Afra Zomorodian

The theory of multidimensional persistence

Persistent homology captures the topology of a filtration - a one-parameter family of increasing spaces - in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.

Technical Section: Fast construction of the Vietoris-Rips complex

The Vietoris-Rips complex characterizes the topology of a point set. This complex is popular in topological data analysis as its construction extends easily to higher dimensions. We formulate a two-phase approach for its construction that separates geometry from topology. We survey methods for the first phase, give three algorithms for the second phase, implement all algorithms, and present experimental results. Our software can be used also for constructing any clique complex, such as the weak witness complex.

Localized homology

In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm and present experiments to validate our approach in practice.

Colon polyp detection using smoothed shape operators: preliminary results.

Computer-aided detection (CAD) algorithms identify locations in computed tomographic (CT) images of the colon that are most likely to contain polyps. Existing CAD methods treat the CT data as a voxelized, volume image. They estimate a curvature-based feature at the mucosal surface voxels. However, curvature is a smooth notion, while our data are discrete and noisy. As a second order differential quantity, curvature amplifies noise. In this paper, we present the smoothed shape operators method (SSO), which uses a geometry processing approach. We extract a triangle mesh representation of the colon surface, and estimate curvature on this surface using the shape operator. We then smooth the shape operators on the surface iteratively. Throughout, we use techniques explicitly designed for discrete geometry. All our computation occurs on the surface, rather than in the voxel grid. We evaluate our algorithm on patient data and provide free-response receiver-operating characteristic performance analysis over all size ranges of polyps. We also provide confidence intervals for our performance estimates. We compare our performance with the surface normal overlap (SNO) method for the same data. A preliminary evaluation of our method on 35 patients yielded the following results (polyp diameter range; sensitivity; false positives/case): (10mm; 100%; 17.5), (5-10 mm; 89.7%, 21.23), (<5 mm; 59.1%; 23.9) and (overall; 80.3%; 23.9). The evaluation of the SNO method yielded: (10 mm; 75%; 17.5), (5-10 mm; 43.1%; 21.23), (<5 mm; 15.9%; 23.9) and (overall; 38.5%; 23.9).

The tidy set: a minimal simplicial set for computing homology of clique complexes

We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the Vietoris-Rips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.

Computing Multidimensional Persistence

The theory of multidimensional persistence captures the topology of a multifiltration --- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence.

Localized Homology

In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm and present experiments to validate our approach in practice.

Editorial: Guest Editors' foreword

This special issue of Computational Geometry: Theory and Applications consists of seven papers, selected from the 25th Annual Symposium on Computational Geometry, held June 8–10, 2009, at Aarhus University, Denmark. We selected the papers to have an algorithmic focus, in consultation with the conference Program Committee, chaired by John Hershberger. The papers reflect the diversity of topics in the conference, tackling combinatorial, geometric, and topological questions. Timothy Chan and Eric Chen give the first optimal, randomized, in-place algorithm for the 3-dimensional convex hull problem. Their algorithm runs in O (n log n) expected time, using O (1) extra space. They also show that their approach yields an optimal algorithm for the 2-dimensional line intersection problem, and also simplifies a known optimal cacheoblivious algorithm for convex hulls. Gary Miller and Donald Sheehy present the first deterministic algorithm for computing an approximate centerpoint of a set S ⊆ Rd with time subexponential in d. Their algorithm derandomizes a previous algorithm by Clarkson et al. and terminates with an O (1/d2)-center. They also leverage higher order Tverberg partitions to improve the running time of the deterministic algorithm as well as the approximation guarantee of the randomized algorithm. Glencora Borradaile, James Lee, and Anastasios Sidiropoulos present a probabilistic planar embedding of a genus g graph with distortion O (g2). Viewing the graph as embedded on a surface with g handles, they cut all handles at once, improving the previous exponential distortion bound of Indyk and Sidiropoulos that was achieved via an iterative algorithm. Vicente Batista, David Millman, Sylvain Pion, and Johannes Singler present several parallel geometric algorithms for multicore computers with shared memory, including 2and 3-dimensional spatial sorting, d-dimensional axis-aligned box intersection, and bulk insertion of points into the 3-dimensional Delaunay triangulation. They present experimental results for their algorithms using implementations based on the Computational Geometry Algorithms Library (CGAL) as a first step toward a parallel mode for the library. Gaiane Panina and Ileana Streinu close the open problem of whether the configuration space of foldable configurations in single-vertex origami with rigid faces is connected. The authors undertake a new approach that utilizes spherical geometry and a Lebesgue measure on the surface of a sphere, which allows them to finish the remaining case of the work of Streinu and Whiteley. Oswin Aichholzer, Wolfgang Aigner, Franz Aurenhammer, Thomas Hackl, Bert Juettler, Margot Rabl, and Elisabeth Pilgerstorfer present a novel divide-and-conquer approach for computing the Voronoi Diagram of fairly general shapes in the plane. They first compute the medial axis of the shapes, inverting the usual order of computation for the medial axis and the Voronoi Diagram. This new approach leads to numerically robust algorithms that work well in practice, even for circular and other sites. Peyman Afshani, Chris Hamilton, and Norbert Zeh present a general framework for cache-oblivious data structures used for approximate range counting and exact range reporting under the condition that the respective range searching problems have appropriate shallow cuttings. A number of problems meet this condition, such as three-sided range searching as well as 3-dimensional dominance or halfspace range searching. The respective data structures need linear space and facilitate worstcase (1 + ε)-approximate answers using optimal O (logB(N/K )) I/Os. This even improves on previous results in internal memory, where the optimal query bound was not achieved in the worst case before, even if superlinear space were allowed. All the papers in this special issue were reviewed through the normal refereeing process reflecting the journal’s high standards. We would like to thank the authors and reviewers for their hard work. Guest Editors Carola Wenk University of Texas at San Antonio, Department of Computer Science, San Antonio, TX, United States E-mail address: carola@cs.utsa.edu