TY - CONF
AB - We introduce TopoCut: a new way to integrate knowledge about topological properties (TPs) into random field image segmentation model. Instead of including TPs as additional constraints during minimization of the energy function, we devise an efficient algorithm for modifying the unary potentials such that the resulting segmentation is guaranteed with the desired properties. Our method is more flexible in the sense that it handles more topology constraints than previous methods, which were only able to enforce pairwise or global connectivity. In particular, our method is very fast, making it for the first time possible to enforce global topological properties in practical image segmentation tasks.
AU - Chen, Chao
AU - Freedman, Daniel
AU - Lampert, Christoph
ID - 3336
SN - 978-1-4577-0394-2
T2 - CVPR: Computer Vision and Pattern Recognition
TI - Enforcing topological constraints in random field image segmentation
ER -
TY - CHAP
AB - Alpha shapes have been conceived in 1981 as an attempt to define the shape of a finite set of point in the plane. Since then, connections to diverse areas in the sciences and engineering have developed, including to pattern recognition, digital shape sampling and processing, and structural molecular biology. This survey begins with a historical account and discusses geometric, algorithmic, topological, and combinatorial aspects of alpha shapes in this sequence.
AU - Edelsbrunner, Herbert
ED - van de Weygaert, R
ED - Vegter, G
ED - Ritzerveld, J
ED - Icke, V
ID - 3311
T2 - Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings
TI - Alpha shapes - a survey
ER -
TY - CONF
AB - This paper presents a method to create a model of an articulated object using the planar motion in an initialization video. The model consists of rigid parts connected by points of articulation. The rigid parts are described by the positions of salient feature-points tracked throughout the video. Following a filtering step that identifies points that belong to different objects, rigid parts are found by a grouping process in a graph pyramid. Valid articulation points are selected by verifying multiple hypotheses for each pair of parts.
AU - Artner, Nicole M.
AU - Ion, Adrian
AU - Kropatsch, Walter G.
ED - Jiang, Xiaoyi
ED - Ferrer, Miquel
ED - Torsello, Andrea
ID - 10907
SN - 0302-9743
T2 - Graph-Based Representations in Pattern Recognition
TI - Spatio-temporal extraction of articulated models in a graph pyramid
VL - 6658
ER -
TY - CONF
AB - In cortex surface segmentation, the extracted surface is required to have a particular topology, namely, a two-sphere. We present a new method for removing topology noise of a curve or surface within the level set framework, and thus produce a cortical surface with correct topology. We define a new energy term which quantifies topology noise. We then show how to minimize this term by computing its functional derivative with respect to the level set function. This method differs from existing methods in that it is inherently continuous and not digital; and in the way that our energy directly relates to the topology of the underlying curve or surface, versus existing knot-based measures which are related in a more indirect fashion. The proposed flow is validated empirically.
AU - Chen, Chao
AU - Freedman, Daniel
ID - 3782
T2 - Conference proceedings MCV 2010
TI - Topology noise removal for curve and surface evolution
VL - 6533
ER -
TY - CHAP
AB - The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.
AU - Edelsbrunner, Herbert
AU - Morozov, Dmitriy
AU - Patel, Amit
ID - 3795
T2 - Topological Data Analysis and Visualization: Theory, Algorithms and Applications
TI - The stability of the apparent contour of an orientable 2-manifold
ER -
TY - CONF
AB - We define the robustness of a level set homology class of a function f:XR as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X=R3 has ramifications in medical imaging and scientific visualization.
AU - Bendich, Paul
AU - Edelsbrunner, Herbert
AU - Morozov, Dmitriy
AU - Patel, Amit
ID - 3848
TI - The robustness of level sets
VL - 6346
ER -
TY - CONF
AB - Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.
AU - Bendich, Paul
AU - Edelsbrunner, Herbert
AU - Kerber, Michael
AU - Patel, Amit
ID - 3849
TI - Persistent homology under non-uniform error
VL - 6281
ER -
TY - CONF
AB - Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape with a disk of fixed radius? If it does, we also seek a preferably simple solution shape P;P’s offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give a decision algorithm for fixed radius in O(nlogn) time that handles any polygonal shape. For convex shapes, the complexity drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one.
AU - Berberich, Eric
AU - Halperin, Dan
AU - Kerber, Michael
AU - Pogalnikova, Roza
ID - 3850
TI - Polygonal reconstruction from approximate offsets
ER -
TY - CONF
AB - Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic meanpayoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.
AU - Chatterjee, Krishnendu
AU - Doyen, Laurent
AU - Edelsbrunner, Herbert
AU - Henzinger, Thomas A
AU - Rannou, Philippe
ID - 3853
TI - Mean-payoff automaton expressions
VL - 6269
ER -
TY - JOUR
AB - We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems.
AU - Bendich, Paul
AU - Edelsbrunner, Herbert
AU - Kerber, Michael
ID - 3901
IS - 6
JF - IEEE Transactions of Visualization and Computer Graphics
TI - Computing robustness and persistence for images
VL - 16
ER -
TY - CONF
AB - We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1-norm of a cycle. Two main results are presented. First, we prove the problem is NP-hard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NP-hard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the ℤ2 field.
AU - Chen, Chao
AU - Freedman, Daniel
ID - 10909
T2 - Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms
TI - Hardness results for homology localization
ER -
TY - CONF
AB - We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.
AU - Edelsbrunner, Herbert
AU - Harer, John
ID - 3968
TI - The persistent Morse complex segmentation of a 3-manifold
VL - 5903
ER -