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BLIND IMAGE RESTORATION VIA RECURSIVE FILTERING USING ...

Figure 1.for ImagesOptumzationAlganthmProposed Blind Deconvolution Schemea novel technique introduced in section 3.. In addition tothe assumptions stated above, the methods of [7]-[9] requirethat the blur also be nonnegative with known finite supportfor proper restoration. In contrast, the only assumption ouralgorithm makes about the blur is that its inverse exists.2. THE PROPOSED METHOD2.1. The Blind Deconvolution ApproachThe proposed NAS-RIF technique consists of a variable FIRfilter u(z, y) with the blurred image g(z, y) as input. Theoutput of this filter represents an estimate of the true imagej(z, y). This estimate is passed through a nonlinear filterwhich uses a non-expansive mapping to project the estimatedimage into the space representing the known characteristicsof the true image. The difference between thisprojected image f ~~(z., y) and f(z, y) is used as the errorsignal to update the variable filter u(z,y). Figure 1 givesan overview of the scheme.We concentrate on the particular algorithm for which theimage is assumed to be nonnegative with known support,so the NL block of Figure 1 represents the projection ofthe estimated image onto the set of images that are nonnegativewith given finite support. This requires replacingthe negative pixel values within the region of support withzero, and pixels values outside the region of support withthe background grey-level LB. The cost function used inthe restoration procedure follows:background colour is black. The third term is used to constrainthe parameters away from the trivial all-zero globalminimum for this situation.It can be shown that equation 1 is convex [ll], so thatconvergence of the algorithm to the global minimum is possibleusing a variety of numerical optimization routines.The conjugate gradient minimization routine is used forminimization of J because its speed of convergence is muchfaster than other descent routines such as the steepestdescentmethod. The recursive algorithm, referred to asthe NAS- RIF method is summarized in Table 1.Table 1. Summary of the proposed NAS-RIF algorithm.I) Definitions:a Jk(z,y): estimate of true image at kth iterationa uk (z, y): FIR filter parameters of dimension NZu x Nyuat iteration ka 6: tolerance used to terminate the algorithma J(gk): cost function at parameter setting gka VJ(gk): gradient of J at gka < ., . >: scalar producta Note: underlined letters represent lexicographically orderedvectors of their two-dimensional counterparts.11) Set initial conditions (k = 0):Set FIR filter uk(z,y) to all zeros with a unit spike in themiddleSet tolerance 6 > 0111) At iteration (k): k = 0,1,2, ...f^k(",Y)= uk(z,Y) *g(z,y)f^NL(Z,Y) = "x,y)lMinimization Routine to update FIR filter parameters.For example: (conjugate gradient routine)3a) F7J(1Lk)lT = r~a * "'where-1r 1 2r 12where j(z, y) = g(z, y) *U(%, y), Dsup is the set of all pixelsinside the region of support, and Dsup is the set of all pixelsoutside the region of support. The variable y in thirdterm of equation 1 is nonzero only when Lg is zero, ie., the2284Authorized licensed use limited to: Texas A M University. Downloaded on December 26, 2008 at 06:13 from IEEE Xplore. Restrictions apply.

2.2. Uniqueness of the SolutionUnder ideal conditions of an infinite extent filter u(z,y),and in the absence of additive noise, the solution to the algorithmmay not be unique. For example, if the backgroundcolour of the image is black, the true image is invertible, andthe support of the image and PSF are identical, a restorationwhich globally minimizes the cost function can be thetrue image, the PSF or many erroneous intermediate solutions.The possibility of these erroneous solutions is one ofthe dilemmas of blind deconvolution algorithms. With thelack of sufficient information, it is difficult to often overcomethis problem.Sufficient conditions for uniqueness of the solution for theNAS-RIF algorithm are developed in [ll], and are analogousto persistence of excitation for system identification.3. DETERMINATION OF THE SUPPORTA method for assessing the optimal support size automaticallyand objectively is proposed. It uses the hold-out(HO) method used for model validation in data analysis.The proposed support-finding algorithm is inspired by theconstraint assessment algorithm of [12], but is modified forblind image restoration.Competing assumptions on the true image, such as differentsupport sizes, can be assessed using the hold-outmethod. A support size for the true image is assumed. Theimage estimate pixels p(z, y) outside the assumed region ofsupport are collectively called the estimation set; they areused to obtain an estimate of the true image. This is accomplishedby minimizing a criterion, called the estimationerror, which incorporates only the pixels within the estimationset. Specifically, the proposed blind deconvolutionalgorithm is applied using the assumed support and excludingthe nonnegativity constraint. The set of pixels withinthe assumed region of support is called the validation set,and is used to assess the “correctness” of the assumed supportsize. This is performed by computing the validationerror which measures the energy of negative pixels of theimage estimate within the assumed region of support. Theassumed support which produces the minimum validationerror is selected as the true image support. The algorithmfollows in Table 2.If the assumed support is exact or larger than the actualsupport a reasonable estimate of the true image can be obtained.Since the true image is nonnegative, the validationerror for such an image estimate should be small. Thus,the assumed support which minimizes the validation erroris intuitively a good estimate of the actual support.4. SIMULATION RESULTS ANDCOMPARISONSThe results of the proposed algorithm and the IBD algorithmdescribed in [7] and modified in [8] are shown in figure2. The original toy and binary images shown in Figures2(a) and 3(a) of support 119 x 81 and 15 x 65 wereblurred using a 21 x 21 truncated Gaussian PSF, and noisewas added for a blurred signal-to-noise ratio (BSNR) of 70dB. The degraded images are displayed in Figures 2(b) and3(b). The proposed support-finding algorithm estimatedthe support of the toy image as 120 x 84, and the binaryimage as 15 x 65. Based on these supports, the NAS-RIFrestorations and mean square error (MSE) plots are shownin Figures 2(c),(e), and 3(c),(e). The proposed NAS-RIFmethod converged to a very good estimate of the solution inTable 2. Summary of the proposed support Andingalgorithm.Assume an equally spaced grid of support parameter values(L,,L,) from (1,1) to the size of the blurredimage (N,,,Nyg).1) Assume a rectangular support S with dimensions (LI, Ly)from the grid. If all values in the grid have been selectedbefore, either1. Go to step 5 if the exhausted grid contains successiveelements.2. Form a finer grid centred about (Lz,mtn,Ly,mln) (theparameters giving the minimum of the validation errorfound so far), and select a parameters (L,,Ly) out ofthis new grid.2) Based on the assumed support S, find the restoration filteru*(z,y) by using the conjugate gradient algorithm, tominimize the following estimation error function: J(g) =&,,E&>y)- LBIZ +Y [Ca(r,u) “(”,Y) - 1]2 wheref(z, y) = u(z, y) * g(z, y) and 3 is the region outside theassumed support.3) Calculate the validation error based on the minimizing filterparameters U*(”,?/) of the estimation error of step 2.where 11 . I/ denotes the number of elements in the argumentset, and the ”restored” image estimate f*’(z,y) = u*(z,y)*S(XClY).4) Save the parameters (L,,,,,, Ly,min), which give the minimumvalue of V(S) found so far. Go to step 1.5) Select the support parameters that minimize V(S) as theoptimal support size for restoration.approximately 300 iterations for the toy image and 100 iterationsfor the binary image. The results of the IBD methodare shown in Figures 2(d),(f) and 3(d),(f). The image estimatewhich showed the minimum energy of negative pixelswithin the region of support and pixels deviating from thebackground grey-level was used as the restored image forboth sets of results of the IBD algorithm. The restorationsof the IBD algorithm, shown in Figures 2(d) and 3(d), arethe image estimates at the 1000th and 3000th iterations,respectively.Although the IBD algorithm produces comparable resultsto the NAS-RIF algorithm for simple binary images, it failsto converge to a reliable image estimate for more complicatedgrey-scale images. The algorithm often exhibits instabilityand, at times, begins to diverge even as it appears tobe converging to a good solution. Simulation results of theIBD algorithm for exact support size at various differentinitial conditions and noise parameter values a producedsimilar results as those shown.The algorithm of [9] produced good results for very smallimages; however, for the images shown in this paper, itwas too computationally time consuming to produce agood estimate. The order of the algorithm per iterationis O(Nf4), where Nj is the number of pixel values of theimage estimate. In contrast, the NAS-RIF method has orderO(NfNuNl,,k) per iteration, where Nu is the numberof FIR filter parameters of u(z, y) and Nl,,k is the number2285Authorized licensed use limited to: Texas A M University. Downloaded on December 26, 2008 at 06:13 from IEEE Xplore. Restrictions apply.