Genus Computing for 3D digital objects: algorithm and implementation

This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components, boundary surfaces and their genus, and homology groups. Due to the fact that genus calculation dominates the entire task for 3D object in 3D space, in this paper, we mainly discuss the calculation of the genus. The new algorithms designed in this paper will perform: (1) pathological cases detection and deletion, (2) raster space to point space (dual space) transformation, (3) the linear time algorithm for boundary point classification, and (4) genus calculation.Comment: 12 pages 7 figures. In Proceedings of the Workshop on Computational Topology in image context 2009, Aug. 26-28, Austria, Edited by W. Kropatsch, H. M. Abril and A. Ion, 200

Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system

A 3D binary image I is called well-composed if the set of points in the topological boundary of the cubes in I is a 2-manifold. Repairing a 3D binary image is a process which produces a well composed image (or a polyhedral complex) from the non-well-composed image I.We propose here to repair 3D images by associating the Body-Centered Cubic grid (BCC grid) to the cubical grid. The obtained polyhedral complex is well composed, since two voxels in the BCC grid either share an entire face or are disjoint. We show that the obtained complex is homotopy equivalent to the cubical complex naturally associated with the image I.To efficiently encode and manipulate the BCC grid, we present an integer 4-valued combinatorial coordinate system that addresses cells of all dimensions (voxels, faces, edges and vertices), and allows capturing all the topological incidence and adjacency relations between cells by using only integer operations.We illustrate an application of this coordinate system on two tasks related with the repaired image: boundary reconstruction and computation of the Euler characteristic

Doctor of Philosophy

dissertationIn this dissertation, we advance the theory and practice of verifying visualization algorithms. We present techniques to assess visualization correctness through testing of important mathematical properties. Where applicable, these techniques allow us to distinguish whether anomalies in visualization features can be attributed to the underlying physical process or to artifacts from the implementation under verification. Such scientific scrutiny is at the heart of verifiable visualization - subjecting visualization algorithms to the same verification process that is used in other components of the scientific pipeline. The contributions of this dissertation are manifold. We derive the mathematical framework for the expected behavior of several visualization algorithms, and compare them to experimentally observed results in the selected codes. In the Computational Science & Engineering community CS&E, this technique is know as the Method of Manufactured Solution (MMS). We apply MMS to the verification of geometrical and topological properties of isosurface extraction algorithms, and direct volume rendering. We derive the convergence of geometrical properties of isosurface extraction techniques, such as function value and normals. For the verification of topological properties, we use stratified Morse theory and digital topology to design algorithms that verify topological invariants. In the case of volume rendering algorithms, we provide the expected discretization errors for three different error sources. The results of applying the MMS is another important contribution of this dissertation. We report unexpected behavior for almost all implementations tested. In some cases, we were able to find and fix bugs that prevented the correctness of the visualization algorithm. In particular, we address an almost 2 0 -year-old bug with the core disambiguation procedure of Marching Cubes 33, one of the first algorithms intended to preserve the topology of the trilinear interpolant. Finally, an important by-product of this work is a range of responses practitioners can expect to encounter with the visualization technique under verification