Image Filter Using with Gaussian Curvature and Total Variation Model

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1 IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 ISSN : (Online) ISSN : (Print) Iage Using with Gaussian Curvature and Total Variation Model 1 Deepak Kuar Gour, Sanjay Kuar Shara 1, Dept. of Electronics & counication Engineering, UIT Bhopal, India Abstract Priors play an essential role in Bayesian theory in iage processing. Geoetric priors are very popular because of their physical explanation. The neighborhood structure of pixel can be described ore accurately by its curvature. In this paper, we deals with the iage restoration algoriths based on Gaussian curvature and total variation odal to achieve sooth denoising preserving the details of iage. There we show that how priors should be iposed for certain types of surfaces and how they can be iposed efficiently in a variational fraework. We first show a novel ethod that can reconstruct a closed surface fro a finite point cloud then by using total variation odal we suppress the noise. We can directly extract prior inforation fro the reconstructed surfaces. We provide paraetric odel and analyze its properties. The new odal can preserve ore details while suppress noise. The experiental results are given to deonstrate the perforance of the proposed ethod. Keywords Bayesian Theory, Gaussian Curvature (GC), Total Variation (TV) Modal, Denoising, Variational Fraework and Priors I. Introduction Digital iage processing covers any aspects. For exaple, iage basic linear transforation and filtering, iage restoration, iage encoding and copression and, iage reconstruction, digital iage, intelligent processing, and orphological processing and so on. Estiating the iage fro observed data is a fundaental task in digital iage processing. During the past few decades, denoising has been a hot field in iage processing and nuerous schees have been proposed for the task, such as variational ethods, wavelet theory, dictionary learning, copressed sensing, etc. Aong these theories, Bayesian Theore can be use to derive variational ethods. There the data I( x n ), where x Ω R is the spatial coordinates, Ω is the sapling doain, and n is the diension, now we want to estiate the signal U( x ). This can be done by Bayesian Theore: PIU ( ) PU ( ) PU ( I) = PIU ( ) PU ( ) (1) P( I) Where p( ) is the probability. By axiizing the probability p(u I) can iniize following energy EU ( ) = log( PU ( I)) () Put it in the Eq. (1), we have EU ( ) = log( PU ( I)) log( PU ( )) (3) Therefore, this sapling odel (derivating I fro U) and soe prior of U have to be assued in this fraework. In general, this links Bayesian Theore and the variational fraework: Where the double headed arrows indicate the counterparts in these two approaches. The is the data fitting energy and the is the regularization energy. Indicates how well an estiation of U fits the data I. This depends on the sapling process. The nor is coonly used because the easureent error usually satisfies Gaussian distribution, which leads to (σ is a paraeter). Another frequent choice is the 1 nor which corresponds to Laplacian noise odel. In ost of odels, is a regularization ter that iposes the prior knowledge of the U, such as Tikhonov, the nor of the gradient, syetry, gradient distribution [5, 7], Total Variation (TV) [1-, 5], Mean Curvature (MC) [5], or Gaussian Curvature (GC) [3-6]. Aong these regularizations, GC and TV are interesting because we have observed that GC filter is better in edge- preserving And TV filter is better in reoving noise. Several researchers have shown that the properties of GC[4-5, 9] and TV [1-, 5, 9]. In this paper, we show that GC and TV filter can be ipose together with soe changes in their inial projection. A. Traditional Solvers Traditional solvers for variational odels, such as gradient decent ethod, Split Bregan Iteration and Prial Dual ethods, usually require coputing the gradient of the total energy functional. Requiring the total energy to be differentiable akes iposing arbitrary iaging odels (noise, blurring, inpainting, supper resolution, scatter light, etc.) difficult. These types of ethods are suffered fro the nuerical stability requireent. Therefore, the step size in each iteration is liited. They usually need a large nuber of iterations to converge. Because of that traditional solvers are usually eory intensive, which akes the coputational expensive. They require the syste eory to be at least several ties larger than the input iage size. This leads to an issue for large iages, where required eory for traditional solvers ay be larger than the syste eory B. Motivation Besides the coplicate equation, large eory requireents, another drawback of traditional ethods is that they are coputationally slow. The ain reason is that these previous approaches start fro the total energy in Eq. (4) without considering the geoetric eaning of iniizing curvature. Another proble for previous solvers is that they are not generic. To accelerate the coputation, the ultigrid strategy is introduced. 98 International Journal of Electronics & Counication Technology

ISSN : 30-7109 (Online) ISSN : 30-9543 (Print) With the ultigrid acceleration, the GC and TV are faster than traditional ethods.

2 ISSN : (Online) ISSN : (Print) With the ultigrid acceleration, the GC and TV are faster than traditional ethods. Fro these coputational ethods and filters we have found that the GC filter use in iage processing for soothing and denoising the iage. GC filter is better in preserving details i.e. GC filter is better in edge- preserving but reduces its ability in noise suppression. And TV filter is better in reoving noise but it also reoves the details of the iage. C. Contribution To overcoe these issues, we have proposed the cobination of GC and TV, with soe odification in these two fraeworks and using these two variational odels. We have adopted the property of these two variational fraeworks and fored a new filter. We have found soe better perforance as shown in the experient section. There to iniize the regularization energy. Our ethod is inspired by the observation that regularization energy is the doinant part during the iniization. As shown in Fig. 1, the regularization energy usually decreases while the energy usually increases if the initial condition is the original iage. Since the total energy has to decrease, ust be the doinant part. Therefore, as long as the decreased aount in is larger than the increased aount in, the total energy E decreases. There are several benefits of doing so. First, we do not require the total energy to be differential. Therefore, it can handle arbitrary coplex noise odel. Second, the edges are preserved. Third, the resulting filter is siple to copute, and its physical eaning is clear. IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 Theore (7) Where K b is the boundary curvature, db a length eleent, and χ the Euler characteristic of ψ. Since total GC is a topological invariant, one can only iniize total absolute GC [5, 9]. Surfaces with zero total absolute GC are called developable. The total absolute GC variational odel is where is the GC energy and e is a given terination threshold. L is the space of square-integrable functions. B. Total Variation We can use a siilar approach to construct a filter to solve TV odels. We use the constrained ROF odel [1, 5] Our filter solver is suarized in Algoriths, which define the GC and TV filter. In the local projection operator respectively, we directly use the piecewise constancy assuption. III. Doain Decoposition Locally iniizing the saller absolute principal curvature is prevented by dependencies between neighboring pixels. We introduce here a doain decoposition algorith to defeasance these dependencies. (8) (9) We decopose the discrete doain of an iage U into two disjoint subsets, the white points W and the black points B. We further split each of these two subsets: white triangles W T, white circles W C, black triangles B T, and black circles B C. This decoposition guarantees that neighbors are in different subsets, as illustrated in Fig.. Fig. 1: Regularization is the Doinant Part in Optiization II. Gaussian Curvature and Total Variation Regularization First, we show the atheatical for of Gaussian Curvature and Total Variation regularization A. Gaussian Curvature Let x Ωdenote the spatial coordinates and I(, i j): x R + denote the given discrete digital iage with coordinates i and j. Let U( x ) denote the unknown signal, i.e., the desired output iage to be estiated. We interpret the signal as a geoetric surface over the space of the data, i.e., GC is defined as ( U( x) ) K = U U U xx yy xy ( 1+ Ux + Uy) Recall that the total GC K of any surface is related to the surface s topology through the Gauss-Bonnet theore: (6) Fig. : Illustration of Disjoint Doain Decoposition This decoposition has several benefits: First, it reoves the dependencies between neighboring pixels. For exaple, when BC needs to be updated, all pixels in BC can be updated siultaneously. Second, this is independence, the update can use the neighbors that have already been updated. This guarantees convergence. w w w. i j e c t. o r g International Journal of Electronics & Counication Technology 99

3 IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 ISSN : (Online) ISSN : (Print) Third, in a 3 3 local window, all tangent planes TS can be fored. Therefore, proxial projection can be used to ake the surface U( x ) ore developable, which eans locally reducing the Gaussian curvature. we need to project U( x) to U( x ) such that U( x ) is on the closest tangent plane of a neighboring pixel. Enueration of all Projections In order to find the tangent plane in N( x ) that has the sallest d i, we find all possible tangent in a 3 3 pixel neighborhood of x that do not include x as a vertex. There are in total 1 such tangents: six through each of the four white neighbors W, six through the four black neighbors. B. New Gauss Curvature Operator We iterate P new over all pixels in each of B T, B C, W T, and W C. Since the pixels within each set are independent, the iteration order does not atter. This yields our curvature filter, as suarized in Algoriths. It is clear that has linear coputational coplexity with respect to the total nuber of pixels. Because of the doain decoposition, all pixels in the sae set are independent of each other and the projection can be applied in parallel. This enables us to prove convergence of Algorith, and also accelerates convergence since each update is based on already updated neighbors. Algorith I Projection Operator P new Require:U(i,j) (a) (b) Fig. 3: di to the Tangent Passing Over x: (a) di for TS (W), (b) di for TS (B) Since soe of the 1 tangent triangles share coon edges over x, and projecting onto these edges is sufficient, there are only 1 different d i, six to the coon edges fro W, six to the coon edges fro B. IV. Algorith Ipleentation In this filter, we use all possible linear fors (that are inial projection) to approxiate the data and choose the inial change to update the current estiation. Since the inial projection of both new GC and TV is used and then it use to update the pixels, this filter is ore efficient in iniizing principle curvature than GC and TV filter separately, as shown in the experient section. A. Minial Projection Operator by odified Gauss operator After coputing all { d, i = 1,...1 i } we use the sallest absolute distance as the iniu projection of the current intensity U(x) to the target intensity U( x) such that U( x) is on one of the tangent planes through the neighboring pixels. More specifically, we find d such that d = in { d, i = 1,...1 i } Then, we let U( x) = U( x) + d. We denote this local update operation by Pnew. It needs iniu operations (plus, inus, divide). This operator, as suarized in Algorith 1, is thus copact and efficient. The set d = in { di, i = 1,...1} is a coplete description of the local geoetry at x. For any given U(x) and its d = in { d,i = 1,...1 i }, U( N( x )) can be obtained by solving a linear syste of equations in Algorith 1. Moreover, di is the linear curvature in the corresponding direction. Based on the Euler Theore, we have where K1, K are the principle curvatures and θi is the angle to the principle plane. Therefore, if the angular sapling θi is dense enough in [ π, π], we have d in { Ki} when KK 1 0. we use d as an approxiation of the inial absolute principle Curvature. 100 International Journal of Electronics & Counication Technology 1: : d = ( U + U )/ 1 i 1j, i+ 1j, ij, d = ( U + U )/ ij, 1 ij, + 1 ij, 3: d = ( U + U )/ 4: d = ( U + U )/ 5: d = ( U + U )/ 5 i 1j, i 1j, 1 ij, 6: d = ( U + U )/ 6 i 1j, i+ 1j, 1 ij, 7: d = ( U + U )/ 7 i+ 1j, ij, 1 ij, 8: d = ( U + U )/ 8 i+ 1j, i+ 1j, + 1 ij, 9: d = ( U + U )/ 9 i 1j, ij, 1 ij, 10 : d = ( U + U )/ U 10 i 1j, ij, + 1 ij, 11 : d = ( U + U )/ U 11 i+ 1j, ij, 1 ij, 1 : d = ( U + U )/ U find 3 i 1j, 1 i+ 1j, + 1 ij, 4 i 1j, + 1 i+ 1j, 1 ij, 1 i+ 1j, i+ 1j, + 1 ij, d, that d = in{ d, i = 1,,1} Ensure : U(, i j) = U(, i j) + d Algorith II New G new Require: U (i, j) 1. x BT, P g. x BC, P g 3. x WT, P g 4. x WC, P g 5. Ensure: U (i, j) C. Proposed Modal According to the previous discussion, as described above the Gaussian Curvature filter and Total Variation filter have soe erits and deerits. In this paper, fro their properties, we have picked up their erits as and apply the together to find batter result. There we have odified the gauss curvature operator as described above in algorith I, after that algorith we have applied the algorith of total variation odal [9]. With that, we have found iproved perforance, as shown in the experient section. V. Experients In this section, we discuss the results of our nuerical experients to define the effectiveness of the proposed ethod. In this i

ISSN : 30-7109 (Online) ISSN : 30-9543 (Print) IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 experient we select the 51x51 pixels of the Lena standard test iage.

1605 For the coparison of perforance and to verify the effectiveness and reliability of the recovery of the proposed filter, we perfored on PC (Intel(R) Core(TM) @ 3.00GHz,.

It is well-known that the GC filter is better in preserving details, especially for the low value of noise and that TV filter is better in reoving noise, it can reove both lower and higher valued

4 ISSN : (Online) ISSN : (Print) IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 experient we select the 51x51 pixels of the Lena standard test iage. It was degraded with Salt and Pepper Noise of 5, 10and 30 % density. As Shown in Fig. 5 and Table 1. Proposed 5% % % For the coparison of perforance and to verify the effectiveness and reliability of the recovery of the proposed filter, we perfored on PC (Intel(R) 3.00GHz,.0 GB eory) with Matlab siulation software 015b algorith prograing. We investigated Gaussian Curvature (GC) and Total Variation (TV) filter. It is well-known that the GC filter is better in preserving details, especially for the low value of noise and that TV filter is better in reoving noise, it can reove both lower and higher valued noise. For a fair coparison we have used sae nuber of iterations i.e. 60. To evaluate perforance of the noise detection, we have used three level of noise density. The results of noise detection for salt and pepper noise are shown in the table 1 and also the experiental output shown in fig. 5. You see in the table and fig.5 (Second row), the perforance of the Gaussian Curvature filter worsened at higher level of noise. And also see in fig. 5(third row) and table the quality filtered iage with total variation filter is worsened. Now the proposed odal of filter is better in preserving edges as well as better in reoving noise as copared with Total Variation (TV) or Gaussian Curvature (GC). As shown in Fig. 5(botto row) and table1. (a). 5% Noise Density, (b) 10% Noise density, (c)30% Noise density (d) (e) (f) The energy profiles of Gaussian Curvature filter, Total Variation filter and proposed filter are shown in the fig. 6 That indicate the regularization energy of the all given filters and their coparison. The fig. 4 shows the perforance and ability of the proposed filter. (g) (h) (i) (a) (b) (c) Fig. 4: (a) Original Iage (left), (b) ed Iage (Mid.), (c) Difference between these original Iage and filtered iage Table 1: Results and Coparison in SNR, PSNR, SSIM, MSE of the Iage Denoising Experient Salt & pepper Noise GC TV Noise Density (%) SNR PSNR SSIM MSE 5% % % % % % % % % (j) (k) (l) Fig. 5: Coparison of the Results With a Real Iage ed by GC (Second Row), TV (Third Row) and Proposed (Botto Row) fro Noisy Iages Having Different Noise Density(Top Row), Colun Wise Fro Top to Botto (a) w w w. i j e c t. o r g International Journal of Electronics & Counication Technology 101

5 IJECT Vo l. 7, Is s u e 3, Ju l y - Se p t 016 ISSN : (Online) ISSN : (Print) (b) [3] D. Firsov, S. H. Lui, Doain decoposition ethods-in iage denoising using gaussian curvature, J. Coput. Appl. Math., Vol. 193, No., pp , Septeber 006. [4] Yuanhao Gong, Ivo F. Sbalzarini, Local weighted Gaussian curvature for iage processing, Intl. Conf. Iage Proc. (ICIP), pp , Septeber 013. [5] Yuanhao Gong,"Spectrally regularized surfaces, Ph.D. thesis, ETH Zurich, Nr. 616, 015. [6] Zhang, Y. Zhang, H. Li, T. S. Huang,"Generative Bayesian iage super resolution with natural iage prior", IEEE Transactions On Iage Processing, 1, pp , 01. [7] Yuanhao Gong, I.F. Sbalzarini, A natural-scene gradient distribution prior for light-icroscopy iage processing, Selected Topics in Signal Processing, IEEE Journal of, Vol. no. 99, pp. 1 1, Dec 015. [8] Adel I. El-Fallah, Gary E. Ford, Mean curvature evolution and surface area scaling in iage filtering, IEEE Trans. Iage Proc., Vol. 6, No. 5, pp , [9] Deepak Kuar Gour, Sanjay Kuar Shara, Experiental Analysis and Coparison of Gaussian Curvature and Total Variation Model, International Journal of Science and Research (IJSR), Vol. 5 Issue 8, August 016 (c) Fig. 6: Energy Profile of (a). Gaussian Curvature (b). Total variation (c) Proposed VI. Conclusion We have proposed new algorith for iage denoising based on Gaussian curvature and total variation odal. It is a fast novel filter that is use the sae assuptions as the respective variational odels, they can iniize the regularization energy for the corresponding variational odels. The proposed filter is guarantees that not only iage edges are are well preserved or other sall-scaled structures but also reove the noise effectively at high level of noise. The filter is uch faster in both runtie and convergence. The filter is paraeter-free and easy to ipleent and parallelize. The experient results of deferent ethod are given. In future, this filter solver can be parallelized to obtained higher perforance, to increase the speed of the filtration and to perfor on the large iages. This filter can also be cobined with wavelet transfor. References [1] L. I. Rudin, S. Osher, Total variation based iage restoration with free local constraints, In Proc. 1st IEEE Int. Conf. Iage Processing, Vol. 1, 1994, pp [] L. Rudin, S. Osher, E. Fatei, Nonlinear total variation based noise reoval algoriths, Phys. D, Vol. 60, No. 1 pp , International Journal of Electronics & Counication Technology

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