A non-local regularization strategy for image deconvolution
Introduction
In the regularized restoration framework, the prior (or regularization) term allows us to both statistically incorporate knowledge concerning the types of restored images a priori defined as acceptable solutions and to stabilize (computationally speaking) the solution of this ill-conditioning inverse problem. That is why the design of efficient image prior models, and especially their ability to summarize the intrinsic properties of an original image to be recovered are crucial in the final image quality and signal-to-noise improvement (ISNR) restoration result.
Over the last two decades, there have been considerable efforts to find an efficient regularization term (or a prior distribution) capable of modeling all the intrinsic properties of a natural image, particularly its edge and textural information. To this end, several edge-preserving local regularization strategies there have been proposed in the spatial domain (Chantas et al., 2006, Foi et al., 2006, Mignotte, 2006, Neelamani et al., 2004, Banham and Katsaggelos, 1996) (e.g., via non-stationary, compound Markov or MRF model with robust potential functions) or in the frequential domain (Guerrero-Colon and Portilla, 2006, Bioucas-Dias et al., 2006, Foi et al., 2006, Figueiredo and Nowak, 2005, Figueiredo and Nowak, 2003, Bioucas-Dias, 2006, Neelamani et al., 2004, Banham and Katsaggelos, 1996) (e.g., via thresholding).
Buades et al. (2005) have recently proposed a natural and elegant extension of the image bilateral filtering paradigm. The basic idea behind the so-called Non-Local means (NL-means) denoising concept is simple. For a given pixel i, its new (denoised) intensity value is computed as a weighted average of grey level values within a search window. The weight of the pixel j in this weighted average is proportional to the similarity (according to the euclidean distance) between the neighborhood configurations of pixels i and j. In this procedure, the denoising process is due to the regularity assumption that self-similarities of neighborhoods exist in a real image1 and that one (or several) neighborhood configuration(s) can efficiently predict the central value of the pixel, as shown by Efros and Leung (1999) for texture synthesis with a (somewhat) similar non-parametric sampling strategy.
In this paper, the idea proposed by Buades et al. (2005) is herein used to derive an efficient image prior distribution. This prior expresses that acceptable restored images are likely the solutions exhibiting similar neighborhood configurations (i.e., images owing a high degree of redundancy or exhibiting numerous similar patterns1). Comparisons with classical deconvolution and restoration approaches using local regularization strategy (in the spatial or frequential domain) are given in order to illustrate the potential of this approach and its pros and cons for some degradation models.
Section snippets
Proposed approach
We herein use the classical penalized likelihood framework leading, in the context of image restoration, to the following cost function to be optimizedwhere y and x represent, respectively, the noisy (degraded by an additive and white Gaussian noise with variance ) and observed blurred image and the undistorted true image. h is the Point Spread Function (PSF) of the imaging system
Regularization parameter estimation
A crucial element in this penalized likelihood framework as expressed by Eq. (1), is the proper choice of the regularization parameter . If is selected as small, the recovered image is dominated by high-frequency noise components (the solution is the so-called under-regularized). If is too large, the effect of the prior will dominate the solution and important information in the data will be lost (leading to a well-known over-regularized estimated image).
Several methods have been presented
Set-up
In all experiments, we have considered the NL-means algorithm with the following parameters: the size of the search window and the neighborhood is set to . The decay of the weights in the similarity measure is set to (as proposed in Buades et al. (2005)) where is the standard deviation of the Gaussian noise and we have considered a classical Euclidean distance (and not a Gaussian weighted Euclidean distance as proposed in (Buades et al., 2005)). We precompute the set of weights
Conclusion
In this paper, we have presented a deconvolution/restoration approach whose regularization term encodes the inherent high redundancy of any natural images. This new prior derived from the denoising algorithm proposed by Buades et al. allows to efficiently constrain a deconvolution procedure, demonstrating its ability to summarize the intrinsic redundancy property of any natural image. In this context, the L-curve based approach proposed by Hansen et al. is well suited to a robust, fast,
Acknowledgements
The author is grateful to all the anonymous reviewers for their many valuable comments and suggestions that helped to improve this paper. In particular, he acknowledges the contribution of the reviewer who suggested we reestimate the weights of the non-local graph from the restored image in order to refine the restoration result. This encouraged the author to do more tests which have come to the comments given in Section 4.2 and finally improved the restoration results.
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