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 -