David Letscher

Topological persistence and simplification

AbstractWe 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 lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.

Delaunay triangulations and Voronoi diagrams for Riemannian manifolds

For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space. Algorithms for constructing these triangulations will also be described.

A simple and robust thinning algorithm on cell complexes

Thinning is a commonly used approach for computing skeleton descriptors. Traditional thinning algorithms often have a simple, iterative structure, yet producing skeletons that are overly sensitive to boundary perturbations. We present a novel thinning algorithm, operating on objects represented as cell complexes, that preserves the simplicity of typical thinning algorithms but generates skeletons that more robustly capture global shape features. Our key insight is formulating a skeleton significance measure, called medial persistence, which identify skeleton geometry at various dimensions (e.g., curves or surfaces) that represent object parts with different anisotropic elongations (e.g., tubes or plates). The measure is generally defined in any dimensions, and can be easily computed using a single thinning pass. Guided by medial persistence, our algorithm produces a family of topology and shape preserving skeletons whose shape and composition can be flexible controlled by desired level of medial persistence.

Detecting filtered cloning in digital images

We present an efficient technique to detect portions of a digital image that have been modified using textural information from another region of the image, for example Photoshops healing brush or Poisson cloning. Common uses of these tools include removing damaged areas, blemishes and sometimes larger objects from a scene. We show that the methods are efficient and accurately identify the use of a large class of image manipulation techniques for uncompressed TIFF images and high quality compressed images. When applied to images that have higher compression levels accurate results are also obtained if the region that has been duplicated is sufficiently large.

Image segmentation using topological persistence

This paper presents a new hybrid split-and-merge image segmentation method based on computational geometry and topology using persistent homology. The algorithm uses edge-directed topology to initially split the image into a set of regions based on the Delaunay triangulations of the points in the edge map. Persistent homology is used to generate three types of regions: p-persistent regions, p-transient regions, and d-triangles. The p-persistent regions correspond to core objects in the image, while p-transient regions and d-triangles are smaller regions that may be combined in the merge phase, either with p-persistent regions to refine the core or with other p-transient and d-triangles regions to potentially form new core objects. Performing image segmentation based on topology and persistent homology guarantees several nice properties, and initial results demonstrate high quality image segmentation.

Optimal Strategies for Sports Betting Pools

Every fall, millions of Americans enter betting pools to pick winners of the weekly NFL football games. In the spring, NCAA tournament basketball pools are even more popular. In both cases, teams that are popularly perceived as “favorites” gain a disproportionate share of entries. In large pools there can be a significant advantage to picking upsets that differentiate your picks from the crowd. In this paper, we present a model of betting pools that incorporates pool participant behavior. We use the model to derive strategies that maximize the expected return on a bet in both football pools and tournament-style pools. These strategies significantly outperform strategies based on maximizing score or number of correct picks---often by orders of magnitude.

Erosion thickness on medial axes of 3D shapes

While playing a fundamental role in shape understanding, the medial axis is known to be sensitive to small boundary perturbations. Methods for pruning the medial axis are usually guided by some measure of significance. The majority of significance measures over the medial axes of 3D shapes are locally defined and hence unable to capture the scale of features. We introduce a global significance measure that generalizes in 3D the classical Erosion Thickness (ET) measure over the medial axes of 2D shapes. We give precise definition of ET in 3D, analyze its properties, and present an efficient approximation algorithm with bounded error on a piece-wise linear medial axis. Experiments showed that ET outperforms local measures in differentiating small boundary noise from prominent shape features, and it is significantly faster to compute than existing global measures. We demonstrate the utility of ET in extracting clean, shape-revealing and topology-preserving skeletons of 3D shapes.

Teaching strategies for reinforcing structural recursion with lists

Recursion is an important concept in computer science and one that possesses beauty and simplicity, yet many educators describe challenges in teaching the topic. Kim Bruce champions the early use of structural recursion in an object-oriented introductory programming course as a more intuitive concept than traditional (functional) recursion. He uses many graphical examples for motivation (e.g., nested boxes, a ringed bullseye, fractals), providing concreteness to the recursive concept. Internally, most of those examples are disguised forms of a basic recursive list pattern. Recursive lists are important in and of themselves and a mainstay within the functional programming paradigm. However, further challenges exist in providing a tangible presentation for pure lists when disassociated from a graphical structure. Recursion is an important concept in computer science and one that possesses beauty and simplicity, yet many educators describe challenges in teaching the topic. Kim Bruce champions the early use of structural recursion in an object-oriented introductory programming course as a more intuitive concept than traditional (functional) recursion. He uses many graphical examples for motivation (e.g., nested boxes, a ringed bullseye, fractals), providing concreteness to the recursive concept. Internally, most of those examples are disguised forms of a basic recursive list pattern. Recursive lists are important in and of themselves and a mainstay within the functional programming paradigm. However, further challenges exist in providing a tangible presentation for pure lists when disassociated from a graphical structure.

On persistent homotopy, knotted complexes and the Alexander module

In this paper techniques from persistent homology are generalized to homotopy groups and to algebraic invariants from knot theory. We define the persistent Alexander module, which can be used to detect knotting in a complex and determine when the knotting changes when viewed from different scales. Algorithms that use the persistent Alexander module are also presented and applied to examples including protein structures. While the basic definition of persistent homotopy is known, this is the first work to use it successfully for computations.

A graphics package for the first day and beyond

We describe cs1graphics, a new Python drawing package designed with pedagogy in mind. The package is simple enough that students can sit down and make use of it from the first day of an introductory class. Yet it provides seamless support for intermediate and advanced lessons as students progress. In this paper, we discuss its versatility in the context of an introductory course. The package is available at www.cs1graphics.org.