Deconvolution of Neutron Degraded Images: Comparative Study

Engineering Letters, 18:3, EL_18_3_03 ______________________________________________________________________________________ Deconvolution of Neutron Degraded Images: Comparative Study between TSVD, Tikhonov Regularization and Particle Swarm Optimization Algorithm Slami Saadi, Maamar Bettayeb, Senior Member, IEEE, and Abderrezak Guessoum, Member, IEEE Abstract—Image deconvolution is an important problem in By selecting the proper filter, one can get a good estimate image processing. It is an ill-posed inverse problem, so of the original image signal, by reducing the effect of noise. regularization techniques are used to solve this problem by Examples of the restoration filters are the deconvolution adding constraints to the objective function. Various popular filter, in which the transfer function of the degraded system algorithms have been developed to solve such problem. This is inverted to produce a restored image, and the Wiener filter paper proposes a new approach to the nonlinear neutron that uses the mean-squared error (MSE) criterion to degraded images restoration problem which is useful in many images enhancement applications, based on swarm intelligence. minimize the error signal between the original and degraded We use the particle swarm optimization (PSO) applied for total images. A limitation in this filter is that it cannot handle variation (TV) minimization, instead of the standard Tikhonov dynamically changing image and noise signals. Regularized regularization method. In this work, we attempt to reconstruct deconvolution can be used effectively when constraints are or recover neutron radiography images that have been applied on the recovered image (e.g., smoothness) and degraded during acquisition; using some a priori knowledge of limited information is known about the additive noise. the degradation phenomenon. The truncated singular value The ill-posedness of this problem arises from the fact that decomposition (TSVD) method is also considered for image the kernel of the blurring function is badly conditioned and deconvolution in this paper. A comparison between the five the degraded image contains noise. As a result, small methods is conducted, using several images. perturbations in additive noise may lead to significant Index Terms—Deconvolution, ill-posed, TV, oscillations in the inversion result when using matrix Tikhonov, TSVD, PSO, regularization inversion solution. Therefore, to correctly recover the unknown original image, regularization is necessary. We can use regularization in frequency domain but this can I. INTRODUCTION result in unlimited amplification of noise. The main shortcoming of the frequency domain regularization By image restoration, we seek to recover the original methods is the difficulty of finding the appropriate ending sharp image using a mathematical model of the blurring frequency at which the blurring function will vanish quickly process. The key issue is that some information on the lost [2]. details is indeed present in the blurred image, but this Spatial domain regularization methods are a wide information is “hidden” and can only be recovered if we research field for image restoration problems. In the spatial know the details of the blurring process. Due to various domain, the canonical regularization method is the unavoidable errors in the recorded image, we can not Tikhonov regularization. The basic regularization theory recover the original image exactly. The most important was first proposed by Tikhonov and Arsenin [3]. errors are fluctuations in the recording process and The solution methods for this optimization problem approximation errors when representing the image with a include singular value decomposition (SVD) based direct limited number of digits [1]. method [3], Newton and quasi-Newton method [4], gradient We can broadly classify restoration techniques into two methods (e.g. steepest descent (SD) method and conjugate classes: the filtering reconstruction techniques and the gradient (CG) method) and various preconditioning algebraic techniques. The filtering techniques are rather techniques. CG method has proven to be an efficient classical and they make use of the fact that noise signals iterative regularization method for recovering the correct usually have higher frequencies than image signals. This image from its degradation [5]. This method overcomes the means that image signals die out faster than noise signals in difficulty of choosing the regularization parameter by high frequencies. controlling the iteration indices but, the optimal stopping iteration index still depends on the noise level. Manuscript received October 25, 2009. Revised December 25, 2009 S. Saadi is with the Department of Instrumentation, Nuclear Research Total variation (TV) is a regularization approach that Center of Birine (CRNB), BP180, Ain-Oussera 17200, Algeria. Phone: performs edge preserving image restoration, but at a high +213772817158, e-mail: [email protected] . computational cost. Total variation regularization requires M. Bettayeb is with the Electrical & Computer Engineering Department, linearization of a highly nonlinear penalty term, which College Of Engineering, University Of Sharjah, P.O.Box 27272, Sharjah, United Arab Emirates, (e-mail: [email protected]) . increases the restoration time considerably for large scale A. Guessoum is the head of Signal and Image Processing Laboratory images. In the total variation method, we will consider an (LATSI), Electronics Department, Blida University, LATSI Laboratory, iterative regularization approach in the spatial domain, B.P. 270 Blida Algeria, (e-mail: [email protected]). which was first addressed in optimization as the Barzilai- Borwein minimization (BB) method [6]. Iterative techniques (Advance online publication: 19 August 2010) Engineering Letters, 18:3, EL_18_3_03 ______________________________________________________________________________________ have a common problem: the error starts increasing after it number of transformation coefficients is maximized. reaches a minimum. The first few iterations restore the low Although applicable in many image restoration applications, frequency components of the signal and, as the number of SVD is severely limited because of a large number of iterations increases, the algorithm attempts to restore the computations required for calculating singular values and high frequency components, which are dominated by noise. singular vectors of large image matrices. Solution to such problem can be attained by adding a The SVD of an mxn matrix A is given by: median filter to maintain a low error by preserving the edge T information while reducing the high frequency error [7]. A=U∑V (2) Most of the optimal techniques that have been proposed ∈ mxm ∈ nxn Where U=( u1, u2,..,un) R and V=(v1, v2,.. vn) R in literature over the past few decades to solve such problem are two column-orthogonal matrices. Σ is a diagonal matrix by iterative optimization procedures are computationally with entries σ1 ≥ σ2 ≥ … ≥ σN 0≥ . For a blurring matrix, all demanding and time consuming. The novel approach the singular values decay gradually to zero and the condition introduced in this paper is to take advantage of particles number cond(A) = σ1 / σN is very large. We use the SVD swarm intelligence in order to facilitate the optimization approach to damp the effects caused by ………... One process in total variation regularized methods. For our test approach to damp the effects caused by division of small images, we consider the physical meaning of the widely singular values is to simply discard all SVD components used 8-bit images; the pixel value varies from 0 to 255. We that are dominated by noise, typically the ones above a then reformulate the optimization problem by imposing the certain truncation parameter. The resulting method is, for nonnegative constraints. obvious reasons, referred to as the truncated SVD, or TSVD method [1]. II. DECONVOLUTION USING A GENERAL LINEAR MODEL We model the blurring of images as a linear process IV. TIKHONOV REGULARIZATION characterized by a blurring matrix H of dimensions NxN, The main objective of regularization is to incorporate with N=mxn and an observed image g which, in vector more information about the desired solution in order to form, are related by the equation: stabilize the problem and find a useful and stable solution. The most common and well-known form of regularization is = gHf (1) that of Tikhonov [8]. Where f is the original image The Tikhonov regularized minimum norm solution of the = +η ℜ∈ N The reason H-1g cannot be used to deblur images is the equation: Hfg is the vector Fδ that amplification of high-frequency components of the noise in minimizes the expression : 2 2 the data, caused by the inversion of very small singular +− λ2 LfgHf (3) values of H. Practical methods for image deblurring need to 2 2 avoid this pitfall [1]. Obtaining f from Equation (1) is not a where λ > 0 is called a regularization parameter straight forward task since, in most cases of interest, the We denote: matrix H is ill-posed. Mathematically this means that certain { 2 2 2 } Fδ = minarg gHf +− λ Lf (4) eigenvalues of this matrix are close to zero, which makes the f 2 2 inversion process very unstable. For practical purposes, this Regularization can be understood as a balance between two implies that the inverse or the pseudo-inverse solutions: requirements: -1 T -1 T f1=H g and f2= ( H Η) Η g amplify the noise and provide 1. f should give a small residual Hf-g. incorrect

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