Abstract
We consider signal and image restoration using convex cost-functions composed of a non-smooth data-fidelity term and a smooth regularization term. First, we provide a convergent method to minimize such cost-functions. Then we propose an efficient method to remove impulsive noise by minimizing cost-functions composed of an ℓ1 data-fidelity term and an edge-preserving regularization term. Their minimizers have the property to fit exactly uncorrupted (regular) data samples and to smooth aberrant data entries (outliers). This method furnishes a new approach to the processing of data corrupted with impulsive noise. A crucial advantage over alternative filtering methods is that such cost-functions can convey adequate priors about the sought signals and images—such as the presence of edges. The numerical experiments show that images and signals are efficiently restored from highly corrupted data.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and Minimization Algorithms, vol. I & II. Springer, Berlin (1996)
Besag, J.E.: Digital image processing: Towards Bayesian image analysis. Journal of Applied Statistics 16, 395–407 (1989)
Bouman, C., Sauer, K.: A generalized Gaussian image model for edge-preserving map estimation. IEEE Transactions on Image Processing 2, 296–310 (1993)
Li, S.: Markov Random Field Modeling in Computer Vision, 1st edn. Springer, New York (1995)
Black, M., Rangarajan, A.: On the unification of line processes, outlier rejection, and robust statistics with applications to early vision. International Journal of omputer Vision 19, 57–91 (1996)
Vogel, C.R., Oman, M.E.: Iterative method for total variation denoising. SIAM Journal of Scientific Computing 17, 227–238 (1996)
Teboul, S., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Variational approach for edge-preserving regularization using coupled pde’s. IEEE Transactions on Image Processing 7, 387–397 (1998)
Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edgepreserving regularization in computed imaging. IEEE Transactions on Image Processing 6, 298–311 (1997)
Alliney, S., Matej, S., Bajla, I.: On the possibility of direct Fourier reconstruction from divergent-beam projections. IEEE Transactions on Medical Imaging MI-12, 173–181 (1993)
Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of absolute norm. IEEE Transactions on Signal Processing 45, 913–917 (1997)
Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM Journal of Numerical Analysis 40, 965–994 (2001)
Nikolova, M.: Minimization of cost-functions with non-smooth data-fidelity terms. application to the processing of impulsive noise. Technical report (CMLA—ENS de Cachan, Report No. 2003-01)
Glowinski, R., Lions, J., Trémoliéres, R.: Analyse numérique des inéquations variationnelles, 1st edn., vol. 1, Dunod, Paris, (1976)
Geman, D.: Random fields and inverse problems in imaging. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427. Springer, Heidelberg (1990)
Abreu, E., Lightstone, M., Mitra, S.K., Arakawa, K.: A new efficient approach for the removal of impulse noise from highly corrupted images. IEEE Transactions on Image Processing 5, 1012–1025 (1996)
Bovik, A.C.: Handbook of image and video processing. Academic Press, New York (2000)
Rudin, L., Osher, S., Fatemi, C.: Nonlinear total variation based noise removal algorithm. Physica 60 D, 259–268 (1992)
Ko, S.J., Lee, Y.H.: Adaptive center weighted median filter. IEEE Transactions on Circuits and Systems 38, 984–993 (1998)
Sun, T., Neuvo, Y.: Detail-preserving based filters in image processing. Pattern- Recognition Letters 15, 341–347 (1994)
Arce, G.R., Hall, T.A., Barner, K.E.: Permutation weighted order statistic filters. IEEE Transactions on Image Processing 4, 1070–1083 (1995)
Yin, L., Yang, R., Gabbouj, M., Neuvo, Y.: Weighted median filters: a tutorial. IEEE Transactions on Circuit Theory 41, 157–192 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nikolova, M. (2003). Minimization of Cost-Functions with Non-smooth Data-Fidelity Terms to Clean Impulsive Noise. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2003. Lecture Notes in Computer Science, vol 2683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45063-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-45063-4_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40498-9
Online ISBN: 978-3-540-45063-4
eBook Packages: Springer Book Archive