Allen van Gelder
Abstract
A popular technique for rendition of isosurfaces in sampled data is to consider cells with sample points as corners and approximate the isosurface in each cell by one or more polygons whose vertices are obtained by interpolation of the sample data. That is, each polygon vertex is a point on a cell edge, between two adjacent sample points, where the function is estimated to equal the desired threshold value. The two sample points have values on opposite sides of the threshold, and the interpolated point is called an intersection point.
When one cell face has an intersection point in each of its four edges, then the correct connection among intersection points becomes ambiguous. An incorrect connection can lead to erroneous topology in the rendered surface, and possible discontinuities. We show that disambiguation methods, to be at all accurate, need to consider sample values in the neighborhood outside the cell. This paper studies the problems of disambiguation, reports on some solutions, and presents some statistics on the occurrence of such ambiguities.
A natural way to incorporate neighborhood information is through the use of calculated gradients at cell corners. They provide insight into the behavior of a function in well-understood ways. We introduce two gradient consistency heuristics that use calculated gradients at the corners of ambiguous faces, as well as the function values at those corners, to disambiguate at a reasonable computational cost. These methods give the correct topology on several examples that caused problems for other methods we examined.
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Index Terms
- Topological considerations in isosurface generation
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Octrees for faster isosurface generation
The large size of many volume data sets often prevents visualization algorithms from providing interactive rendering. The use of hierarchical data structures can ameliorate this problem by storing summary information to prevent useless exploration of ...
Reviews
Nickolas S. Sapidis
The authors describe techniques for generating topologically correct approximate isosurfaces from sample data where the underlying function is not available for resampling. Section 2 lists desirable features of a general-purpose polygonal isosurface algorithm. Section 3 briefly reviews existing approaches and introduces concepts and definitions related to isosurface generation. It defines topological ambiguities, discusses the problem of disambiguation, and introduces functions useful in testing an algorithm's ability to determine correct topology. Also, section 3 establishes that “it is impossible (in general) to determine correct surface topology in a cell solely by examination of the voxel values at the vertices of that cell.” The last part of section 3 focuses on insuring the continuity of isosurfaces and proves that “if the method of disambiguation for ambiguous faces employs only values in the plane of the face, and is invariant under rotations and reflections, then the isosurface…is continuous.” Section 4 discusses in detail approaches to disambiguation of sampled data, which are categorized as “simple boolean,” “extended boolean,” “simple metric,” or “extended metric.” In section 5, these techniques are applied on realistic data, and performance measurements are reported. Finally, section 6 attempts to evaluate the topological accuracy of the various heuristics by applying them on data from known underlying functions and by evaluating the smoothness of volumes with the aid of Fourier analysis. This paper offers a comprehensive treatment of its subject, including an interesting evaluation of both old and new techniques. This work will be useful to graduate students, researchers, and practitioners, because the text is clearly written and understandable even by nonspecialists.
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