Elsevier

Pattern Recognition Letters

Volume 29, Issue 16, 1 December 2008, Pages 2206-2212
Pattern Recognition Letters

A non-local regularization strategy for image deconvolution

https://doi.org/10.1016/j.patrec.2008.08.004Get rights and content

Abstract

In this paper, we propose an inhomogeneous restoration (deconvolution) model under the Bayesian framework exploiting a non-parametric adaptive prior distribution derived from the appealing and natural image model recently proposed by Buades et al. [Buades, A., Coll, B., Morel, J.-M., 2005. A review of image denoising algorithms, with a new one. SIAM Multiscale Model. Simul. (SIAM Interdisc. J.), 4(2), 490–530] for pure denoising applications. This prior expresses that acceptable restored solutions are likely the images exhibiting a high degree of redundancy. In other words, this prior will favor solutions (i.e., restored images) with similar pixel neighborhood configurations. In order to render this restoration unsupervised, we have adapted the L-curve approach (originally defined for Tikhonov-type regularizations), for estimating our regularization parameter. The experiments herein reported illustrate the potential of this approach and demonstrate that this regularized restoration strategy performs competitively compared to the best existing state-of-the art methods employing classical local priors (or regularization terms) in benchmark tests.

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 E(x) to be optimizedxˆ=argminx{y-hx2+γΩ(x)E(x)}where y and x represent, respectively, the noisy (degraded by an additive and white Gaussian noise with variance σ2) 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 γ. If γ is selected as small, the recovered image is dominated by high-frequency noise components (the solution is the so-called under-regularized). If γ 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 (S) is set to 7×7. The decay of the weights in the similarity measure is set to h=10σ (as proposed in Buades et al. (2005)) where σ 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.

References (27)

  • A. Jalobeanu et al.

    Hyperparameter estimation for satellite image restoration using a MCMC maximum likelihood method

    Pattern Recognition

    (2002)
  • R. Molina et al.

    Restoration of severely blurred high range images using stochastic and deterministic relaxation algorithms in compound Gauss–Markov random fields

    Pattern Recognition

    (2000)
  • M.R. Banham et al.

    Spatially adaptive wavelet-based multiscale image restoration

    IEEE Trans. Image Process.

    (1996)
  • Batu, O., Etin, M., 2008. Hyper-parameter selection in advanced synthetic aperture radar imaging algorithms. In: IEEE...
  • J. Bioucas-Dias

    Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors

    IEEE Trans. Image Process.

    (2006)
  • Bioucas-Dias, J., Figueiredo, M., Oliveira, J., 2006. Adaptive total-variation image deconvolution: A...
  • A. Buades et al.

    A review of image denoising algorithms, with a new one

    SIAM Multiscale Model. Simulat. (SIAM Interdiscip. J.)

    (2005)
  • G. Chantas et al.

    Bayesian restoration using a new nonstationary edge-preserving image prior

    IEEE Trans. Image Process.

    (2006)
  • P. Charbonnier et al.

    Deterministic edge-preserving regularization in computed imaging

    IEEE Trans. Image Process.

    (1997)
  • Efros, A.A., Leung, T.K., 1999. Texture synthesis by non-parametric sampling. In: 7th Internat. Conf. on Computer...
  • M.A.T. Figueiredo et al.

    An EM algorithm for wavelet-based image restoration

    IEEE Trans. Image Process.

    (2003)
  • Figueiredo, M., Nowak, R., 2005. A bound optimization approach to wavelet-based image deconvolution. In: IEEE Internat....
  • Foi, A., Dabov, K., Katkovnik, V., Egiazarian, K., 2006. Shape-adaptive DCT for denoising and image reconstruction. In:...
  • Cited by (71)

    • Adaptive Non-Local Regression Prior based on Transformer for Image Deblurring

      2023, ACM International Conference Proceeding Series
    • Ultrasound Medical Image Deconvolution Using L2 Regularization Method and Artificial Bee Colony Optimization Algorithm

      2022, 2022 7th International Conference on Image and Signal Processing and their Applications, ISPA 2022 - Proceedings
    View all citing articles on Scopus
    View full text