Abstract
This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and vice-versa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtained on noisy synthetic data and real images will illustrate the capabilities of this original and promising approach.
Similar content being viewed by others
References
T. Aach and A. Kaup, “Bayesian algorithms for adaptive change detection in image sequences using markov random fields,” Signal Processing: Image Communication, Vol. 7, pp. 147–160, 1995.
R. Acart and C.R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Problems, Vol. 10, pp. 1217–1229, 1994.
L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal of Numerical Analysis, Vol. 29, pp. 845–866, 1992.
L. Alvarez and L. Mazorra, “Signal and image restoration using shock filters and anisotropic diffusion,” SIAM Journal of Numerical Analysis, Vol. 31, No. 2, pp. 590–605, 1994.
L. Ambrosio, “A compactness theorem for a new class of functions of bounded variation,” Boll. Unione Mat. Ital.,Vol. 7, No. 4, pp. 857–881, 1989.
L. Ambrosio and G. Dal Maso, “A general chain rule for distributional derivatives,” Proc. Am. Math. Soc., Vol. 108, No. 3, pp. 691–702, 1990.
G. Anzellotti, “The euler equation for functionals with linear growth,” Trans. Am. Math. Soc., Vol. 290, No. 2, pp. 483–501, 1985.
G. Aubert, M. Barlaud, L. Blanc-Feraud, and P. Charbonnier, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Imag. Process., Vol. 5, No. 12, 1997.
G. Aubert, R. Deriche, and P. Kornprobst, “Amathematical study of the regularized optical flow problem in the space BV (Ω),” Technical Report 503, Université de Nice-Sophia Antipolis, Dec. 1997.
G. Aubert, R. Deriche, and P. Kornprobst, “A variational method and its mathematical study in image sequence analysis,” Technical Report 3415, INRIA, April 1998.
G. Aubert and L. Vese, “Avariational method in image recovery,” SIAM J. Numer. Anal., Vol. 34, No. 5, pp. 1948–1979, 1997.
M.J. Black, G. Sapiro, D.H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Imag. Proc., Vol. 7, No. 3, pp. 421–432, 1998. Special Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Processing and Analysis.
A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, 1987.
P. Blomgren and T.F. Chan, “Color Tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Imag. Proc., Vol. 7, No. 3, pp. 304–309, 1998. Special Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Processing and Analysis.
G. Bouchitté, I. Fonseca, and L. Mascarenhas, “A global method for relaxation,” Technical report, Université de Toulon et du Var, May 1997.
J. Boyce, “Noise reduction of image sequences using adaptative motion compensated frame averaging,” in IEEE ICASSP, 1992, Vol. 3, pp. 461–464.
J.C. Brailean and A.K. Katsaggelos, “Simultaneous recursive displacement estimation and restoration of noisy-blurred image sequences,” IEEE Transactions on Image Processing, Vol. 4, No. 9, pp. 1236–1251, 1995.
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Comp: Amsterdam-London, 1973.
O. Buisson, B. Besserer, S. Boukir, and F. Helt, “Deterioration detection for digital film restoration,” in Computer Vision and Pattern Recognition, Puerto Rico, June 1997, pp. 78–84.
V. Caselles, J.M. Morel, G. Sapiro, and A. Tannenbaum, “Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis,” IEEE Transactions on Image Processing, Vol. 7, No. 3, pp. 269–273, 1998.
A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math., Vol. 76, No. 2, pp. 167–188, 1997.
I. Cohen, “Nonlinear variational method for optical flow computation,” in Proceedings 8th SCIA, 1993, Vol. 1, pp. 523–530.
L.D. Cohen, Auxiliary variables and two-step iterative algorithms in computer vision problems, ICCV, 1995.
G.-H. Cottet and L. Germain, “Image processing through reaction combined with nonlinear diffusion,” Mathematics of Computation, Vol. 61, No. 204, pp. 659–673, 1993.
F. Demengel and R. Temam, “Convex functions of a measure and applications,” Indiana University Mathematics Journal, Vol. 33, pp. 673–709, 1984.
R. Deriche and O. Faugeras, “Les EDP en traitement des images et vision par ordinateur,” Technical report, INRIA, Nov. 1995. A more complete version of this Research Report has appeared in the French Revue, Traitement du Signal, Vol. 13, No. 6, 1996 (special issue).
E. Dubois and S. Sabri, “Noise reduction in image sequences using motion-compensated temporal filtering,” IEEE Transactions on Communications, Vol. 32, No. 7, pp. 826–831, 1984.
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC, 1992.
H. Federer, Geometric Measure Theory, “Classics in Mathematics,” Springer-Verlag, Berlin, Heidelberg, New York, 1969.
D. Geman and G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 14, No. 3, pp. 367–383, 1993.
S. Geman, D.E. McClure, and D. Geman, “A nonlinear filter for film restoration and other problems in image processing,” CVGIP: Graphical Models and Image Processing, Vol. 54, No. 4, pp. 281–289, 1992.
E. De Giorgi and T. Franzoni, “Su un tipo di convergenza variazionale,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., Vol. 68, pp. 842–850, 1975.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984.
C. Goffman and J. Serrin, “Sublinear functions of measures and variational integrals,” Duke Math J., Vol. 31, pp. 159–178, 1964.
F. Guichard, “Axiomatisation des analyses multi-échelles d'images et de films,” Ph.D. thesis, Université Paris IX Dauphine, 1994.
K. Karmann, A. Brandt, and R. Gerl, “Moving object segmentation based on adaptive reference images,” Signal Processing: Theories and Applications, Vol. 5, pp. 951–954, 1990.
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.
A. Kokaram, “Reconstruction of severely degraded image sequences,” in International Conference on Image Applications and Processing, Florence, Italy, 1997.
A.C. Kokaram and S.J. Godsill, “A system for reconstruction of missing data in image sequences using sampled 3D SAR models and mrf motion priors,” in Proceedings of the 4th European Conference on Computer Vision, Bernard Buxton (Ed.), Cambridge, UK, April 1996, pp. 613–624.
P. Kornprobst, “Contributions à la restauration d'images et à l'analyse de séquences: Approches Variationnelles et Equations aux Dérivées Partielles,” Ph.D. thesis, Université de Nice-Sophia Antipolis, 1998.
P. Kornprobst, R. Deriche, and G. Aubert, “Image restoration via PDE',” in First Annual Symposium on Enabling Technologies for Law Enforcement and Security-SPIE Conference 2942: Investigative Image Processing, Boston, Massachusetts, USA., Nov. 1996.
P. Kornprobst, R. Deriche, and G. Aubert, “Image coupling, restoration and enhancement via PDE',” in International Conference on Image Processing, vol. II of III, Santa-Barbara, California, Oct. 1997, pp. 458–461.
P. Kornprobst, R. Deriche, and G. Aubert, “Nonlinear operators in image restoration,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition, Puerto-Rico, IEEE, June 1997, pp. 325–331.
P. Kornprobst, R. Deriche, and G. Aubert, “Edp, débruitage et réhaussement en traitement d'image: Analyse et contributions,” in 11 ème Congres RFIA, AFCET, Jan. 1998.
S. Liou and R. Jain, “Motion detection in spatio-temporal space,” Computer Vision, Graphics and Image Understanding, Vol. 45, pp. 227–250, 1989.
R. Malladi and J.A. Sethian, “Image processing: Flows under min/max curvature and mean curvature,” Graphical Models and Image Processing, Vol. 58, No. 2, pp. 127–141, March 1996.
G. Dal Maso, An introduction to Г-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhauser, 1993.
N.G. Meyers, “An l p-estimate for the gradient of solutions of second order elliptic divergence equations,” Ann. Scuola Norm. Sup. Pisa, Vol. 3, No. 17, pp. 189–206, 1963.
N.G. Meyers and A. Elcrat, “Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions,” Duke math. J., Vol. 42, pp. 121–136, 1975.
L. Moisan, “Traitement numérique d'images et de films: équations aux dérivées partielles préservant forme et relief,” Ph.D. thesis, Université Paris IX Dauphine, June 1997.
J.-M. Morel and S. Solimini, “Segmentation of images by variational methods: A constructive approach,” Rev. Math. Univ. Complut. Madrid, Vol. 1, pp. 169–182, 1988.
R.D. Morris, “Image Sequence Restoration using Gibbs Distributions,” Ph.D. thesis, Cambridge University, England, 1995.
D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Comm. Pure Appl. Math., Vol. 42, pp. 577–684, 1989.
N. Nordström, “sed anisotropic diffusion-A unified regularization and diffusion approach to edge detection” Image and Vision Computing, Vol. 8, No. 11, pp. 318–327, 1990.
N. Paragios and R. Deriche, “A PDE-based level set approach for detection and tracking of moving objects,” in Proceedings of the 6th International Conference on Computer Vision, Bombay, India, Jan. 1998. IEEE Computer Society Press.
N. Paragios and G. Tziritas, “Detection and localization of moving objects in image sequences,” FORT-Hellas Technical Report, Accepted for publication in Signal Processing: Image Communication, Oct. 1996.
N. Paragios and R. Deriche, “Detecting multiple moving targets using deformable contours,” in International Conference on Image Processing, Vol. II of III, Santa-Barbara, California, Oct. 1997, pp. 183–186.
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 12, No. 7, pp. 629–639, 1990.
M. Proesmans, E. Pauwels, and L. Van Gool, “Coupled geometry-driven diffusion equations for low-level vision,” Computational Imaging and Vision, Kluwer Academic Publishers, 1994, pp. 191–228.
L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” in International Conference on Image Processing, Nov. 1994, Vol. 1, pp. 31–35.
G. Sapiro and V. Caselles, “Contrast enhancement via image evolution flows,” Graphical Models and Image Processing, Vol. 59, No. 6, pp. 407–416, 1997.
G. Sapiro and D.L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Transactions on Image Processing, Vol. 5, No. 11, pp. 1582–1585, 1996.
G. Sapiro, A. Tannenbaum, Y.L. You, and M. Kaveh, “Experiments on geometric image enhancement,” in International Conference on Image Processing, 1994.
C. Schnörr, “Unique reconstruction of piecewise-smooth images by minimizing strictly convex nonquadratic functionals,” Journal of Mathematical Imaging and Vision, Vol. 4, pp. 189–198, 1994.
J. Shah, “A common framework for curve evolution, segmentation and anisotropic diffusion,” IEEE, 1996.
R.L. Stevenson, B.E. Schmitz, and E.J. Delp, “Discontinuity preserving regularization of inverse visual problems,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 24, No. 3, pp. 455–469, 1994.
D.M. Strong and T.F. Chan, “Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing,” Technical Report 46, UCLA, Nov. 1996.
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-posed Problems, Winston and Sons, Washington, DC, 1977.
L. Vese, “Problèmes variationnels et EDP pour l'analyse d'images et l'évolution de courbes,” Ph.D. thesis, Université de Nice Sophia-Antipolis, Nov. 1996.
A.I. Vol'pert, “The spaces BV and quasilinear equations,” Math. USSR-Sbornik, Vol. 2, No. 2, pp. 225–267, 1967.
J. Weickert, “Anisotropic diffusion in image processing,” Ph.D. thesis, University of Kaiserslautern, Germany, Laboratory of Technomathematics, Jan. 1996.
J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, 1998.
O. Wenstop, “Motion detection from image information,” Proceedings in Scandianvian Conference on Image Analysis, 1983, pp. 381–386.
Y.L. You, M. Kaveh, W.Y. Xu, and A. Tannenbaum, “Analysis and design of anisotropic diffusion for image processing,” in International Conference on Image Processing, 1994, Vol. 2, pp. 497–501.
W.P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, 1989.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kornprobst, P., Deriche, R. & Aubert, G. Image Sequence Analysis via Partial Differential Equations. Journal of Mathematical Imaging and Vision 11, 5–26 (1999). https://doi.org/10.1023/A:1008318126505
Issue Date:
DOI: https://doi.org/10.1023/A:1008318126505