This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
A Variational Model for Image Classification and Restoration
May 2000 (vol. 22 no. 5)
pp. 460-472

Abstract—Herein, we present a variational model devoted to image classification coupled with an edge-preserving regularization process. The discrete nature of classification (i.e., to attribute a label to each pixel) has led to the development of many probabilistic image classification models, but rarely to variational ones. In the last decade, the variational approach has proven its efficiency in the field of edge-preserving restoration. In this paper, we add a classification capability which contributes to provide images composed of homogeneous regions with regularized boundaries, a region being defined as a set of pixels belonging to the same class. The soundness of our model is based on the works developed on the phase transition theory in mechanics. The proposed algorithm is fast, easy to implement, and efficient. We compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a Potts regularization.

[1] S. Allen and J. Cahn, “A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening,” Acta Metallurgica, vol. 27, pp. 1,085-1,095, 1979.
[2] L. Alvarez, P.L. Lions, and J.M. Morel, “Image Selective Smoothing and Edge Detection by Nonlinear Diffusion,” SIAM J. Numeric Analysis, vol. 29, pp. 845-866, 1992.
[3] L. Ambrosio and V. Tortorelli, “Approximation of Functionals Depending on Jumps by Elliptic Functionals via$\Gamma$-Convergence,” Comm. Pure Applied Math., vol. 43, pp. 999-1,036, 1990.
[4] L. Ambrosio and V. Tortorelli, “On the Approximation of Functionals Depending on Jumps by Quadratic, Elliptic Functionals,” Boll. Un. Mat. Ital., vol. 46-b, pp. 105-123, 1992.
[5] S. Angenent and M.E. Gurtin, “Multiphase Thermomechanics with Interfacial Structure 2. Evolution of an Isothermal Interface,” Arch. Rational Mech. Anal., vol. 108, pp. 333-391, 1989.
[6] G. Aubert and L. Vese, “A Variational Method in Image Recovery,” SIAM J. Numerical Analysis, vol. 34, no. 5, pp. 1,948-1,979, 1997.
[7] S. Baldo, “Minimal Interface Criterion for Phase Transitions in Mixtures of Cahn-Hilliard Fluids,” Annals Inst. Henri Poincaré, vol. 7, pp. 67-90, 1990.
[8] G. Barles, L. Bronsard, and P.E. Souganidis, “Front Propagation for Reaction-Diffusion Equations of Bistable Type,” Annals Inst. Henri Poincaré, vol. 9, pp. 479-496, 1992.
[9] G. Bellettini, M. Paolini, and C. Verdi, “Numerical Minimization of Geometrical Type Problems Related to Calculus of Variations,” Calcolo, vol. 27, pp. 251-278, 1991.
[10] M. Berthod, Z. Kato, S. Yu, and J. Zerubia, “Bayesian Image Classification Using Markov Random Fields,” Image and Vision Computing, vol. 14, no. 4, pp. 285-293, 1996.
[11] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.
[12] C.A. Bouman and M. Shapiro, “A Multiscale Random Field Model for Bayesian Image Segmentation,” IEEE Trans. Image Processing, vol. 3, pp. 162-176, 1994.
[13] L. Bronsard and F. Reitich, “On Three-Phase Boundary Motion and the Singular Limit of a Vector-Valued Ginzburg-Landau Equation,” Arch. Rational Mech. Anal., vol. 124, pp. 355-379, 1993.
[14] G. Caginalp, “Stefan and Hele-Shaw Type as Asymptotic Limits of the Phase-Field Equations,” Physical Review, vol. 39, no. 11, pp. 5,887-5,896, 1989.
[15] J.W. Cahn and J.E. Hilliard, “Free Energy of a Nonuniform System. I. Interfacial Free Energy,” J. Chemical Physics, vol. 28, no. 1, pp. 258-267, 1958.
[16] V. Caselles, F. Catte, T. Coll, and F. Dibos, “A Geometric Model for Active Contours,” Numerische Mathematik, vol. 66, pp. 1-31, 1993.
[17] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic Active Contours,” Int'l J. Computer Vision, vol. 22, no. 1, pp. 61-79, 1997.
[18] P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic Edge-Preserving Regularization in Computed Imaging,” IEEE Trans. on Image Processing, vol. 6, no. 2, pp. 298-311, Feb. 1997.
[19] X. Descombes, R. Morris, and J. Zerubia, “Some Improvements to Bayesian Image Segmentation Part One: Modeling,” Traitement du Signal, vol. 14, no. 4, pp. 373-382, 1997 (in french).
[20] X. Descombes, R. Morris, and J. Zerubia, “Some Improvements to Bayesian Image Segmentation Part Two: Classification,” Traitement du Signal, vol. 14, no. 4, pp. 383-395, 1997, (in french).
[21] X. Descombes, R. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov Random Field Prior Parameters Using Markov Chain Monte Carlo Maximum Likelihood,” IEEE Trans. Image Processing, 1999.
[22] I. Fonseca and L. Tartar, “The Gradient Theory of Phase Transitions for Systems with Two Potential Wells,” Proc. Royal Soc. Edinburgh, vol. 111A, no. 11, pp. 89-102, 1989.
[23] D. Geman and G. Reynolds, "Constrained Restoration and the Recovery of Discontinuities," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 367-383, 1992.
[24] S. Geman and D.-E. McCLure, “Bayesian Image Analysis: An Application to Single Photon Emission Tomography,” Proc. Am. Statistics Assoc., Statistics Computer Section, pp. 11-18, 1985.
[25] E. De Giorgi, “Convergence Problems for Functionals or Operators,” Proc. Int'l Meeting Recent Methods in Nonlinear Analysis, 1978.
[26] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, 1984.
[27] P.J. Green, Bayesian Reconstruction from Emission Tomography Data Using a Modified Em Algorithm IEEE Trans. Medical Imaging, vol. 9, pp. 84-93, 1990.
[28] T. Hebert and R. Leahy, “A Generalized EM Algorithm for 3D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Medical Imaging, vol. 8, no. 2, pp. 194-202, 1989.
[29] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active Contour Models,” Int'l J. Computer Vision, vol. 1, pp. 321-331, 1987.
[30] Z. Kato, “Multiresolution Markovian Modeling for Computer Vision. Application to SPOT Image Segmentation,” PhD thesis, Universitéde Nice-Sophia Antipolis, France 1994, (in French and English).
[31] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. YezziJr., “Conformal Curvature Flows: From Phase Transitions to Active Vision,” Arch. Rational Mech. Anal., vol. 134, pp. 275-301, 1996.
[32] S. Lakshmanan and H. Derin, “Simultaneous Parameter Estimation and Segmentation of Gibbs Random Fields Using Simulated Annealing,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 799-813, 1989.
[33] R. Malladi,J. A. Sethian,, and B. C. Vemuri,“Evolutionary fronts for topology-independent shape modeling and recovery,” in Proc. of Third European Conf. on Computer Vision, LNCS vol. 800, pp. 3-13,Stockholm, Sweden, May 1994.
[34] B.S. Manjunath and R. Chellappa, “Unsupervised Texture Segmentation Using Markov Random Field Models,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 478-482, 1991.
[35] R. March, “Visual Reconstruction Using Variational Methods,” Image and Vision Computing, vol. 10, pp. 30-38, 1992.
[36] R. March and M. Dozio, “A Variational Method for the Recovery of Smooth Boundaries,” Image and Vision Computing, vol. 15, pp. 705-712, 1997.
[37] L. Modica, “The Gradient Theory of Phase Transitions and the Minimal Interface Criterion,” Arch. Rational Mech. Anal., vol. 98, pp. 123-142, 1987.
[38] J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation. Birkhäuser, 1995.
[39] D. Mumford and J. Shah, “Boundary Detection by Minimizing Functionals,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1985.
[40] C. Owen and P. Sternberg, “Nonconvex Variational Problems with Anisotropic Perturbations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 16, no. 7, pp. 705-719, 1991.
[41] T. Pavlidis and Y.-T. Liow, “Integrating Region Growing and Edge Detection,” Proc. IEEE Computer Vision and Pattern Recognition, 1988.
[42] F. Reitich and H.M. Soner, “Three-Phase Boundary Motions Under Constant Velocities Part One: The Vanishing Surface Tension Limit,” Proc. Royal Soc. Edinburgh, vol. 126A, pp. 837-865, 1996.
[43] J. Rubinstein, P. Sternberg, and J.B. Keller, “Fast Reaction, Slow Diffusion, and Curve Shortening,” SIAM J. Applied Math., vol. 49, pp. 116-133, 1989.
[44] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear Total Variation Based Removal Algorithm,” Physica D, vol. 60, pp. 259-268, 1992.
[45] C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “Image Classification Using a Variational Approach,” INRIA Research Report RR-3,523, Oct. 1998. (http://www.inria.fr/RRRTRR-3523.html).
[46] P. Sternberg, “Vector-Valued Local Minimizers of Nonconvex Variational Problems,” J. Math., vol. 21, pp. 799-807, 1991.
[47] P. Sternberg and W.P. Zeimer, “Local Minimisers of a Three-Phase Partition Problem with Triple Junctions,” Proc. Royal Soc. Edinburgh, vol. 124A, pp. 1,059-1,073, 1994.
[48] S. Teboul and L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Variational Approach for Edge-Preserving Regularization Using Coupled PDEs,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 387-397, 1998.
[49] P. Teo, G. Sapiro, and B.A. Wandell, “Creating Connected Representations of Cortical Gray Matter for Functional MRI Visualization,” IEEE Trans. Medical Imaging, vol. 17, 1998.
[50] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. Winston and Wiley, 1977.
[51] S.C. Zhu and A. Yuille, “Region Competition: Unifying Snakes, Region Growing and Bayes/MDL for Multiband Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, pp. 884-900, 1996.

Index Terms:
Variational model, classification, labeling, phase transition theory, edge-preserving regularization, minimization, satellite images.
Citation:
Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia, "A Variational Model for Image Classification and Restoration," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 5, pp. 460-472, May 2000, doi:10.1109/34.857003
Usage of this product signifies your acceptance of the Terms of Use.