, e.g. invalid topological skeletons, and thus lead to misinterpretations of the data. We propose an alternate representation for triangulated vector fields that exchanges vector values with an encoding of the transversal flow behavior of each triangle. We call this representation edge maps. This work focuses on the mathematical properties of edge maps; a companion paper discusses some of their applications [1]. Edge maps allow for a multi-resolution approximation of flow by merging adjacent streamlines into an interval based mapping. Consistency is enforced at any resolution if the merged sets maintain an order-preserving property. At the coarsest resolution, we define a notion of equivalency between edge maps, and show that there exist 23 equivalence classes describing all possible behaviors of piecewise linear flow within a triangle.
Figure 1: Edge maps enable new views of vector field stability, illustrated with a vector field on this wavy surface. Top row (middle right): A visualization of some colored regions where flow shares the same source (green spheres) and sink (red spheres) is augmented to show how these regions overlap when error is introduced. Bottom row (middle right): Streamwaves (colored green to red as they grow) show the advection of a single particle. In the presence of error, waves can widen and narrow, and bifurcate or merge. ABSTRACTRobust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Many analysis techniques rely on computing streamlines, a task often hampered by numerical instabilities. Approaches that ignore the resulting errors can lead to inconsistencies that may produce unreliable visualizations and ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with linear maps defined on its boundary. This representation, called edge maps, is equivalent to computing all possible streamlines at a user defined error threshold. In spite of this error, all the streamlines computed using edge maps will be pairwise disjoint. Furthermore, our representation stores the error explicitly, and thus can be used to produce more informative visualizations. Given a piecewise-linear interpolated vector field, a recent result [15] shows that there are only 23 possible map classes for a triangle, permitting a concise description of flow behaviors. This work describes the details of computing edge maps, provides techniques to quantify and refine edge map error, and gives qualitative and visual comparisons to more traditional techniques. Keywords MOTIVATIONSVector fields are a common form of simulation data appearing in a wide variety of applications ranging from computational fluid dynamics (CFD) and weather prediction to engineering design. Visualizing and analyzing the flow behavior of these fields can help to provide critical insights into simulated physical processes. However, achieving a consistent and rigorous interpretation of vector fields is difficult, in part because traditional numerical techniques for integration do not preserve the expected invariants of vector fields.To better understand this inherent issue of traditional numerical techniques, we start with a description of a common way to store vector fields. Both a discretization of the domain of the field (often in the form of a triangulated mesh) as well as a set of sample vectors (defined at the vertices of the mesh) are required. The vector field on the interior of a triangle is approximated by interpolating vector values from the samples at the triangle's corners. Computing properties that then require integrating these vector values presents a significant computational challenge. For example, consider computing the flow paths (streamlines) of massless particles that travel using the ...
Wide-field microscopes are commonly used in neurobiology for experimental studies of brain samples. Available visualization tools are limited to electron, two-photon, and confocal microscopy datasets, and current volume rendering techniques do not yield effective results when used with wide-field data. We present a workflow for the visualization of neuronal structures in wide-field microscopy images of brain samples. We introduce a novel gradient-based distance transform that overcomes the out-of-focus blur caused by the inherent design of wide-field microscopes. This is followed by the extraction of the 3D structure of neurites using a multi-scale curvilinear filter and cell-bodies using a Hessian-based enhancement filter. The response from these filters is then applied as an opacity map to the raw data. Based on the visualization challenges faced by domain experts, our workflow provides multiple rendering modes to enable qualitative analysis of neuronal structures, which includes separation of cell-bodies from neurites and an intensity-based classification of the structures. Additionally, we evaluate our visualization results against both a standard image processing deconvolution technique and a confocal microscopy image of the same specimen. We show that our method is significantly faster and requires less computational resources, while producing high quality visualizations. We deploy our workflow in an immersive gigapixel facility as a paradigm for the processing and visualization of large, high-resolution, wide-field microscopy brain datasets.
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