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Region-Based Active Contours with Exponential Family Observations

  • François Lecellier
  • Jalal Fadili
  • Stéphanie Jehan-Besson
  • Gilles Aubert
  • Marinette Revenu
  • Eric Saloux
Article

Abstract

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. In the framework developed in this paper, we consider the general case of region-based terms involving functions of parametric probability densities, for which the anti-log-likelihood function is a special case. Using shape derivative tools, our effort focuses on constructing a general expression for the derivative of the energy (with respect to a domain), and on deriving the corresponding evolution speed. More precisely, we first show by an example that the estimator of the distribution parameters is crucial for the derived speed expression. On the one hand, when using the maximum likelihood (ML) estimator for these parameters, the evolution speed has a closed-form expression that depends simply on the probability density function. On the other hand, complicating additive terms appear when using other estimators, e.g. method of moments. We then proceed by stating a general result within the framework of multi-parameter exponential family. This result is specialized to the case of the anti-log-likelihood function with the ML estimator and to the case of the relative entropy. Experimental results on simulated data confirm our expectations that using the appropriate noise model leads to the best segmentation performance. We also report preliminary experiments on real life Synthetic Aperture Radar (SAR) images to demonstrate the potential applicability of our approach.

Keywords

Segmentation Region-based active contours Exponential families Shape derivation Maximum likelihood Relative entropy 

Notations

ΩI

The image domain

Ω

A region of the image

Ω

The boundary of the region Ω

|Ω|

Region size Ω d x

I(x)

The intensity of the pixel at the location x

k(x,Ω)

The region descriptor of Ω

kb(x)

The boundary descriptor of Ω

Ωin

Inside region

Ωout

Outside region

J′(Ω),V

The Eulerian derivative of domain energy criterion J(Ω)

p(y,η)

Probability density function (pdf) of the random variable Y

η

Hyper-parameter vector of the pdf

\(\mathbb{E}{[Y]}\)

Expectation of the random variable Y

k′(x,Ω,V)

The domain derivative of the function k

speed(x,Ω)

Evolution speed of the active contour

References

  1. 1.
    Achim, A., Kuruoglu, E., Zerubia, J.: SAR image filtering based on the heavy-tailed Rayleigh model. IEEE Trans. Image Process. 15(9), 2686–2693 (2006) CrossRefGoogle Scholar
  2. 2.
    Aubert, G., Barlaud, M., Faugeras, O., Jehan-Besson, S.: Image segmentation using active contours: calculus of variations or shape gradients? SIAM J. Appl. Math. 63(6), 2128–2154 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aujol, J.F., Aubert, G., Blanc-Féraud, L.: Wavelet-based level set evolution for classification of textured images. IEEE Trans. Image Process. 12(12), 1634–1641 (2003) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Banerjee, A., Dhillon, I., Ghosh, J., Merugu, S.: An information theoretic analysis of maximum likelihood mixture estimation for exponential families. In: ICML, pp. 57–64 (2004) Google Scholar
  5. 5.
    Barndorff-Nielsen, O.: Information and Exponential Families in Statistical Theory. Wiley, New York (1978) zbMATHGoogle Scholar
  6. 6.
    Bickel, P., Docksum, K.: Mathematical Statistics: Basic Ideas and Selected Topics, vol. I, 2nd edn. Prentice-Hall, London (2001). ISBN 013850363-X Google Scholar
  7. 7.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997) zbMATHCrossRefGoogle Scholar
  8. 8.
    Chakraborty, A., Staib, L., Duncan, J.: Deformable boundary finding in medical images by integrating gradient and region information. IEEE Trans. Med. Imaging 15, 859–870 (1996) CrossRefGoogle Scholar
  9. 9.
    Cheng, L., Yang, J., Fan, X.: A new region-based active contour for object extraction using level set method. Pattern Recognit. Image Anal. 285–291 (2005) Google Scholar
  10. 10.
    Chesnaud, C., Réfrégier, P., Boulet, V.: Statistical region snake-based segmentation adapted to different physical noise models. IEEE Trans. Pattern Anal. Mach. Intell. 21(11), 1145–1157 (1999) CrossRefGoogle Scholar
  11. 11.
    Cohen, L., Bardinet, E., Ayache, N.: Surface reconstruction using active contour models. In: SPIE Conference on Geometric Methods in Computer Vision, San Diego (1993) Google Scholar
  12. 12.
    Cohen, L.D.: On active contour models and balloons. Comput. Vis. Graph. Image Process. Image Underst. 53(2), 211–218 (1991) zbMATHGoogle Scholar
  13. 13.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991) zbMATHCrossRefGoogle Scholar
  14. 14.
    Cremers, D., Rousson, M., Deriche, R.: A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. Int. J. Comput. Vis. 72(2), 195–215 (2007) CrossRefGoogle Scholar
  15. 15.
    Cremers, D., Tischhäuser, F., Weickert, J., Schnörr, C.: Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional. Int. J. Comput. Vis. 50, 295–313 (2002) zbMATHCrossRefGoogle Scholar
  16. 16.
    Delfour, M.C., Zolésio, J.P.: Shape and geometries. In: Advances in Design and Control. SIAM, Philadelphia (2001) Google Scholar
  17. 17.
    Dutt, V.: Statistical analysis of ultrasound echo envelope. PhD thesis, Mayo Clinic College of Medicine (1995) Google Scholar
  18. 18.
    Dydenko, I., Friboulet, D., Magnin, I.: A variational framework for affine registration and segmentation with shape prior: application in echocardiographic imaging. In: IEEE Workshop on V.G.L.S.M. in Computer Vision, vol. 1, pp. 201–208 (2003) Google Scholar
  19. 19.
    Foulonneau, A., Charbonnier, P., Heitz, F.: Geometric shape priors for region-based active contours. In: ICIP (2003) Google Scholar
  20. 20.
    Galland, F., Bertaux, N., Réfrégier, P.: Multi-component image segmentation in homogeneous regions based on description length minimization: application to speckle. Poisson and Bernoulli noise. Pattern Recognit. 38, 1926–1936 (2005) CrossRefGoogle Scholar
  21. 21.
    Gastaud, M., Barlaud, M., Aubert, G.: Tracking video objects using active contours and geometric priors. In: IEEE 4th E.W.I.A.M.I.S., pp. 170–175 (2003) Google Scholar
  22. 22.
    Goodman, J.: Some fundamental properties of speckle. J. Opt. Soc. Am. 66, 1145–1150 (1976) CrossRefGoogle Scholar
  23. 23.
    Gudbjartsson, H., Patz, S.: The Rician distribution of noisy MRI data. Magn. Reson. Med. 34, 910–914 (1995) CrossRefGoogle Scholar
  24. 24.
    Hintermuller, M., Ring, W.: A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64(2), 442–467 (2004) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Jehan-Besson, S., Barlaud, M., Aubert, G.: Video object segmentation using Eulerian region-based active contours. In: International Conference on Computer Vision. Vancouver, Canada (2001) Google Scholar
  26. 26.
    Jehan-Besson, S., Barlaud, M., Aubert, G.: DREAM2S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation. Int. J. Comput. Vis. 53, 45–70 (2003) CrossRefGoogle Scholar
  27. 27.
    Karoui, I., Fablet, R., Boucher, J.M., Augustin, J.M.: Region-based image segmentation using texture statistics and level-set methods. In: ICASSP, vol. 2, pp. 693–696 (2006) Google Scholar
  28. 28.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1, 321–332 (1988) CrossRefGoogle Scholar
  29. 29.
    Koopman, P.: On distributions admitting a sufficient statistic. Trans. Am. Math. Soc. 39, 399–409 (1936) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Lau, P.Y., Ozawa, S.: A region-based approach combining marker-controlled active contour model and morphological operator for image segmentation. In: IEEE EMBS, pp. 1651–1655 (2004) Google Scholar
  31. 31.
    Lecellier, F., Fadili, J., Jehan-Besson, S., Aubert, G., Revenu, M.: Region-based active contours and sparse representations for texture segmentation. In: ICPR (2008) Google Scholar
  32. 32.
    Lecellier, F., Jehan-Besson, S., Fadili, J., Aubert, G., Revenu, M.: Statistical region-based active contours with exponential family observations. In: ICASSP, vol. 2, pp. 113–116 (2006) Google Scholar
  33. 33.
    Lecellier, F., Jehan-Besson, S., Fadili, J., Aubert, G., Revenu, M., Saloux, E.: Region-based active contours with noise and shape priors. In: ICIP, vol. 1, pp. 1649–1652 (2006) Google Scholar
  34. 34.
    Leventon, M.: Statistical models for medical image analysis. PhD thesis, MIT (2000) Google Scholar
  35. 35.
    Martin, P., Réfrégier, P., Goudail, F., Guérault, F.: Influence of the noise model on level set active contour segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 26(6), 799–803 (2004) CrossRefGoogle Scholar
  36. 36.
    Osher, S.J., Sethian, J.A.: Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Paragios, N., Deriche, R.: Coupled geodesic active regions for image segmentation: a level set approach. In: European Conference in Computer Vision, Dublin, Ireland (2000) Google Scholar
  38. 38.
    Paragios, N., Deriche, R.: Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. J. Vis. Commun. Image Represent. 13, 249–268 (2002) CrossRefGoogle Scholar
  39. 39.
    Paragios, N., Deriche, R.: Geodesic active regions and level set methods for supervised texture segmentation. Int. J. Comput. Vis. 46(3), 223 (2002) zbMATHCrossRefGoogle Scholar
  40. 40.
    Precioso, F., Barlaud, M., Blu, T., Unser, M.: Robust real-time segmentation of images and videos using a smooth-spline snake-based algorithm. IEEE Trans. Image Process. 14, 910–924 (2005) CrossRefGoogle Scholar
  41. 41.
    Ronfard, R.: Region-based strategies for active contour models. Int. J. Comput. Vis. 13(2), 229–251 (1994) CrossRefGoogle Scholar
  42. 42.
    Rousson, M., Lenglet, C., Deriche, R.: Level set and region based surface propagation for diffusion tensor MRI segmentation. In: Computer Vision Approaches to Medical Image Analysis (CVAMIA) and Mathematical Methods in Biomedical Image Analysis (MMBIA) Workshop (2004) Google Scholar
  43. 43.
    Slabaugh, G., Unal, G., Fang, T., Wels, M.: Ultrasound-specific segmentation via decorrelation and statistical region-based active contours. In: International Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 45–53 (2006) Google Scholar
  44. 44.
    Sokolowski, J., Zolésio, J.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992) zbMATHGoogle Scholar
  45. 45.
    Tsai, A., Yezzi, A., Wells, W.: A shape-based approach to the segmentation of medical imagery using level sets. IEEE Trans. Med. Imaging 22, 137–154 (2003) CrossRefGoogle Scholar
  46. 46.
    Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(1), 271–293 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Zhu, S., Lee, T., Yuille, A.: Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. In: International Conference on Computer Vision, pp. 416–423 (1995) Google Scholar
  48. 48.
    Zhu, S., Yuille, A.: Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996) CrossRefGoogle Scholar
  49. 49.
    Zhu, W., Jiang, T., Li, X.: Local region based medical image segmentation using j-divergence measures. In: IEEE EMBS, pp. 7174–7177 (2005) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • François Lecellier
    • 1
  • Jalal Fadili
    • 1
  • Stéphanie Jehan-Besson
    • 2
  • Gilles Aubert
    • 3
  • Marinette Revenu
    • 1
  • Eric Saloux
    • 4
  1. 1.GREYC UMR CNRS 6072Caen CedexFrance
  2. 2.LIMOS UMR CNRS 6158Université Blaise PascalAubiereFrance
  3. 3.Laboratoire J.A. Dieudonné UMR CNRS 6621Nice CedexFrance
  4. 4.CHU de CaenCaenFrance

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