We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its life-time or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.

We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.

We define the Morse-Smale complex of a Morse function over a 3-manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3-dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian kernel of Euclidean space. As an important special case, kernels based on the geometry of multinomial families are derived, leading to kernel-based learning algorithms that apply naturally to discrete data. Bounds on covering numbers and Rademacher averages for the kernels are proved using bounds on the eigenvalues of the Laplacian on Riemannian manifolds. Experimental results are presented for document classification, for which the use of multinomial geometry is natural and well motivated, and improvements are obtained over the standard use of Gaussian or linear kernels, which have been the standard for text classification.

Many high-resolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axis-aligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate out-of-core execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.

Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models—the ability to automatically handle topology changes—turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved. In this paper, we present a new class of geometric deformable models designed using a novel topology-preserving level set method, which achieves topology preservation by applying the simple point concept from digital topology. These new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces. Moreover, since the topology-preserving constraint is enforced efficiently through local computations, the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models. Several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm.

We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is based on the input and output energy of a system. We deal with the input and output energy function of a stable nonlinear systems and propose a method to use these functions to get a balanced form for a stable nonlinear system. It is a local result, but gives `broader' results then we obtain by just linearizing the system. Keywords: balancing, nonlinear systems, energy functions, Hamilton-Jacobi equations, Hankel singular values. 1 Introduction Balancing for linear systems is a well known subject on which there has been a lot of research in the last decade. It started with a paper of Moore [6] in 1981, where balancing is introduced with the aim of using it as a tool for model reduction. If a linear system is in balanced form the Hankel singular values are a measure for the importance of state components. This means that the influence of the correspondin...

This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R 3 . The surface is modeled as the skin defined by a finite set of spheres, as defined in [9]. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface. Keywords. Computational geometry, differential geometry, adaptive meshing, deformation, metamorphoses, algorithms, proofs. 1 Introduction This paper develops a fully dynamic algorithm for maintaining a triangulation of a surface embedded in R 3 that changes its local and global shape, curvature, and topology with time. Motivation. Deforming surfaces arise in moving boundary problems of physical simulation, where they act as boundaries of spatial domains that grow and shrink with time. An example is the boundary between the solid and the liquid portions of metal during solidification [17]. Another is the phase boundary in a solid alloy that goes through the nucleation, growth and coarse...

Abstract — We combine topological and geometric methods to construct a multi-resolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by canceling pairs of critical points. Based on a simple notion of dependency among these cancellations we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometrybased representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at run-time. Index Terms — Critical point theory, Morse-Smale complex, terrain data, simplification, multi-resolution data structure. I.