Abstract

Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their formulation on arbitrary geometric graphs, which place no restrictions in the spatial configuration of samples, is still at an early stage. In this paper, we introduce geometric approximations of gradient and curvature on arbitrary graphs, which enable a straightforward extension of active contour models that are formulated through level sets to such general inputs. We prove convergence in probability of our gradient approximation to the true gradient value and derive an asymptotic upper bound for the error of this approximation for the class of random geometric graphs. Two different approaches for the approximation of curvature are presented, and both are also proved to converge in probability in the case of random geometric graphs. We propose neighborhood-based filtering on graphs to improve the accuracy of the aforementioned approximations and define two variants of Gaussian smoothing on graphs which include normalization in order to adapt to graph nonuniformities. The performance of our active contour framework on graphs is demonstrated in the segmentation of regular images and geographical data defined on arbitrary graphs, using geodesic active contours and active contours without edges as representative models in our experiments.

Keywords

  1. active contours
  2. graph segmentation
  3. random geometric graphs
  4. image segmentation

MSC codes

  1. 35Q68
  2. 35R02
  3. 60D05
  4. 65D25
  5. 68U10

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Supplementary Material

Index of Supplementary Materials

Title of paper: Theoretical Analysis of Active Contours on Graphs

Authors: Christos Sakaridis, Kimon Drakopoulos, and Petros Maragos

File: M110010_01.pdf

Type: PDF file

Contents: 1) Illustrations of gradient and curvature approximations of analytical functions on random geometric graphs with smoothing filters, 2) Segmentation results on graphs with synthetic binary data, 3) Proof of Theorem 3 in the main paper about convergence in probability of the proposed geometric curvature approximation.

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Information & Authors

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Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 1475 - 1510
ISSN (online): 1936-4954

History

Submitted: 24 October 2016
Accepted: 26 June 2017
Published online: 7 September 2017

Keywords

  1. active contours
  2. graph segmentation
  3. random geometric graphs
  4. image segmentation

MSC codes

  1. 35Q68
  2. 35R02
  3. 60D05
  4. 65D25
  5. 68U10

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