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.


  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.


E. Arias-Castro, B. Pelletier, and P. Pudlo, The normalized graph cut and Cheeger constant: From discrete to continuous, Adv. Appl. Probab., 44 (2012), pp. 907--937.
M. Belkin and P. Niyogi, Convergence of Laplacian eigenmaps, in Advances in Neural Information Processing Systems (NIPS), 2006, pp. 129--136.
S. Bougleux, A. Elmoataz, and M. Melkemi, Discrete regularization on weighted graphs for image and mesh filtering, in Proc. International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), 2007, pp. 128--139.
Y. Boykov and V. Kolmogorov, Computing geodesics and minimal surfaces via graph cuts, in Proc. International Conference on Computer Vision, vol. 1, 2003, pp. 26--33.
Y. Boykov, O. Veksler, and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Mach. Intell., 23 (2001), pp. 1222--1239.
V. Caselles, F. Catté, T. Coll, and F. Dibos, A geometric model for active contours in image processing, Numer. Math., 66 (1993), pp. 1--31.
V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis., 22 (1997), pp. 61--79.
T. F. Chan, S. Esedoḡlu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), pp. 1632--1648, https://doi.org/10.1137/040615286.
T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266--277.
C. Couprie, L. Grady, L. Najman, J.-C. Pesquet, and H. Talbot, Dual constrained TV-based regularization on graphs, SIAM J. Imaging Sci., 6 (2013), pp. 1246--1273, https://doi.org/10.1137/120895068.
C. Couprie, L. Grady, L. Najman, and H. Talbot, Power watershed: A unifying graph-based optimization framework, IEEE Trans. Pattern Anal. Mach. Intell., 33 (2011), pp. 1384--1399.
C. Couprie, L. Grady, H. Talbot, and L. Najman, Combinatorial continuous maximum flow, SIAM J. Imaging Sci., 4 (2011), pp. 905--930, https://doi.org/10.1137/100799186.
J. Cousty, G. Bertrand, L. Najman, and M. Couprie, Watershed cuts: Minimum spanning forests and the drop of water principle, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), pp. 1362--1374.
J. Cousty, G. Bertrand, L. Najman, and M. Couprie, Watershed cuts: Thinnings, shortest path forests, and topological watersheds, IEEE Trans. Pattern Anal. Mach. Intell., 32 (2010), pp. 925--939.
J. Cousty, L. Najman, and J. Serra, Some morphological operators in graph spaces, in Proc. International Symposium on Mathematical Morphology, 2009, pp. 149--160.
X. Desquesnes, A. Elmoataz, and O. Lézoray, Eikonal equation adaptation on weighted graphs: Fast geometric diffusion process for local and non-local image and data processing, J. Math. Imaging Vision, 46 (2013), pp. 238--257.
K. Drakopoulos and P. Maragos, Active contours on graphs: Multiscale morphology and graphcuts, IEEE J. Sel. Topics Signal Process., 6 (2012), pp. 780--794.
A. Elmoataz, F. Lozes, and M. Toutain, Nonlocal PDEs on graphs: From tug-of-war games to unified interpolation on images and point clouds, J. Math. Imaging Vision, 57 (2017), pp. 381--401.
A. Elmoataz, O. Lézoray, and S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing, IEEE Trans. Image Process., 17 (2008), pp. 1047--1060.
A. Elmoataz, M. Toutain, and D. Tenbrinck, On the \(p\)-Laplacian and \(\infty\)-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), pp. 2412--2451, https://doi.org/10.1137/15M1022793.
N. García Trillos and D. Slepcev, Continuum limit of total variation on point clouds, Arch. Ration. Mech. Anal., 220 (2016), pp. 193--241.
L. Grady, Random walks for image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2006), pp. 1768--1783.
L. Grady and C. Alvino, The piecewise smooth Mumford-Shah functional on an arbitrary graph, IEEE Trans. Image Process., 18 (2009), pp. 2547--2561.
L. Grady and J. Polimeni, Discrete Calculus: Applied Analysis on Graphs for Computational Science, Springer Science & Business Media, 2010.
H. J. A. M. Heijmans, P. Nacken, A. Toet, and L. Vincent, Graph morphology, J. Visual Commun. Image Represent., 3 (1992), pp. 24--38.
A. N. Hirani, Discrete Exterior Calculus, Ph.D. thesis, Caltech, 2003.
J. Jaromczyk and G. Toussaint, Relative neighborhood graphs and their relatives, Proc. IEEE, 80 (1992), pp. 1502--1517.
M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vis., 1 (1988), pp. 321--331.
N. Kolotouros and P. Maragos, A finite element computational framework for active contours on graphs, submitted.
D. D. Lee and H. S. Seung, Algorithms for non-negative matrix factorization, in Advances in Neural Information Processing Systems (NIPS), 2001, pp. 556--562.
F. Lozes, A. Elmoataz, and O. Lézoray, Partial difference operators on weighted graphs for image processing on surfaces and point clouds, IEEE Trans. Image Process., 23 (2014), pp. 3896--3909.
O. Lézoray, A. Elmoataz, and V.-T. Ta, Nonlocal PdEs on graphs for active contours models with applications to image segmentation and data clustering, in Proc. 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012, pp. 873--876.
O. Lézoray and L. Grady, Image Processing and Analysis with Graphs: Theory and Practice, Digital Imaging and Computer Vision, CRC Press, 2012.
M. Maier, U. von Luxburg, and M. Hein, How the result of graph clustering methods depends on the construction of the graph, ESAIM Probab. Statist., 17 (2013), pp. 370--418.
D. Martin, C. Fowlkes, D. Tal, and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in Proc. International Conference on Computer Vision, vol. 2, 2001, pp. 416--423.
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577--685.
S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12--49.
M. Penrose, Random Geometric Graphs, Oxford University Press, 2003.
D. Pollard, Strong consistency of (k)-means clustering, Ann. Statist., 9 (1981), pp. 135--140.
J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), pp. 888--905.
T. Snijders and K. Nowicki, Estimation and prediction for stochastic blockmodels for graphs with latent block structure, J. Classification, 14 (1997), pp. 75--100.
V.-T. Ta, A. Elmoataz, and O. Lézoray, Nonlocal PDEs-based morphology on weighted graphs for image and data processing, IEEE Trans. Image Process., 20 (2011), pp. 1504--1516.
D. Ting, L. Huang, and M. I. Jordan, An analysis of the convergence of graph Laplacians, in Proc. International Conference on Machine Learning, 2010, pp. 1079--1086.
L. Vincent, Graphs and mathematical morphology, Signal Process., 16 (1989), pp. 365--388.
U. von Luxburg, M. Belkin, and O. Bousquet, Consistency of spectral clustering, Ann. Statist., 36 (2008), pp. 555--586.
Z. Yang, T. Hao, O. Dikmen, X. Chen, and E. Oja, Clustering by nonnegative matrix factorization using graph random walk, in Advances in Neural Information Processing Systems (NIPS), 2012, pp. 1079--1087.
Z. Yuan and E. Oja, Projective nonnegative matrix factorization for image compression and feature extraction, in Proc. Scandinavian Conference on Image Analysis, 2005, pp. 333--342.

Information & Authors


Published In

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


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


  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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