Image Deblurring in the Presence of Impulsive Noise

Abstract

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 L 1 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.

Reconstruction of noisy signals by minimization of non-convex functionals

“…The high quality of the minimization results for some functionals of the class F(·) is asserted in the works of Nikolova [17] and Bar et al [15], which successfully deals with both PWC or PWS signals corrupted with Impulsive and/or White noise by using edge-preserving regularization methods with non-smooth fidelity terms such as L 1 , Student's t-distribution, and Mestimators.…”

“…The first part is aimed at proving (14), which means that the potential value does not increase. The second part is devoted to demonstrate (15), which means that the fitting term decreases.…”

“…A theoretical study on the existence of minimizers for a family of nonconvex functionals in the discrete setting is shown in [10], while [15] provides empirical evidence on the usefulness of a particular non-convex functional for image denoising and deblurring in the presence of impulsive noise. However, no proof on the existence of minimizers for nonconvex variational methods with both non-quadratic fidelity terms and edge-preserving potential terms in the continuous domain has been reported.…”

Coarse to over-fine optical flow estimation

“…Brox et al (2004); Bar et al (2006). The incorporation of the constant ε makes the approximation differentiable at s = 0; the value of ε sets the level of approximation.…”

Mumford-Shah Regularizer with Contextual Feedback

“…It is possible to interpret v 2 as an analog form of the line process introduced by Geman and Geman [41]. As shown by Bar, Kiryati and Sochen [7] and Teboul et al [78], the AT approximation of the MS functional defines an extended line process regularization where the regularizer has an additional constraint introduced by the term ρ|∇v| 2 . This term mildly forces some spatial organization by demanding the edges to be smooth.…”