Image Deblurring in the Presence of Impulsive Noise

Consider the problem of image deblurring in the presence of impulsive noise. Standard image deconvolution methods rely on the Gaussian noise model and do not perform well with impulsive noise. The main challenge is to deblur the image, recover its discontinuities and at the same time remove the impulse noise. Median-based approaches are inadequate, because at high noise levels they induce nonlinear distortion that hampers the deblurring process. Distinguishing outliers from edge elements is difficult in current gradient-based edge-preserving restoration methods. The suggested approach integrates and extends the robust statistics, line process (half quadratic) and anisotropic diffusion points of view. We present a unified variational approach to image deblurring and impulse noise removal. The objective functional consists of a fidelity term and a regularizer. Data fidelity is quantified using the robust modified L1 norm, and elements from the Mumford-Shah functional are used for regularization. We show that the Mumford-Shah regularizer can be viewed as an extended line process. It reflects spatial organization properties of the image edges, that do not appear in the common line process or anisotropic diffusion. This allows to distinguish outliers from edges and leads to superior experimental results.

[1]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[2]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[3]  Joachim Weickert,et al.  Coherence-Enhancing Diffusion Filtering , 1999, International Journal of Computer Vision.

[4]  Curtis R. Vogel,et al.  Ieee Transactions on Image Processing Fast, Robust Total Variation{based Reconstruction of Noisy, Blurred Images , 2022 .

[5]  F. Santosa,et al.  ENHANCED ELECTRICAL IMPEDANCE TOMOGRAPHY VIA THE MUMFORD{SHAH FUNCTIONAL , 2001 .

[6]  Jayant Shah,et al.  A common framework for curve evolution, segmentation and anisotropic diffusion , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[7]  Thomas Brox,et al.  High Accuracy Optical Flow Estimation Based on a Theory for Warping , 2004, ECCV.

[8]  Mila Nikolova,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004, Journal of Mathematical Imaging and Vision.

[9]  Jacques Froment,et al.  Reconstruction of Wavelet Coefficients Using Total Variation Minimization , 2002, SIAM J. Sci. Comput..

[10]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[11]  Jayant Shah,et al.  Free-discontinuity problems via functionals involving the L1-norm of the gradient and their approximations , 1999 .

[12]  Stuart Geman,et al.  Statistical methods for tomographic image reconstruction , 1987 .

[13]  Tony F. Chan,et al.  Total variation blind deconvolution , 1998, IEEE Trans. Image Process..

[14]  Richard A. Haddad,et al.  Adaptive median filters: new algorithms and results , 1995, IEEE Trans. Image Process..

[15]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

[16]  Dana H. Brooks,et al.  Electrical imaging of the heart , 1997, IEEE Signal Process. Mag..

[17]  Donald Geman,et al.  Constrained Restoration and the Recovery of Discontinuities , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Antonin Chambolle,et al.  A l1-Unified Variational Framework for Image Restoration , 2004, ECCV.

[19]  Raymond H. Chan,et al.  Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization , 2005, IEEE Transactions on Image Processing.

[20]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[22]  L. Ambrosio,et al.  Approximation of functional depending on jumps by elliptic functional via t-convergence , 1990 .

[23]  Ron Kimmel,et al.  Variational Restoration and Edge Detection for Color Images , 2003, Journal of Mathematical Imaging and Vision.

[24]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Jyh-Charn Liu,et al.  Selective removal of impulse noise based on homogeneity level information , 2003, IEEE Trans. Image Process..

[26]  Nahum Kiryati,et al.  Image Deblurring in the Presence of Salt-and-Pepper Noise , 2005, Scale-Space.

[27]  Michel Barlaud,et al.  Variational approach for edge-preserving regularization using coupled PDEs , 1998, IEEE Trans. Image Process..

[28]  H. Wu,et al.  Space variant median filters for the restoration of impulse noise corrupted images , 2001 .

[29]  M. Rosati,et al.  Asymptotic Behavior of a Geman and McClure Discrete Model , 2000 .

[30]  Guillermo Sapiro,et al.  Robust anisotropic diffusion , 1998, IEEE Trans. Image Process..

[31]  Alexander Brook.  Variational Segmentation for Color images , 2002 .

[32]  Gilles Aubert,et al.  Gamma-Convergence of Discrete Functionals with Nonconvex Perturbation for Image Classification , 2004, SIAM J. Numer. Anal..

[33]  François Malgouyres,et al.  Minimizing the total variation under a general convex constraint for image restoration , 2002, IEEE Trans. Image Process..

[34]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[35]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[36]  Mila Nikolova,et al.  Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms. Application to the Processing of Outliers , 2002, SIAM J. Numer. Anal..

[37]  Michael J. Black,et al.  On the unification of line processes , 1996 .

[38]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[39]  Michael J. Black,et al.  On the unification of line processes, outlier rejection, and robust statistics with applications in early vision , 1996, International Journal of Computer Vision.

[40]  P. Wilson.  Restoration of wavelet coefficients by minimizing a specially designed objective function , 2003 .

[41]  Niklas Nordström.  Biased Anisotropic Diffusion - A Unified Regularization and Diffusion Approach to Edge Detection , 1990, ECCV.

[42]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

[43]  Nahum Kiryati,et al.  Variational Pairing of Image Segmentation and Blind Restoration , 2004, ECCV.

[44]  Andrea Braides.  Approximation of Free-Discontinuity Problems , 1998 .

[45]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[46]  Michel Chipot,et al.  Analysis of a Nonconvex Problem Related to Signal Selective Smoothing , 1997 .

[47]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[48]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .