A New Technique For Enhanced Regularized Image Restoration Moh& M. Hadhoud *, Member IEEE,Moawad I. Dessouky *,

Fathi E. Abd El-Samie" and Said E. Ei-Khamy**, Fellow IEEE

* Department of electronics and electrical communications, faculty of Electronic Eng., Menoufia University,

32952, Menouf, EGYPT,Email: fathi sayecl(iiNahoo.com **Department of electrical Eng., Faculty of Engineering, Alexandria University,

Alexandria 21544, Egypt, Email: elkhamy@ieee.org

Abstract

The pivoting problem encountered in regularized image restoration is how to

evaluate the regulurization parameter. Existing mefhods for evalua ting this

regularization parameter are categorized to two main groups; methods that require

knowledge of the noise variance and methods that don't require knowledge of the

noise variance in ,tFte degraded image. This paper suggests a new approach for

choosing the regmtarization parameter. I t depends on maximizing the power in tlzv

restored image by the coincidence of the passband nfthe regularization filter with

the frequency band in which most of the image power exists. The suggested method

has an iterative natirre and doesn V require a priori knowledge of the noise variance.

which may be inexact. Results show that the estimated value ojthe regularization

paramefer coincides with the minimum of the mean squared error and results in

enhanced image restoration.

I. INTRODUCTION

The linear shift invariant image degradation model is described using the following equation [ 1,2,3]:

Where f , g and n are lexicographic orders by either column or raw of the two dimensional image, the degraded image and the noise, respectively, each of size MxI1f. The matrix H is the discrete representation of the degradation of dimension M2,%kf2 . For linear shift invariant systems, matrix M is a block Toeplitz matrix. The image restoration problem is to cstiniate f given the samples of the recorded image g .

g = H f + n (1)

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The image restoration problem can be solved directly by inverting the blurring operator. Mathematically, this can be represented by seeking i that minimizes the following cost function [ 31 :

Y (i) = ((g - Hi((

This leads to [3]:

Equation ( 3 ) can be written in an equivalent forin

i. = H-'g 1 f + H-'n

approximation [4]:

(3) using the Toeplitz to circulant

where F(u, v) is the Fourier transform of the original iinage , and G(u, v ) ~ , ? ( U , v) , H(u, v) and N(u,v) are the Fourier transforms of the degradecJ.,image, the estimate of the original image , the point spread function PSF of the blurring operator, and the noise sequence, respectively. 3 *+

An inverse problem is characterized as ill $';sed when there is no guarantee for the existence, uniqueness and stability of the solylion based on direct inversion. In the presence of noise the problem has an ill posed nature. Thus, severe deteriorations are observed in the restored images in the cases of low signal to noise ratio. On the other hand, this direct restoration scheme can be implemented perfectly in the absence of noise.

....- ..

11. REGULARIZED IMAGE RESTORATION 1 1

Regularization theory, which was basically introduced by Tikhonov and Miller, provides a formal basis for the development of regularized solutions of ill posed problems[5]. The stabilizing function approach is one of the basic methodologies for the development of regularized solutions. According to this approach, an ill-posed problem can be formulated as the constrained minimization of a certain function, called stabilizing function[ 51. The specific constraints imposed by the stabilizing function approach on the solution depend on the form and the properties of the stabilizing function used. From the nature of the problem, these constraints are necessarily related to the a priori information regarding the expected regularized solution.

According to the regularization approach, the solution of Eq.( 1) is obtained by the minimization of the following cost function [ 3 ] :

Y ( i ) = /lg - Hill2 + AllCil(2 ( 5 )

where C is the regularization operator and A is the regularization parameter . Solving for that i. that provides the minimum of the cost function yields [3] :

where, i. = (H'H + AC'C)-'H'g = A(A)g (6)

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-__ _ _ _ _ _ _ _ _ ----- --

A(A.) = (H'H + A.C'C)-' H' (7) The rule of the regularization operator C is to move the sinall eigenvalues of H away

from zero while leaving the large eigenvalues unchanged. The amount of change of these eigenvalues is dependent on the choice of the regularization parameter A . Equation (6) can be written in the following Form using the Toeplitz to circulant approximation [4] :

(8) H * ( u v)

F ( u v) =: G(u,v) = D(tr, v.A)G(u,v) Ill(u,v)12 + AIC(u,v)/

where C(u, v) is the Fourier transform of the 2-D sequence o€ the regularization operator .

where Diu, v, A) is the transfer function of the 2-D regularized image restoration jilter

111. METHODS OF CHOOSING THE REGULARIZATION PARAMETER

The problem of choosing the regularization parameter is addressed using a diversity oi. techniques [6,7]. Some of these techniques require knowledge of the noise variance ofthe degraded image like the constrained least squares (CLS) approach, the equivalent degrees of freedom (EDF) approach, the mean square error (MSE) approach and the se1 theoretic (ST) approach [7] . In many practical situations, the noise variance o2 is not Imov~n and its estimation may be a tedious problem. So the methods mentioned above can't be used. ?'hiis. Other techniques don't necessitate this knowledge of tlie noise variance like the cross validation (CV) approach and the maximum likelihood (ML) approach are usedl 71.

The different methods mentioned above for choosing the regulai~ization parameter yield different values of the regularization parameter 1 . The regularization parameter A dictates, in general, the level of smoothness in the restored image. ' h i s , if two different regularization parameters A., and A.,, are used in restoring a degraded image with A(, > A, , thc

resulting restored iniage f ( A z , ) will be smoother than j ' (2,) [ 7 ] . The smooihness criterion doesn't reveal any thing about tlie MSE of the solution.

Based on the above motioned smoothness criterion, the smoothness of the solutions obtained using the different approaches for evaluating the regularization parameter differ:: according to the value of A. obtained in each approach. The constrained leas1 squares (c'L,S I approach yields over smoothed estimate of the restored image, since it has ii large ialue of A [7]. The cross validation approach yields the best results from the mean square error point o f view for restoring highly correlated images that exhibit low pass characteristics only. O n the other hand, it is not prFferred when images don't exhibit this behavior and the mean square error (MSE) approach is the best approach used in this case. The maximum likelihoocl approach is very sensitive to the choice of the regularization operator C , hi t is less sensitive, in comparison to the MSE and CV approaches, to the characteristics ofthe image [71.

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Accordingly, a unified approach is needed to estimate the regularization parameter A . In the next section, we propose a method for selecting the regularization parameter that coincide with the minimum mean square error MMSE in the restored image .

IV. THE SUGGESTED APPROACH FOR CHOOSING THE REGULARIZATION PARAMETER

The regularized image restoration filter transfer function is expressed using Eq.(9). It is clear from this equation that the parameter A controls the pass band of this restoration filter. This is illustrated in Figs (1) to (5 ) . Figure (1) illustrates the frequency response of a low pass blurring operator H(u , v) . Figure (2) illustrates the frequency response of the inverse filter. It is clear that the inverse filter has singularities at certain frequencies. In Figs.(3) to ( 5 ) , the effect of regularization is illustrated on the inverse filter with regularization parameter h=O. 1 , 0.01 and 0.001 respectively. It is clear that the regularized filter has a band pass nature and the parameter h controls the location of the passband in the frequency spectrum.

In general most images to be restored using this regularized restoration filter have their spatial frequencies concentrated near the low frequency region of the spectrum. When the filter passband coincides with the spectral region at which the frequency contents of the image to be restored lie, good restoration results are obtained due to no missing frequency components of the restored image. The noise frequency components, which extend to infinite frequencies lie outside the filter pass band and thus are rejected.

The suggested approach used here to evaluate the regularization parameter is based on the above-mentioned concept and is illustrated in Fig.(G) . In this approach an initial estimate of the regularization parameter is used and the restoration process is performed on the degraded image g(n, , n2 ) using the regularized filter D( U, v, A) . The power spectrum of the restored image is estimated and then the total power in the restored image is calculated. This process is repeated iteratively by updating the regularization parameter until maxi mum value of the total power is obtained in the restored image. The value of A which yields this maximum total power is the best estimate of A which is expected to yield minimum mean square error in the restored image.

The auto correlation function of the restored image g' (n , , n2 ) is evaluated using the following relation[2] :

where R g , ( n , , n ) is the correlation at position ( n , y1 ) . w is an arbitrary window length taken in the upper left corner of the image [2 ] .

The power spectrum of the restored image g ' (n , , n 2 ) is evaluated using the following relation:

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An integration process. which may be approximated using a sunmiation precess, is performed over the power spectrum to evaluate the total power in the restored image g ' (n , ..nZ j . The approximation of this integration process is given using the following equalion :

When this total power is maximum, the restored image g ' ( n , ,U,) can be considered thc hest

estimate of the image f ( n , ,n2). If the power is not maximurn, the regularization parameter is updated and the process is repeated.

V. SIMULATION RESULTS

The suggested approach for choosing the regularization parameier I S tested ill this section for two different images with different spatid activities, the cii~neraman arid the mandrel images illustrated in Figs. (7) and (S), respectively. In this experiment, the restoration process is perhrmed in the presence of'additivc noise and ii b h d g operator of' 5x5 with SNR=4O dB . The total power and MSE in the restored image:; are plotted in Pigs.(9) and (1 l), respectively, for the cameraman image and in I'igs (10) L113d

(1 2) for the mandrel image . [t is clear that the minimum MSE coincides with the maximum pclwer in the case of

the cameraman image, which is mainly an image of low frequcncy nature. 'rlius the 1~ropo~c.d algorithm succeeds in choosing the best regularization piuameier.

In the case of the mandrel image which is of high fiequency nature. the MSE: r e d m its minimum value with the maxiinuin of the tola1 power in the restored ini:qy and I<cey)s l h i ~

minimum value for a long range of A .

VI. CONCLUSION The suggested approach for choosing the regularization paranieter in regularized image

restoration is tested. Results show that this approach succeeds in estimating a val ue for thc regularization parameter regardless of any knowledge of the noise variance h r both additive noise environments. Results also show that the sensitivity ol'lhe restoration process to the choice of the regularization parameter is higher in the cave sf images with low lieqiiencq nature than in the case of images with high frequency nai tire.

VII. REFERENCES [ 13 William K.Pratt, Digital Image I'rocessing. John Wiley & Sons . [nc. , 199 1

[2] Nikclas. P. Galatsanos. and ftonald. T. Chin. '* Digital R.estoration 0 1 ' Multich,iiii~c.l Images," IEEE Trans. Acoustics, Speech and Signal l"rocessjtrg, 101. 37, N0.3, pp 31.5-42 I

March 1989.

[3 J H.C Anderws and B.R. Hunt, Digital Image Rcstoliitiol1. Engleivooti Cliffs, N.I. I'reniice-

Hall, 1977.

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[4] Reginald L. Lagendijk, Jan Biemond and Dick E. Boekee, “ Regularized Iterative Image

Restoration With Ringing Reduction,” IEEE Trans. Acoustics, Speech and Signal Processing,

vol. 36, No.12. ,pp 1874-1888, Dec 1988.

[5] Nicolas B. Karayiannis and Anastasios N. Venetsanopoulos, “ Regularization Theory In

Image Restoration- The Stabilizing Functional Approach,” IEEE Trans. Acoustics, Speech

and Signal Processing, Vol. 38 , No.7, pp.1155-1179, July 1990.

[6] Nikolas. P. Galatsanos, Vladimir Z. MesaroviC, Rafael Molina and Aggelos K.

Katasaggelos, “ Hierarchical Bayesian Image Restoration From Partially Known Blurs,” IEEE

Trans. Image Processing , Vol. 9, No. 10, pp. 1784-1 797, Oct. 2000.

[7] Nikolas. P. Galatsanos and Aggelos K. Katasaggelos, “ Methods Of Choosing The

Regularization Parameter And Estimating The Noise Variance In Image Restoration And

Their Relation,” IEEE Trans. Image Processing , Vol. 1, No.3, pp.322-336, July 1992.

Fig.( 1) Low pass filter frequency response.

response h=O. 1.

Fig.(2) Inverse Filter frequency response.

Fig.(4) Regularized Filter frequency response h=0.01

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Fig.(S) Regularized Filter frequency response h=0.001.

DCgEd&+ image Regularized image Power Spectrum

restoration g'(nl7 ' 2 ) estimation

Fig.(6) Block diagram of the suggested approach for choosing the regularization parameter.

A a

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I v Update

regularization parameter

Integrate over spatial frequencies

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~. ._ , .. . , . . . . . ,

Fig.(7) Original Cameraman image.

Fig.(8) Original Mandrel image.

Fig.(9) Pt versus h for Fig.(lO) Pt versus h for restoring the kamerainan restoring the inonkey image image degraded by 5x5 degraded by 5x5 operator operator SNR=4O dB. SNR=40 dB.

rest6ring the cameraman restoring the monkey image image degraded by 5x5 degraded by 5x5 operator operator SNR=40 dB. SNR=40 dB.

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