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The Elastic Ratio: Introducing Curvature Into Ratio-Based Image Segmentation

Schoenemann, Thomas LU ; Masnou, Simon and Cremers, Daniel (2011) In IEEE Transactions on Image Processing 20(9). p.2565-2581
Abstract
We present the first ratio-based image segmentation method that allows imposing curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature of the curve. The key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment. Among ratios whose discrete counterparts can be globally minimized with our approach, we focus in particular on the elastic ratio integral(L(C))(0) del I(C(S)) . (C'(S)(perpendicular to) ds/nu L(C) + integral(L(C))(0) broken vertical bar kappa(C)(S)broken vertical bar(q) ds that depends,... (More)
We present the first ratio-based image segmentation method that allows imposing curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature of the curve. The key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment. Among ratios whose discrete counterparts can be globally minimized with our approach, we focus in particular on the elastic ratio integral(L(C))(0) del I(C(S)) . (C'(S)(perpendicular to) ds/nu L(C) + integral(L(C))(0) broken vertical bar kappa(C)(S)broken vertical bar(q) ds that depends, given an image I, on the oriented boundary C of the segmented region candidate. Minimizing this ratio amounts to finding a curve, neither small nor too curvy, through which the brightness flux is maximal. We prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions. We also prove that the discrete minimizers provided by our graph-based algorithm converge, as the resolution increases, to continuous minimizers. In contrast to most existing segmentation methods with computable and meaningful, i.e., nondegenerate, global optima, the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes. Numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations. Furthermore, our results allow drawing conclusions about global optima of a parameterization-independent version of the snakes functional: the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Contour-based segmentation, curvature, global optimization, graph, cycles, image segmentation, snakes model, unsupervised segmentation
in
IEEE Transactions on Image Processing
volume
20
issue
9
pages
2565 - 2581
publisher
IEEE--Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • wos:000294132800014
ISSN
1941-0042
DOI
10.1109/TIP.2011.2118225
language
English
LU publication?
yes
id
5d1c52fa-7b2a-4946-a3a7-f5edce068ee8 (old id 2161365)
date added to LUP
2011-09-21 13:55:06
date last changed
2016-09-20 05:38:34
@article{5d1c52fa-7b2a-4946-a3a7-f5edce068ee8,
  abstract     = {We present the first ratio-based image segmentation method that allows imposing curvature regularity of the region boundary. Our approach is a generalization of the ratio framework pioneered by Jermyn and Ishikawa so as to allow penalty functions that take into account the local curvature of the curve. The key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment. Among ratios whose discrete counterparts can be globally minimized with our approach, we focus in particular on the elastic ratio integral(L(C))(0) del I(C(S)) . (C'(S)(perpendicular to) ds/nu L(C) + integral(L(C))(0) broken vertical bar kappa(C)(S)broken vertical bar(q) ds that depends, given an image I, on the oriented boundary C of the segmented region candidate. Minimizing this ratio amounts to finding a curve, neither small nor too curvy, through which the brightness flux is maximal. We prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions. We also prove that the discrete minimizers provided by our graph-based algorithm converge, as the resolution increases, to continuous minimizers. In contrast to most existing segmentation methods with computable and meaningful, i.e., nondegenerate, global optima, the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes. Numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations. Furthermore, our results allow drawing conclusions about global optima of a parameterization-independent version of the snakes functional: the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution.},
  author       = {Schoenemann, Thomas and Masnou, Simon and Cremers, Daniel},
  issn         = {1941-0042},
  keyword      = {Contour-based segmentation,curvature,global optimization,graph,cycles,image segmentation,snakes model,unsupervised segmentation},
  language     = {eng},
  number       = {9},
  pages        = {2565--2581},
  publisher    = {IEEE--Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Image Processing},
  title        = {The Elastic Ratio: Introducing Curvature Into Ratio-Based Image Segmentation},
  url          = {http://dx.doi.org/10.1109/TIP.2011.2118225},
  volume       = {20},
  year         = {2011},
}