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Creators/Authors contains: "Schenfisch, Anna"

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  1. ; ; (, Proc. Canadian Conference on Computational Geometry)
    Recent developments in shape reconstruction and comparison call for the use of many different (topological) descriptor types, such as persistence diagrams and Euler characteristic functions. We establish a framework to quantitatively compare the strength of different descriptor types, setting up a theory that allows for future comparisons and analysis of descriptor types and that can inform choices made in applications. We use this framework to partially order a set of six common descriptor types. We then give lower bounds on the size of sets of descriptors that uniquely correspond to simplicial complexes, giving insight into the advantages of using verbose rather than concise topological descriptors. 
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    Accepted Manuscript Full Text Available
  2. ; ; (, EuroCG)
    Recent developments in shape reconstruction and comparison call for the use of many different types of topological descriptors (persistence diagrams, Euler characteristic functions, etc.). We establish a framework that allows for quantitative comparisons of topological descriptor types and therefore may be used as a tool in more rigorously justifying choices made in applications. We then use this framework to partially order a set of six common topological descriptor types. In particular, the resulting poset gives insight into the advantages of using verbose rather than concise topological descriptors. We then provide lower bounds on the size of sets of descriptors that are complete discrete invariants of simplicial complexes, both tight and worst case. This work sets up a rigorous theory that allows for future comparisons and analysis of topological descriptor types. 
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    Accepted Manuscript Full Text Available
  3. ; (, Homology, Homotopy and Applications)
    Full Text Available 
  4. ; ; ; ; (, Canadian Conference on Computational Geometry (CCCG))
    Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in R d , the (augmented) persistent homology transform ((A)PHT) is a family of persistence diagrams, parameterized by directions in the ambient space. A recent advance in understanding the PHT used the framework of reconstruction in order to find finite a set of directions to faithfully represent the shape, a result that is of both theoretical and practical interest. In this paper, we improve upon this result and present an improved algorithm for graph— and, more generally one-skeleton—reconstruction. The improvement comes in reconstructing the edges, where we use a radial binary (multi-)search. The binary search employed takes advantage of the fact that the edges can be ordered radially with respect to a reference plane, a feature unique to graphs. 
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    Accepted Manuscript Full Text Available
  5. ; ; ; ; ; ; ; ; (, Computational Geometry)
    Full Text Available 
  6. ; (, Young Researcher's Forum (CG Week))
    Although gland curvature features are an important element in current prostate cancer diagnostic tools, they are typically evaluated qualitatively rather than quantitatively. We propose a method of approximating the curvature of prostate gland cross sections by using the nuclei as data points. By investigating the relationship between features of a gland's estimated curvature and its severity of cancer, we show that our curvature estimates may be used as a tool to add a more objective element into the current diagnostic process. 
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    Accepted Manuscript Full Text Available
  7. ; ; ; ; ; ; ; ; (, Canadian Conference on Computational Geometry)
    Topological Data Analysis (TDA) studies the “shape” of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K = (V, E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 
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    Accepted Manuscript Full Text Available