A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives

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1 East Asian Journal on Applied Mathematics Vol. 8, No. 3, pp doi: /eajam August 2018 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives Donghuan Jiang, Xuejian Wang, Guangbao Xu and Jianqiang Lin College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, , China. Received 13 September 2017; Accepted (in revised version) 15 February Abstract. A new image denoising-decomposition model is proposed. It is based on the TV minimisation and fractional derivatives of various order. It is shown that the texture details are better described by fractional derivatives of order greater than 1, whereas the noise part by fractional derivative of order smaller than 1, and this effect is used in image denoising. The model is able to eliminate staircase effect and keeps the edge and texture information of the image. Various experiments confirm the efficiency of the model. AMS subject classifications: 68U10, 58E50 Key words: Image denoising, total variation model, fractional derivatives, partial differential equations. 1. Introduction It is well known that noise is always present during the image acquisition, transmission and storage, affecting the quality of the image. Therefore, very often the observed image should be processed before we will use it. The corresponding process, called image denoising, aims to reduce noise and improve the observed image. Image denoising is one of the most important tools in image processing. It suppresses noise and provides more accurate information for subsequent processing. To restore an image, various variational methods have been used, including one of the most popular total variation (TV) image denoising model[18]. The TV model is a regularising criterion for solving inverse problems. This model uses the bounded variation space and requires an assumption that ideal image has a bounded total variation and allows jumps. Therefore, the TV model is quite efficient in regularisation of images without smoothing the edges. There are various numerical algorithms developed for the TV model e.g. an iterative method[19], a primal-dual method[6, 7], projection methods[2, 5], the split Bregman method[12] and so on. On the other hand, the TV model has a number of deficiencies. Corresponding author. addresses: ¹¾¼¼¾½ ºÓÑ (D. Jiang) c 2018 Global-Science Press

2 448 D. Jiang, X. Wang, G. Xu and J. Lin The preservation of the image texture details is not satisfactory, and while working with smooth regions, it can consider noise as an edge producing the staircase effect. One way to overcome the drawbacks of this model is to introduce the adaptivity. An adaptive variant of TV model has been recently proposed by Lenzen and Berger[14], who established existence results for a class of adaptive TV regularisers in continuous setting. Another way to improve the TV model is the use of high-order derivatives. Thus Lysaker et al.[15] presented a four-order partial differential equation (PDE) denoising model (LLT model) with a relatively fast evolution of the shock signal. This model efficiently overcomes the staircase effect, restores smooth regions and preserves the texture information but the sharp jumps can not be well retained. Taking into account the deficiencies of the variational model with integer order derivatives, a fractional derivative variational denoising model has been developed by Chen et al. [8]. The Bioucas majorisation-minimisation algorithm was used to decompose complex fractional TV optimisation problems into a set of linear optimisation problems that can be solved by the conjugate gradient algorithm. Zhang et al.[20] considered an adaptive fractional-order multi-scale denoising model that uses the local variance measures and the wavelet based estimation of singularity. A new class of fractional-order anisotropic diffusion equations for noise removal was introduced by Bai and Feng[3], who used the discrete Fourier transform to construct an iterative scheme in the frequency domain. Various experimental results show that fractional-order variational models efficiently diminishes the staircase effect, preserve the image details and improve its quality. Moreover, Dong and Chen[21] discussed theoretical aspects of the total fractional order variation model and suitable approximation methods. They employed the fractional derivatives of different order in the regularisation term of the objective function and proposed a unified variational framework for noise removal cf. Ref.[10]. We note that images usually consist of a mix of edges, texture regions and a noise. To obtain an optimal denoising effect, we can use different approach to different parts of the image. Here we develop a denoising model, which combines total variation with fractional derivatives. The texture and noise parts are, respectively, represented by the fractional derivatives of the order greater and smaller than 1. This model inherits the advantages of TV and fractional derivative models, while avoiding some of their deficiencies. The rest of the paper is organised as follows. Section 2 introduces the TV model and fractional derivatives. Section 3 presents a new denoising model and a minimisation algorithm. In Section 4 we discuss the results of numerical experiments for real and synthetic images and compare various denoising models. A brief summary and conclusions are in Section Preliminaries Let u :Ω R be an ideal undistorted image and f :Ω Rthe observed version of u. The TV model can be formulated as an optimisation problem viz. one has to find the minimum

3 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives 449 min u λ 2 u f u d x, Ω where =( / x, / y) is the gradient operator,ωthe image domain andλ>0the regularisation parameter. The first term is used to keep the edge information close to original image while the second one deals with the noise. The Euler-Lagrange equation of the TV model has the form u +λ(u f)=0. u It can be solved by the gradient descent method. We note that 1/ u is the diffusion coefficient. Since at the image edge u is large, the diffusion coefficient is small, so that the diffusion along the edge is weak and the edge can be retained. For smooth parts of the image, the term u is small and the strong diffusion can efficiently remove the noise. Let us now introduce the fractional derivatives. Such derivatives play an important role in various fields of mathematics cf.[4, 9, 16]. They can be introduced in a different way, but here we will use the Riemann-Liouville definition[16, 17]. Letα 0be a fraction which lies in between two integers n 1 and n. Here u(x) is a function on[a, b] such that u(x)=0for x< a. For the function u, the left-sided fractional derivative of orderαis defined by G a Dα x u(x) := 1 d n x Γ(n α) d x a u(t) d t. (x t) α n+1 To define discrete derivative, we consider equidistant nodes with the step h in the interval [a, b] i.e x i = ih, i= 0,1,, N, where x 0 = a and x N = b. Theα-th derivative at x i is approximated by the backward fractional difference G a Dα x u(x) α u(x i ) i 1 = h α g i h α k u i k, i= 0,1,, N, k=0 where g k =( 1) k (α)(α 1)(α 2) (α k+1)/k!=( 1) k C k α is a constant. In particular, ifα=1, then G a D1 x u(x) is the usual first order derivative of u(x) at the point x i i. Let us generalise the above notion to functions of two variables. A bounded discrete image is denoted by u=(u(i, j)) 1 i,j N. Following the Riemann-Liouville definition, we define the discrete fractional gradient and the divergence by where α u=(( α u) i j ) N i,j=1 :=((Dα 1 u) i j,(d α 2 u) i j) N i,j=1, K 1 K 1 (D α 1 u) i j= ( 1) k C k α u(i k, j), (Dα 2 u) i j= ( 1) k C k αu(i, j k). k=0 According to[17], the operator D α 1 can be written as k=0 (D α 1 u)(:, j)=b u(:, j), 1 j N,

4 450 D. Jiang, X. Wang, G. Xu and J. Lin with the matrix B= N 1. N and :=( 1) k C k k α, 0 k N 1. Similar representation is available for the operator. On the other hand, the discrete fractional divergence can be defined by D α 2 where div α p=((div α p) i j ) N i,j=1,, K 1 (div α p) i j =( 1) α ( 1) k C k α [p 1(i+ k, j)+ p 2 (i, j+ k)]. k=0 Now we want to discuss the filtering properties of the fractional derivatives. Let us start with the definition of the fractional derivative using the Fourier transform {f}= {f}(ω)= If 1 is the inverse Fourier transform, then It follows that + + e iωx f(x)d x. D α f(x)= D α 1 { {f}}(x) = D α e iωx e iωx f(x)d xdω 2π = 1 2π = 1 2π + + D α e iωx e iωx f(x)d xdω + (iω) α e iωx e iωx f(x)d xd t = 1 {(iω) α {f}}(x). {D α f(x)}=(iω) α {f(x)}. In the frequency domain, the amplitude of D α f(x) is {D α f(x)} = (iω) α {f(x)}, hence one can introduce the fractional derivative filtering H by H(ω) := (iω) α = ω α.

5 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives α=0.2 α=0.4 α=0.7 α=1 α=2 1.5 H(ω) ω ÙÖ ½ Ö Ø ÓÒ Ð Ö Ú Ø Ú ÐØ Ö Ò α>0º Fig. 1 contains the fractional derivative filtering for different values of α > 0. It shows that H(ω)>1 grows for high frequencies and is bounded by 1 for low frequencies. Besides, in high frequencies it grows faster for α > 1. Therefore the edge and the textures corresponding for high frequency components can be well preserved. For low frequencies, the values of H are larger for smallerαand we use this property to detect the noise components of the image. 3. A New Model and Its Algorithm As was already mentioned, in order to obtain a quality image, we can use different approach to different parts of the image. The structure of the image can be modeled by the TV norm since it is able to preserve sharp edges. The results of[10] show that higher order fractional derivatives can eliminate the staircase effect and preserve textures. On the other hand, it is difficult to separate the noise from the image details. Taking into account the fact that the noise part can be considered as weakly high frequency components, it can be described by fractional derivative of small order. Let us consider the minimisation problem min u,v,w λ 1 u d xd y+λ 2 D α v d xd y+λ 3 D β w d xd y, f= u+ v+ w, (3.1) Ω Ω Ω where f is a noisy image,α>1, 0<β< 1 andλ 1,λ 2,λ 3 are positive parameters. To find the solution of the problem (3.1), we use a penalty method replacing (3.1) by the equation

6 452 D. Jiang, X. Wang, G. Xu and J. Lin min u,v,w λ 1 u d xd y+λ 2 D α v d xd y+λ 3 D β w d xd y+ λ Ω Ω Ω 2 u+ v+w f 2 2. (3.2) Let us note a few special cases. 1. Ifλ 2 =λ 3 = 0, then (3.2) is the classical total variational model[18]. 2. Ifλ 1 =λ 3 = 0, then (3.2) is equivalent to the fractional total variation mode[21]. 3. Ifλ 1 = 0, then (3.2) is a special case of a model in[10]. In order to find the solution of the problem (3.2) we use the alternating minimisation algorithm i.e. we have to find the solution of three auxiliary sub-problems. 1. Assume that v and w are fixed and solve the minimisation problem: Defining function F as min u λ 2 u+ v+w f 2 2 +λ 1 u d xd y. (3.3) Ω F := λ 2 (u+ v+w f)2 +λ 1 u = λ u 2 u 2 1/2 2 (u+ v+ w f)2 +λ 1 +, x y and setting p= u/ x, q= u/ y we can write F u = u+ v+ w f, F p = u/ x u, F q = u/ y. u The variational lemma[13] implies that or λ(u+ v+ w f) λ 1 x u/ x u + y u/ y u = 0, λ(u+ v+w f) λ 1 x, 1 u y u x, u = 0, y and the corresponding Euler-Lagrange equation has the form λ(u+ v+ w f) λ 1 div u = 0. u 2. Assume that u and w are fixed and solve the minimisation problem: min v λ 2 u+ v+w f 2 2 +λ 2 D α v d xd y. (3.4) Ω

7 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives 453 Taking into account the results in Ref.[1], we write the fractional Euler-Lagrange equation for (3.4) as λ(u+ v+w f) λ 2 ( 1) α div α α v = 0. α v 3. Assume that u and v are fixed and solve the minimisation problem: min w It follows from (3.5) that λ 2 u+ v+ w f 2 2 +λ 3 D β w d xd y. (3.5) Ω λ(u+ v+w f) λ 3 ( 1) β div β β w β = 0. w Now we can write the iterative algorithm for the minimisation problem (3.1). Algorithm 3.1 Solution of Minimisation Problem (3.1) Initialize: k=0, u 0 = 0, v 0 = 0, w 0 = 0; while uk u k 1 2 2> tol or vk v k 1 2 2> tol do u k 2 2 v k 2 2 u k+1 =(f v k w k )+ λ 1 λ div u k, u k v k+1 =(f u k+1 w k )+ λ 2 λ ( 1) α div α α v k, α v k w k+1 =(f u k+1 v k+1 )+ λ 3 λ ( 1)β div β β w k k=k+1. end while u+ v is the denoised image. β w k, 4. Numerical Experiments We carried out a few experiments to compare our approach with classical TV minimisation model and fractional PDE method. All numerical experiments are performed in Matlab environment on PC with 2.4 GHz Intel Core i3 and 4GB RAM. Signal to noise ratio (SNR) and peak signal to noise ratio (PSNR) are used to evaluate the denoised image quantitatively. The stop criterion tol of the algorithm is always set to In addition,λ=1, λ 1 = 0.3,λ 2 =λ 3 = 5. The parametersαandβ are used to, respectively, model the texture and the noise parts of the image. We start with the selection of the parametersαandβ. Taking into account the results presented in Fig. 1, we note that the derivatives of order 0<α<1 can be used to detect weakly high frequency components. Let us recall that the texture is a function of spatial variation in pixel intensities. It can be considered as the repeating patterns of local variations in image intensity. Our previous

8 454 D. Jiang, X. Wang, G. Xu and J. Lin analysis shows that fractional derivatives can enhance the textures nonlinearly. Therefore, fractional derivatives of orderα>1 can be used to model textures. We fix β and perform numerical experiments choosing various α to examine the denoising effect. In particular, we consider real images with a rich texture such as Girl, Lena, Boat and Barbara, adding a Gaussian white noise to them. We run the algorithm with different α to remove the noise. Optimal values ofαfor various images have been manually selected according to the test results. The parameterβ takes values 0.2,0.4,0.6 for noisy images withσ=5,10,15. Figs. 2 and 3 demonstrate the original, noisy and denoised Girl and Cameraman images with different fractionalα. In Table 1, we show the results for six different images which are obtained with different values of α and different noise levels. The experiments show that the denoising effect got better when α increases in intervals (1,1.3) or (1,1.4). The PSNR attains the maximum atα=1.3 orα=1.4. Forα>1.4, the PSNR Original image Noisy image =1.1 =1.2 =1.3 =1.4 =1.5 =1.6 ÙÖ ¾ ÒÓ Ò Ø ÖÐ Ñ σ=10º Original image Noisy image =1.1 =1.2 =1.3 =1.4 =1.5 =1.6 ÙÖ ÒÓ Ò Ø Ñ Ö Ñ Ò Ñ σ=10º

9 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives PSNR α ÙÖ Ì PSNR ÓÖ Ö Ö Ñ ÓÖ Ú Ö ÓÙ αº Ì Ð ½ ÓÑÔ Ö ÓÒ Ó PSNR Ó Ñ Û Ø Ö ÒØ ÓÖ Ö αº Image noise α=1.1 α=1.2 α=1.3 α=1.4 α=1.5 α=1.6 Girl σ = σ = σ = Boat σ = σ = σ = Peppers σ = σ = σ = Lena σ = σ = σ = Cameraman σ = σ = σ = Barbara σ = σ = σ =

10 456 D. Jiang, X. Wang, G. Xu and J. Lin decreases rapidly. The dependance of PSNR of Barbara image with noise of invariance 10 onαis shown in Fig. 4. Our analysis indicates that fractional derivatives of order smaller than 1 can balance the noise removal and preserve small scale details. In order to find suitable parameter β, we run a number of tests withβ= 0.2,0.4 and 0.7 using the image shown in Fig. 5. The results of the test for the image are presented in Fig. 6. Note thatβ= 0.2 still keeps a lot of noise in the texture part. Forβ= 0.7 there are more textures in the noisy part, whereas forβ= 0.4 we have a compromising result. Thus ifβ is close to 1, some small scale details are considered as a noise, but ifβ is close to 0, no much noise is removed. As a result, β= 0.4 seems to be the most appropriate parameter in the caseσ= 10. Fig. 7 shows the dependance on PSNR on the parameterβ for a fixedα. If there are too much noise present, thenβ has to be closer to the right end of the interval(0,1). ÙÖ Ì Ø Ø Ñ Ò Ø ÒÓ Ý Ñ Ù Ò ÒÓ Û Ø Ú Ö Ò σ=10µº =0.2 =0.4 =0.7 ÙÖ ÓÑÔÓ Ø ÓÒ Ó Ø Ø Ø Ñ º Ø ÖÓÛ ÆÓ Ë ÓÒ ÖÓÛ Ì ÜØÙÖ º

11 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives 457 In experiments we tested our approach on the images Girl, Lena, Boat and Barbara and compared it with the projection algorithm for TV minimisation model[5], TV model using split Bregman algorithm[11,12] and the fractional partial differential equation method[3]. Figs show the denoising results of Lena, Peppers and Baboon images with Gaussian noiseσ=10. Table 2 contains PSNR and SNR for all experiments mentioned here. It is easily seen that new model provides a better denoising than other methods. It maintains PSNR β ÙÖ Ì PSNR ÙÖÚ Ó Ö Ö Ñ Û Ø Ö ÒØβº (a) (b) (c) (d) (e) (f) ÙÖ ÒÓ Ò Ø Ä Ò Ñ º µ ÇÖ Ò Ð Ñ º µ ÆÓ Ý Ñ σ=10º µ ÈÖÓ Ø ÓÒ Ð ÓÖ Ø Ñ º µ Ö Ø ÓÒ Ð È Å Ø Ó º µ ËÔÐ Ø Ö Ñ Ò ÌÎ ÒÓ Ò º µ ÇÙÖ Å Ø Ó º

12 458 D. Jiang, X. Wang, G. Xu and J. Lin (a) (b) (c) (d) (e) (f) ÙÖ ÒÓ Ò Ø È ÔÔ Ö Ñ º µ ÇÖ Ò Ð Ñ º µ ÆÓ Ý Ñ σ=10º µ ÈÖÓ Ø ÓÒ Ð ÓÖ Ø Ñ º µ Ö Ø ÓÒ Ð È Å Ø Ó º µ ËÔÐ Ø Ö Ñ Ò ÌÎ ÒÓ Ò º µ ÇÙÖ Å Ø Ó º ÙÖ ½¼ ÒÓ Ò Ø ÓÓÒ Ñ º µ ÇÖ Ò Ð Ñ º µ ÆÓ Ý Ñ σ=10º µ ÈÖÓ Ø ÓÒ Ð ÓÖ Ø Ñ º µ Ö Ø ÓÒ Ð È Å Ø Ó º µ ËÔÐ Ø Ö Ñ Ò ÌÎ ÒÓ Ò º µ ÇÙÖ Å Ø Ó º ÙÖ ½½ Ì Ö Ò f u v ÓÖ Ø ÓÓÒ Ñ º ÖÓÑ Ð Ø ØÓ Ö Ø Ñ ÓÐÐ ÌÎ ÒÓ Ò Ö Ø ÓÒ Ð Ö ÒØ Ð ÕÙ Ø ÓÒ ËÔÐ Ø Ö Ñ Ò ÌÎ ÒÓ Ò ÇÙÖ ÅÓ Ðº

13 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives 459 Image Ì Ð ¾ ÓÑÔ Ö ÓÒ Ó PSNR Ò SNR ØÛ Ò ÓÙÖ Ò Ó ÒÓ Ò Ñ Ø Ó º Index Noisy Image Chambolle TV Fractional PDE Split Bregman TV New Model Lena PSNR SNR Peppers PSNR SNR boat PSNR SNR Girl PSNR SNR Barbara PSNR SNR Cameraman PSNR SNR test image PSNR SNR test image PSNR SNR Baboon PSNR SNR Ì Ð PSNR Ò SNR ÓÖ Ö Ø ÓÒ Ð Ö Ú Ø Ú ÑÓ Ð º Image index noisy image model in Ref.[13] model in Ref.[14] New model Lena PSNR SNR Peppers PSNR SNR Girl PSNR SNR Cameraman PSNR SNR the edge details and retains the texture details, especially for flat regions. Comparing it with other algorithms, we note that in most cases, PSNR and SNR of new algorithm are higher cf. Table 2. The difference f u v is presented in Fig. 11. It shows that new model has less residual than others i.e. fractional derivatives properly distinguish between the texture details and the noise. Our model was also compared to the state-of-art models of[10, 21] based on fractional derivations cf. Table 3. In the experiments we used the following sets of parameters: iter= 500,λ=2000,α=1.6 for the model[21] andα=2.2,β= 0.8,µ=0.2 for the

14 460 D. Jiang, X. Wang, G. Xu and J. Lin ÙÖ ½¾ ÓÑÔÓ Ø ÓÒ Ó Ø Ø Ñ Û Ø Ö ÒØ ÒÓ º ÖÓÑ Ð Ø ØÓ Ö Ø ÖØÓÓÒ u Ø ÜØÙÖ v Ò ÒÓ w Ô ÖØ º ÌÓÔ ÖÓÛ ÒÓ Ó ÒÚ Ö Ò 5 α=1.4 β= 0.2º Å Ð ÖÓÛ ÒÓ Ó ÒÚ Ö Ò 10 α=1.4,β= 0.4º ÓØØÓÑ ÊÓÛ ÒÓ Ó ÒÚ Ö Ò 15 α=1.4,β= 0.6º Relative error of u /Iterations Relative error of v /Iterations ÙÖ ½ Ê Ð Ø Ú ÖÖÓÖº ÒÓ Ò Ó Ø Ä Ò Ñ σ=5º

15 A Denoising-Decomposition Model Combining TV Minimisation and Fractional Derivatives 461 model[10]. Note that all fractional derivative models demonstrate a good denoising effect. In addition, the images containing different Gaussian noise have been decomposed into cartoon, texture and noisy parts. The test images having cartoon part and texture details, were well separated by our model cf. Fig. 12. In order to examine the convergence of the algorithm, Gaussian noise with zero mean and varianceσ=5was added to Lena image. The relative error of structure and texture parts for the method with the parameters α=1.3,β= 0.2 are given in Fig Conclusions and Discussion In this paper, we introduced a new image denoising and decomposition model based on TV minimisation and the fractional order derivatives. The texture details are better described by fractional derivatives of order greater than 1 and the noise part by fractional derivative of order smaller than 1. This model is able to eliminate staircase effect and keep the edge and texture information of the image. Various experiments confirm the efficiency of the model. The model demonstrates a better denoising effect than Chambolle s TV, fractional differential equation and Split Bregman TV denoising methods. The combination of TV minimisation with fractional order functionals can efficiently restore both piecewise constant parts and smooth subregions of the image. This method can be also used in other image problems, including image deblurring, image inpainting, image segmentation and so on. However, it still does not completely distinguish between noise and textures. Acknowledgements The authors would like to thank the referees for the helpful suggestions. This work started when one of us (DJ) visited Professor Haomin Zhou at Georgia Tech, whose hospitality and support is gratefully acknowledged. This work is also supported by the international cooperation project of the Shandong Provincial Education Department and partially supported by the National Natural Science Fund of China (Grants No ), the Shandong Provincial Natural Science Foundation (ZR2015AQ001) and the Qingdao Postdoctoral Research Project. References [1] O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl. 272, (2002). [2] J.F. Aujol, Some algorithms for total variation based image restoration, CMLA Preprint , (2008). [3] J. Bai and X.C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process. 16, (2007). [4] Z.B. Bai, S. Zhang, S.J. Sun and C. Yin, Monotone iterative method for fractional differential equations, Electron. J. Differential Equations 2016, 1 8 (2016).

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