Medial/skeletal linking structures for multi-region configurations

@article{Damon2014MedialskeletalLS,
  title={Medial/skeletal linking structures for multi-region configurations},
  author={James N. Damon and Ellen Gasparovic},
  journal={arXiv: Differential Geometry},
  year={2014}
}
We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{\Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the… 

Modeling Multi-object Configurations via Medial/Skeletal Linking Structures

We introduce a method for modeling a configuration of objects in 2D or 3D images using a mathematical “skeletal linking structure” which will simultaneously capture the individual shape features of

Shape and Positional Geometry of Multi-Object Configurations

Numerical invariants for positional properties which measure the closeness of neighboring objects, including identifying the parts of the objects which are close, and the "relative significance" of objects compared with the other objects in the configuration are introduced.

Rigidity Properties of the Blum Medial Axis

  • J. Damon
  • Mathematics
    J. Math. Imaging Vis.
  • 2021
It is shown that in the generic case, along a Y -branching submanifold, that there are three cross ratios involving the three limiting tangent planes of the three smooth sheets and each of the hyperplanes defined by one of the radial lines and the tangent space to the Y - Branches sub manifold at the point, which must be preserved.

Rigidity Properties of the Blum Medial Axis

  • J. Damon
  • Mathematics
    Journal of Mathematical Imaging and Vision
  • 2020
We consider the Blum medial axis of a region in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}

Skeletons, Object Shape, Statistics

This paper lays out the definition of s-reps, their advantages and limitations, their mathematical properties, methods for fitting s- reps to single- and multi-object boundaries, method for measuring the statistics of these object andMulti-object representations, and examples of such applications involving statistics.

Medial Fragments for Segmentation of Articulating Objects in Images

This work proposes a method for combining fragments of the medial axis, generated from the Voronoi diagram of an edge map of a natural image, into a coherent whole using techniques from persistent homology and graph theory to aggregate parts of the same object into a larger whole.

CGWeek Young Researchers Forum 2020

An algorithm for Boolean operations on two polygonal regions designed so that degeneracies typically requiring special handling are treated as general cases is detailed.

References

SHOWING 1-10 OF 57 REFERENCES

Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case

  • J. Damon
  • Mathematics
    Compositio Mathematica
  • 2004
A skeletal structure (M, U) in ${\mathbb R}^{n+1}$ is a special type of n-dimensional Whitney stratified set M on which is defined a multivalued ‘radial vector field’ U. This is an extension of the

Global geometry of regions and boundaries via skeletal and medial integrals

For a compact region Ω in Rn+1 with smooth generic boundary B, the Blum medial axis M is the locus of centers of spheres in Ω which are tangent to B at two or more points. The geometry of Ω is

The Blum Medial Linking Structure for Multi-Region Analysis

ELLEN GASPAROVIC: The Blum Medial Linking Structure for Multi–Region Analysis (Under the direction of James Damon) The Blum medial axis of a region with smooth boundary in R is a skeleton-like

Determining the Geometry of Boundaries of Objects from Medial Data

  • J. Damon
  • Mathematics
    International Journal of Computer Vision
  • 2005
This work considers a region Ω in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued “radial vector field” U from points x on M to the points of tangency of the sphere at x with B, and defines a “geometric medial map” on M which corresponds to the differential geometric properties of B.

A formal classification of 3D medial axis points and their local geometry

  • P. GiblinB. Kimia
  • Computer Science
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 2004
This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis, and derives a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function.

Nested Sphere Statistics of Skeletal Models

A method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods.

Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations

  • P. GiblinB. Kimia
  • Mathematics
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 2009
This work is inspired by that of Bogaevsky, who obtained the transitions as part of an investigation of viscosity solutions of Hamilton-Jacobi equations by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions.

Deformable M-Reps for 3D Medical Image Segmentation

The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper and the accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

Flux invariants for shape

This measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclideans distance function, as well as to explicitly reconstruct the boundary from it.

Symmetry sets

Synopsis For a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea
...