An Enhanced Primal Dual Method For Total Variation-Based Wavelet Domain Inpainting

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1 An Enhanced Primal Dual Method For Total Variation-Based Wavelet Domain Inpainting P. Sravan Kumar 1, H.V. Ram Ravi Kishore 2, V. Ravi Kumar 3 1 MTech, Spatial Information Technology, from JNTU, research scholar. 2 MTech, Electronics & communication engineering, academic scholar. 3 MTech, Electronics & communication engineering, Associate Professor, Mallareddy institute of technology & sciences, Hyderabad, experience 8 years of in teaching, India. Abstract The filling in of missing region in the image is known as image inpainting. Recently various methods have been proposed to tackle the inpainting problem. But they had their own disadvantages, but one of the best method chosen for inpainting problem is primal dual method for total variation-based wavelet domain inpainting [9]. But it is not up-to-the mark, because it has lag-city-of time (i.e. it take more number of iterations) due to this there is a delay in time to produce the output. Therefore in this paper we proposed an efficient scheme called an enhanced primal dual method for total variation based wavelet domain inpainting which put fourths the work done in image inpainting within a mean time to get the output by using the primal dual vertex cover algorithm (EPDVC) [11]. Keywords image compression, image restoration, inpainting, pixel-domain, total variation, wavelet transform. I. INTRODUCTION The filling-in of missing region in an image is known as Image Inpainting[7]. Inpainting is the art of modifying an image or video in a form that is not easily detectable by an ordinary observer. Image Inpainting has become a fundamental area of research in image processing. The modification of images in a way that is non-detectable for an observer who does not know the original image is a practice as old as artistic creation itself. Medieval artwork started to be restored as early as the renaissance. The motive was simple, to bring medieval pictures, up to date as to fill any gaps. This practice is called retouching or Inpainting. Also image Inpainting has been widely investigated in the applications of digital effect (i.e. object removal, image editing, image resizing), image restoration (e.g. Scratch or text removal in photograph), image coding and transmission (e.g. recovery of missing blocks) etc. Inpainting in wavelet [12] domains is a completely different problem since there are no well defined inpainting regions in the pixel domain. After the release of the new image compression standard JPEG2000, many images are formatted and stored in terms of wavelet coefficients. During storing or transmitting, some wavelet coefficients may be lost or corrupted. This prompts the need of restoring the missing information in the wavelet domain. 549 Inspired by the practical need, Chan, Shun and Zhou studied image inpainting problems in wavelet domains. Let us denote the standard orthogonal wavelet expansion of g and f by Where { } denotes the wavelet basis, and { } { } are the wavelet coefficients of g and f given by For For convenience, it is denoted as by f when there is no ambiguity. Assume that the wavelet coefficients in the index region I are known, that is, the available wavelet coefficients are given by { The aim of wavelet domain inpainting is to reconstruct the wavelet coefficients of f using the given coefficients ξ. It is well-known that the inpainting[7] problem is ill-posed, i.e. it admits more than one solution. There are many different ways to fill in the missing coefficients, and therefore one may have different reconstructions in the pixel domain. A regularization method can be used to fill in the missing wavelet coefficients. Numerous methods have been proposed to solve saddlepoint [9] problems. They include the finite envelope method, the primal dual steepest descent algorithm, the alternating direction method, primal dual iteration algorithm and so on each of them has its own advantages and disadvantages. In the above primal dual iterative algorithm is suited to solve but the main difficulty & disadvantage arises from the forward & backward wavelet transformation, the gradient operation & the divergence operator, which causes a lag-city-of-time.

2 Therefore in this paper we demonstrate a simple & efficient algorithm called EPDVC to solve the above problem. II. TOTAL VARIATION A. Why is the total variation useful for images? The total variation has been introduced for image denoising and reconstruction, the command line program tvrestore performs TV-regularized image restoration[9] by solving the minimization problem ( ) - Including denoising, deconvolution, and inpainting as special cases. The important options are If the input f is a color image, then TV is replaced with vectorial TV ( ) --7 Compared to processing each channel independently, the advantage of vectorial TV is that it forces the channels to stay aligned, which prevents false color artifacts near edges. B. TOTAL VARIATION APPROACH:- The total variation [7] has been introduced in Computer Vision first by Rudin, Osher and Fatemi, as a regularizing criterion for solving inverse problems. It has proved to be quite efficient for regularizing images without smoothing the boundaries of the objects. In digital image processing, an image is represented by a matrix or by a vector formed by stacking up the columns of the matrix. In the latter representation of an image, the n n pixel becomes the (r,s)th pixel becomes the((r -1)n+s) entry of the vector. The acquired image g is modeled as The original image f plus an additive Gaussian noise η. i.e In the following, it is denoted by f r,s the ((r -1)n+s)th entry of f In the JPEG 2000 format, images are transformed from the pixel domain[12] to a wavelet domain through a wavelet transform. Wavelet coefficients of an image under an orthogonal wavelet transform W are designated by ^:.Thus, the wavelet coefficients of the noisy image g are given by. The observed wavelet coefficients are given by g but with the missing ones set to zero, i.e. ^ 550 ( ) Here, - is the complete index set in the wavelet domain, and I ½ - is the set of available wavelet coefficients. Then, an observed image is obtained by the inverse wavelet transform. The aim of wavelet-domain inpainting is to reconstruct the original image f from the given coefficients. Durand and Formant considered to reconstruct the wavelet coefficients using the total variation (TV) norm. Rune et al. in considered wavelet domain inpainting in wireless networks. The method classifies the corrupted blocks of coefficients into texture blocks and structure blocks, which are then reconstructed by using texture synthesis and wavelet-domain inpainting, respectively. To fill in the corrupted coefficients in the wavelet domain in such a way that sharp edges in the image domain are preserved, Chan et al. proposed the following in to minimize the objective function: ( ) Where ( ) is the TV norm, Dz is the forward difference operator in the z-direction for z fx; yg, and is a regularization parameter. This is a hybrid method that combines coefficients fitting in the wavelet domain and regularization in the image domain. In, it was shown that such a hybrid method can reduce the staircase effect of the TV norm. Various alternative objective functions with distinctive properties were proposed in, and. A number of numerical methods have been proposed for solving TV minimization problems in the primal [11, 9] setting. They include 1. Time marching scheme. 2. fixed-point-iteration method. 3. majorization-minimization approach and many others. Solving TV regularization problems using these methods faces numerical difficulty due to the non differentiability of the TV norm. This difficulty can be overcome by introducing a small positive parameter " in the TV norm that makes the TV norm differentiable, i.e., ( ) ( )

3 In, Chan et al. derived the Euler-Lagrange equation and used an explicit gradient descent scheme to solve equation in the primal setting. Numerical results showed that, as decreases, the reconstructed edges become sharper. Since the exact TV norm (i.e.,"=0) can often produce better images than its smoothed version, one would prefer the original formulation, i.e., equation, to the smoothed one. an efficient optimization transfer algorithm[9] (OTA) is introduced to solve original problem equation. The method introduces an auxiliary variable yielding a new objective function, i.e. ( ( ) ) Where t is a positive control parameter. An alternating minimization algorithm is used to find a solution of the bivariate functional. For the auxiliary variable ς, the minimizer has a closed-form solution; for the original variable f, the minimization problem is a classical TV denoising problem, i.e., * Where ϒ= λ/(1+t) and STV (.) is known as the Moreau proximal map, which can be efficiently computed using dual based approaches, primal-dual hybrid methods (i.e. EPDVC algorithm). for minimizing the total variation based on a dual formulation is quite fast. In Chambolles approach [9], the TV norm is reformulated using duality, i.e Where A ½ R2n2 is defined by ( ) The discrete divergence of p is a vector defined as With px 0;s = py r;0 = 0 for r, s = 1,...,n.Chambolle showed that solving equation requires the orthogonal projection of the observed image onto the convex constraints appearing in equation. Hence, computing the solution of equation hinges on computing a nonlinear projection [10]. III. EPDVC ALGORITHM So far, we have seen many algorithms based on linear program (LP) relaxations, typically involving rounding a given fractional LP solution to an integral solution of approximately the same objective value. 551 In this article, we will look at another approach to LP relaxations, in which we will construct a feasible integral solution to the LP from scratch, using a related LP to guide our decisions. Our LP will be called the Primal LP, and the guiding LP will be called the Dual LP. As we shall see, the PD method is quite powerful. Often, we can use the Primal- Dual (PD) [11] method to obtain a good approximation algorithm, and then extract a good combinatorial algorithm from it. Conversely, sometimes we can use the PD method to prove good performance for combinatorial algorithms, simply by reinterpreting them as PD algorithms. We begin with a generic covering LP, and illustrate the ideas later with Vertex Cover as an example. Let [k]:= {1, 2... k}. Suppose we have matrix and vectors. We can represent the primal LP as, Now suppose we want to develop a lower bound on the optimal value of this LP. One way to do this is to find constraints that look like j c j x j z, for some z. using the constraints in the LP. To do this, note that any convex combination of constraints from the LP is also a valid constraint. Therefore, if we have non-negative multipliers y i on the constraints, we get a new constraint which is satisfied by all feasible solutions to the primal LP. That is, if for all i, j a ij x j b i then Note that we require the y i s to be non-negative, because multiplying an inequality (in this case j a ij x j b i ) By a negative number switches the sign of the inequality. We will obtain a lower bound o optimal value pf the primal LP. Switching the order of summation, we get And can ensure this sum is at most jc j x j by requiring the y i s to satisfy [ ] Putting it all together. If the y i s are non-negative and satisfy constraint, then Note that the constraints on the y i s are linear, as is the lower bound we obtain. Thus we can write down an LP. To find the y i s giving the best lower bound. This is the dual LP.

4 [ ] In more compact notation: Primal [ ] Dual Then integral solutions to VC-Dual-LP correspond exactly to matching s in the input graph. Thus, by weak duality, we conclude that the minimum vertex cover in an unweighted instance is at least the size of the maximum cardinality matching in the input graph. That is, in nondecreasing order of cost, we have the maximum cardinality matching (equal to VC-Dual-IP-OPT), VC-Dual-LP-OPT, VC-LP-OPT, and finally VC-IP-OPT (equal to the Vertex Cover OPT). By now we have covered everything you need to know to mechanically find the dual of any LP. However, we start with a covering linear program, P, like the primal above. First, assign a variable y i to each constraint in p (excluding the x j 0 constraints). Writing down the objective, max i y i b i, is easy. The only tricky part is the dual constraints, y T A c T. for now, let us fix a coordinate of c, say j, and figure out the constraint of the form ( ) c j in the dual. Note y T A is an row vector whose j th coordinate is the dot product of y and the j th column of A, which we will denote by a j c. this column contains the coefficients for variable x j. thus we get the constraint y T a j c c j. Lastly, force the y i s to be non-negative. To make this concrete, consider the natural LP for vertex cover. Here, we are given an undirected graph G=(V,E) whose edges are sets of vertices, each of size two. We associate a variable x v with each vertex, and interpret x v =1 as including v in the solution. Enhanced Primal-dual vertex cover algorithm (EPDVC):- Input: undirected graph G= (V, E) and vertex costs c. Initialize x =0, y =0 While E Ǿ Select nonempty E E arbitrarily. Raise y e E uniformly until some dual constraint goes tight. Let S be the set of vertices corresponding to dual constraints that just went tight. Set x v = 1 for each v S, and delete all edges incident on vertices in S from E. Output (x, y) and buy vertex set A= {v x v =1} We assign dual variable y e to the constraint v e x v 1. Since b e =1 for all e, the dual objective is max e y e. now consider the dual constraint corresponding to v, and a e,v =0 otherwise, we can write this constraint as e (v) y e c v. the final result is Before we get to primal-dual algorithms, observe that strong duality is useful as a min-max relation. In fact many min-max relations can be proven from it relatively easily. For example, Von Neumann s minimax theorem follows easily from it. The max-flow/min-cut theorem also falls right out of LP duality, if you realize that the natural maxflow LP is dual to the natural min-cut LP and the min-cut LP has integral basic feasible solutions. We will prove another min-max relation using the Weak Duality Theorem and the dual LP for Vertex Cover given above. Suppose we are given an unweighted vertex cover instance, so that cv = 1 for all v. IV. NUMERICAL RESULTS Here, we report some experimental results to illustrate the performance of the proposed approach. The experiments were performed under Windows XP and MATLAB (R2010b) running on a desktop machine with an Intel Core 2 Duo central processing unit (CPU; E6750) at 2.67 GHz and 3 GB of memory. The quality of a restored image is assessed using the signal-to-noise-ratio (SNR) defined as Where f true and f are the original and restored images respectively. We consider the original clean images shown in Fig. 1(a & b). Fig (a) shows the original image while fig (b) shows the inpainted image. They are corrupted by an additive white Gaussian noise [13] with standard deviation σ=10. And fig (2) shows the restored image with the proposed algorithm & we prove the performance evaluation of the proposed method i.e. with the EPDVC algorithm. 552

5 In order to reduce the size of an image, the JPEG 2000 standard can reduce the resolution automatically through its multiresolution decomposition structure. The wavelet coefficients [7] of all high frequency sub bands are discarded, and only that of low-frequency sub bands are kept. In this scheme, we obtain a low-resolution image. The reconstruction of the original image from this lowresolution image is reported in the third experiment Fig (a): original image Fig (3) Comparison of proposed and previous algorithm for evolution of the SNR against the CPU time for the above images Fig (b): inpainted image Fig [1(a & b)] Fig (a): noisy image Fig (b): restored image Fig [2(a & b)] 553 V. CONCLUSION We have presented an efficient EPDVC algorithm in which it produce an output within an affordable time unlike a primal dual method for total variation based wavelet domain inpainting where it produces output, by taking large number of iterations, where it in turns take large time. REFERENCES [1 ] J. Carter, Dual methods for total variation-based image restoration, Ph.D. dissertation, Dept. Math., UCLA, Los Angeles, CA, [2 ] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Image. Vis., vol. 20, no. 1/2, pp , Jan [3 ] T. Chan, G. Glob, and P. Mullet, A nonlinear primal dual method for total variation-based image restoration, SIAM J. Sci. Comput., vol. 20, no. 6, pp , [4 ] Y. Huang, M. Ng, and Y. Wen, Fast image restoration methods for impulse and Gaussian noise removal, IEEE Signal Process. Lett., vol. 16, no. 6, pp , Jun [5 ] L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, vol. 60, no. 1 4, pp , Nov [6 ] M. Zhu and T. Chan, An efficient primal dual hybrid gradient algorithm for total variation image restoration UCLA, Los Angeles, CA, CAM Rep , [7 ] Shubhangi,N.Ghate, algorithm for total variation image inpainting International Journal of Computer and Electronics Research [Volume 1, Issue 3, October 2012].

6 [8 ] An introduction to total variation for image analysis, A. Chambolle, V. Ceaseless. [9 ] a primal dual method for total variation based wavelet domain inpainting, You-Wei Wen, Raymond H. Chan, and Andy M. Yip, ieee transactions on image processing, vol. 21, no. 1, January [10 ] L.I. Rudin, S. Osher, and E. Fatemi. \Nonlinear total variation based noise removal algorithms." Physica D, vol. 60, p. 259{268, 1992}. [11 ] Danielgloving, approximations algorithms, Nov. 21, [12 ] D.Ruikar,Nalawade, inpaintingusingwavelet,transform,internationl journal of engineering and technology, eissn [13 ] S. esakkirajan, digital image processing, TMH, ISBN: , volume-1. APPENDIX Theorem (1): Let x, y be vectors output by the algorithm above. Then x is primal feasible, and y is dual feasible. Proof: Each edge deleted from E is incident on some vertex v such that x v = 1. The algorithm only terminates when every edge has been deleted. Thus v ex v 1 for all e E, and x is feasible. As for y, no constraint is violated, since once a constraint goes tight, the edges in that constraint are deleted and thus their ye values are not raised any further. Theorem (2): Let x, y be vectors output by the algorithm above. Then c T x 2.(y T.ī). Proof: let A={v x v =1}. then ( ) The second line follows from the fact that we set x v =1 only for vertices v corresponding to tight dual constraints. That is, v A implies e (v) y e =c v. The third line is simply switching the order of summation, and the last line follows from the fact that e = 2 for all e. 554