Effective Image Restorations Using a Novel Spatial Adaptive Prior

Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2010, Article ID 508089, 10 pages doi:10.1155/2010/508089 Research Article Effective Image Restorations Using a Novel Spatial Adaptive Prior Yang Chen,1, 2 Yinsheng Li,1, 2 Yingmei Dong,3 Liwei Hao,2 Limin Luo,1 and Wufan Chen2 1 The Laboratory of Image Science and Technology, Southeast University, Nanjing 210096, China 2 The School of Biomedical Engineering, Southern Medical University, Guangzhou 510515, China 3 Cadre Reset Institute, The Joint Logistics Department, Chengdu Military Region, Chengdu 610083, China Correspondence should be addressed to Wufan Chen, Received 20 October 2009; Revised 29 December 2009; Accepted 16 February 2010 Academic Editor: Liang-Gee Chen Copyright © 2010 Yang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Bayesian or Maximum a posteriori (MAP) approaches can effectively overcome the ill-posed problems of image restoration or deconvolution through incorporating a priori image information. Many restoration methods, such as nonquadratic prior Bayesian restoration and total variation regularization, have been proposed with edge-preserving and noise-removing properties. However, these methods are often inefficient in restoring continuous variation region and suppressing block artifacts. To handle this, this paper proposes a Bayesian restoration approach with a novel spatial adaptive (SA) prior. Through selectively and adaptively incorporating the nonlocal image information into the SA prior model, the proposed method effectively suppress the negative disturbance from irrelevant neighbor pixels, and utilizes the positive regularization from the relevant ones. A twostep restoration algorithm for the proposed approach is also given. Comparative experimentation and analysis demonstrate that, bearing high-quality edge-preserving and noise-removing properties, the proposed restoration also has good deblocking property. 1. Introduction 1.1. Problem Formulation. As one of the most classical linear inverse problems, image restoration has its wide applications in remote sensing, radar imaging, tomographic imaging, microscopic imaging, astronomic imaging, digital photography, and so forth [1–4]. For linear and shiftinvariant imaging systems, the transformation from f to g is well described by following additive linear degradation model [3, 4]: g = A ∗ f + ε, (1) where g, f , and ε represent, respectively, the degraded observed image, the original true image, and the corrupting white Gaussian noise with variance σ 2 . Point-spread function (PSF) A is the imaging system and ∗ is the linear convolution operator. Throughout this paper we assume that the degradation model PSF A and noise variance σ 2 are known for they could be numerically estimated or calibrated. Based on the Gaussian statistics of the additive noise, maximum log-likelihood (ML) restoration method could be applied to find the least-squares estimation of f . However, such ML restoration method often leads to unacceptable restoration solutions due to the ill-posedness of the inverse problems. Small changes in the data due to noise can cause large changes in the solution, and the knowledge of the degradation model is not sufficient to determine a restoration result with acceptable accuracy [2–5]. Bayesian or Maximum a posteriori (MAP) approach, within rigorous Markov random fields (MRFs) framework, can provide a stable regularized solution through the incorporation of a priori image information about the geometrical properties of an image [3–8]. Through modeling the unknown parameters in the prior probability density functions, such prior information measures the extent to which the contextual constraint assumption is violated by the image or surface. Bayesian approach is able to distinguish good solutions from less desirable ones by transforming the original ill-posed problem into a well-posed one. It is also noted that most regularization restoration approaches can find their Bayesian interpretation with different forms [6, 7]. 2 EURASIP Journal on Advances in Signal Processing We can build following posterior probability P( f | g) for image restoration       P f |g ∝P g| f P f , (2)   2      1  P g | f = exp L g, f = exp − κg − A ∗ f  , 2 (3) ⎞   U j f ⎠. ⎛      P f = Z −1 × exp −αU f = Z −1 × exp⎝−α j (4) Here, P(g | f ) is the likelihood distribution and P( f ) is the prior distribution. The partition function Z is a normalizing constant. U( f ) is the prior energy function, and U j ( f ) is the notation for the value of the energy function U evaluated on the f at pixel index j. α is the global hyperparameter that controls the degree of the prior’s influence on the image f . The energy function U( f ) in (4) attains its minimum and the corresponding prior distribution (4) attains its maximum when the image meets the prior assumptions ideally. So from (2)–(4), we can build the log-posterior energy function as       log P f | g = L g, f − αU f 2   1  = − κ g − A ∗ f  − α U j f . 2 j (5) The reconstructed image f can be obtained through iteratively maximizing function ψ( f ). The simple and widely used quadratic membrane (QM) prior or the Tikhonov L2 regularization, which smoothes both noise and edge details equally, leads to a linear inversion process and tends to produce an unfavorable oversmoothing effect [5]. To solve this oversmoothing problem, many edgepreserving Bayesian restoration methods were proposed in the past twenty years. We generalize them into three main classes: wavelet Bayesian restoration, Bayesian restoration with nonquadratic prior energies, and Bayesian restoration with total variation (TV) prior/regularization. Wavelet complexity regularization restorations, using a multiscale stochastic prior model, have also been proposed for edge-preserving restoration [8–11]. Priors based on wavelet decompositions and heavy-tailed pdfs along with the EM restoration algorithm have been proposed in [10]. These wavelet regularization methods rely on statistical modeling of the distribution of wavelet coefficients and strategy of coefficient thresholding. We choose not to include the comparison of wavelet regularization methods in this paper for the consideration that they can also be reinterpreted as some wavelet domain forms for Wiener filters, TV variational methods, or some Tikhonov regularization procedures [11]. Edge-preserving priors with nonquadratic energies were also proposed to preserve edge details by assuming a nonlinear inverse-proportional relationship between the existence of edges and intensity differences [5, 12, 13]. The weighting matrices in nonquadratic prior Bayesian restoration preserve edges by turning off or suppressing smoothing at appropriat (...truncated)