Variable exponent functionals in image restoration

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Abstract

We study a functional with variable exponent, 1<p(x)2, which provides a model for image denoising and restoration. Here p(x) is defined by the gradient information in the observed image. The diffusion derived from the proposed model is between total variation based regularization and Gaussian smoothing. The diffusion speed of the corresponding heat equation is tuned by the variable exponent p(x). The minimization problem and its associated flow in a weakened formulation are discussed. The existence, uniqueness, stability and long-time behavior of the proposed model are established in the variable exponent functional space W1,p(x). Experimental results illustrate the effectiveness of the model in image restoration.

Introduction

Image denoising is one of the fundamental problems in image processing with numerous applications. The aim of image denoising is to design methods which can selectively smooth a noisy image without losing significant features such as edges.

Variational denoising methods are widely studied numerically and theoretically in recent years. In variational framework, the denoising problem can be expressed as follows: given an original image f, it is assumed that it has been corrupted by some additive noise n. Then the problem is to recover the true image u fromf=u+n.Let us consider the following representative minimization problemminE(u)=Ω|u|pdx+λ2Ω(u-f)2dx,where 1p2 is a constant and λ is a scalar parameter. The first term in the energy functional of (1.1) is a regularization term and the second term is a fidelity term. As p=1, it is the widely used Rudin–Osher–Fatemi (ROF) model proposed in 1992 [12]. The considerable advantage of the ROF model is that it can well preserve edge sharpness and location while smooth out noise. Mathematically, it is reasonable since its solution belongs to bounded variation (BV) space which allows discontinuities in functions. However, the ROF model favors solutions that are piecewise constant which often causes the staircasing effect [11], [14], [15]. The staircasing effect creates false edges which are misleading and not satisfactory in visual effects.

Choosing p=2 in (1.1) results in isotropic diffusion which solves the staircasing effect problem but it oversmoothes images such that the edges are blurred and dislocated. A fixed value of 1<p<2 results in anisotropic diffusion between the ROF model and the isotropic smoothing. However, there is a trade-off between piecewise smooth regions reconstruction and edge preservation.

Since different values of p should have different advantages, it encourages one to combine their benefits with a variable exponent. Blomgren et al. proposed the following minimization problem in [1]min{E(u)=Ω|u|p(|u|)dx},where lims0p(s)=2,limsp(s)=1, and p is a monotonically decreasing function. This model is a variable exponent model. It chooses diffusion speed through exponent and then can reduce the staircasing effect. Since p depends on u, it is hard to establish the lower semi-continuity of the energy functional. Bollt et al. proved that this problem with an L1 or L2 norm fidelity term has a minimizer in [2], however, nothing about the associated heat equations was discussed.

Later, Chen et al. proposed the following model in [3]minuBV(Ω)L2(Ω)E(u)=Ωφ(x,Du)+λ2Ω(u-f)2dx,whereφ(x,r)=1q(x)|r|q(x),|r|β,|r|-βq(x)-βq(x)q(x),|r|>β,q(x)=1+11+k|Gσf(x)|,Gσ(x)=12πσexp|x|22σ2 is the Gaussian kernel, k>0,σ>0 are fixed parameter, and β is a user-defined threshold. Mathematically, the energy minimization problem and the associated heat flow were discussed.

Inspired by the above models, we propose the modelminuW1,p(x)(Ω)L2(Ω)Ep(x)(u)=Ω1p(x)|u|p(x)dx+λ2Ω(u-f)2dx,where p(x)=1+g(x) and g(x)=11+k|Gσf(x)|.

Clearly in the regions with edges, g0 since the image gradient is large, model (1.4) approximates the ROF model, so the edges will be preserved; In relatively smooth regions g1 since image gradient is small, model (1.4) approximates isotropic smoothing, so they will be processed into piecewise smooth regions. In other regions, the diffusion is properly adjusted by the function p(x).

The proposed model (1.4) is simpler than (1.3) in the formulation. Meanwhile, model (1.4) is more automatic than (1.3) since no user-defined threshold β is needed in (1.4). Chen et al. studied problem (1.3) in BV framework [3], however, in this paper we will study problem (1.4) in the variable exponent space W1,p(x).

The paper is organized as follows: in Section 2 we give some important lemmas and then prove the existence and uniqueness of the solution of the minimization problem (1.4). In Section 3 we prove the existence, uniqueness and stability of the solution of the heat flow problem and discuss the long-time behavior. In Section 4 we provide our numerical algorithm and experimental results to illustrate the effectiveness of our model in image restoration. Finally, we conclude the paper in Section 5.

Section snippets

The minimization problem

Let ΩRn be a bounded open set with Lipschitz boundary, fL(Ω). By the definition of g(x) and Gaussian convolution, we obtain GσfC(Ω). Then there exists a constant M>0, such that |Gσf|M. Therefore, g(x)11+M2 and p(x)1+11+M2>1. Meanwhile, since g(x)1, we get 1<p(x)2 in the proposed model (1.4).

Variable exponent spaces. Let p(x):Ω[1,+) be a measurable function, called variable exponent on Ω. By P(Ω) we denote the family of all measurable functions on Ω. Let p-essinfΩp(x),p+esssupΩp

The associated heat flow to problem (1.4)

Using the steepest descent method, the associated heat flow to problem (1.4) is given byut=div(|u|p(x)-2u)-λ(u-f),(x,t)ΩT,uN=0,(x,t)ΩT,u(0)=f,(x,t)Ω×{t=0}.Firstly, we derive another definition of weak solution of problem (3.1), (3.2), (3.3). DenoteF(u,u,x)=1p(x)|u|p(x)+λ2(u-f)2.Then (3.1) is equivalent to ut=-F(u,u,x), where F(u,u,x) denotes the Gateaux derivative of F about u.

Suppose u be a classical solution of (3.1), (3.2), (3.3). For each vL2(0,T;W1,p(x)(Ω)L2(Ω)),

Numerical results

We consider dimension n=2. Suppose the image size is N×N. Set τ be the time step and h=1 be the space step. Let xi=ih,yj=jh,i,j=0,1,,N,tn=nτ,n=0,1,,ui,jn=u(xi,yj,tn),uij0=f(xi,yj). Define(Dx±u)i,j=±[ui±1,j-ui,j],(Dy±u)i,j=±[ui,j±1-ui,j],|(Dxu)i,j|=(Dx+(ui,j))2+(m[Dy+(ui,j),Dy-(ui,j)])2+0.001,|(Dyu)i,j|=(Dy+(ui,j))2+(m[Dx+(ui,j),Dx-(ui,j)])2+0.001,where m[a,b]=signa+signb2·min(|a|,|b|). Then the finite difference scheme of the heat flow (3.1), (3.2), (3.3) is given byuk+1=uk+τDx-Dx+uk|Dxuk|1-g+

Conclusion

In this paper, we have studied a variational exponent (1<p(x)2) functional to recover images based on the models (1.2), (1.3). The significant difference between our model and (1.3) is that in our model (1.4) p(x) can approximate 1 (but larger than 1) while in (1.3) p(x) will be equal to 1 in regions with large gradient. However, theoretically, the two models are discussed in different spaces. (1.3) is studied in BV space while (1.4) is studied in variable exponent Sobolev space W1,p(x).

The

Acknowledgements

This work is partially supported by the National Science Foundation of Shanghai (10ZR1410200), the National Science Foundation of China (10901104, 10871126) and the Research Fund for the Doctoral Program of Higher Education (200802691037).

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