Shearlet Based Total Variation for Denoising

IEEE TRANSACTIONS IMAGE PROCESSING 1 Shearlet Based Total Variation for Denoising Glenn R. Easley, Demetrio Labate, and Flavia Colonna Abstract—We propose a shearlet formulation of largest wavelet coefficients and a denoised esti- the total variation (TV) method for denoising im- mate of the image can be made by removing the ages. Shearlets have been mathematically proven to represent distributed discontinuities such as wavelet coefficients whose absolute value is below edges better than traditional wavelets and are a specified noise level [1]. This approach, how- a suitable tool for edge characterization. Com- ever, often leads to the formation of Gibbs-type mon approaches in combining wavelet-like repre- (or ringing) artifacts around sharp discontinu- sentations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts af- ities, due to the elimination of small wavelet coef- ter obtaining a nearly optimal estimate. We show ficients that should have been retained. In addi- that it is possible to obtain much better estimates tion, this technique as well as other sophisticated from a shearlet representation by constraining the residual coefficients using a projected adaptive to- wavelet coefficient reduction schemes (e.g. [2]) tal variation scheme in the shearlet domain. We do not necessarily remove all high-noise values also analyze the performance of a shearlet-based (outliers). Although new wavelet extensions such diffusion method. Numerical examples demonstrate that these schemes are highly effective at as curvelets [3], [4], [5], [6] (which inspired the denoising complex images and outperform a re- source of many of these extensions) and shear- lated method based on the use of the curvelet lets [7], [8] have a better approximation rate, they transform. Furthermore, the shearlet-TV scheme may also suffer from the same types of effects. requires far fewer iterations than similar competi- tors. TV and diffusion-based methods are other Index Terms—Shearlets, curvelets, total varia- powerful tools for denoising and greatly reduce tion, diffusion, regularization, denoising. these ringing effects. It is generally understood that they have superior denoising performance I. Introduction when applied to simple classes of images with no textures, such as images of conic shapes with Restoring images contaminated by measure- flat colors. These methods, however, often pro- ment errors that cause noise is an important duce approximations that are reminiscent of oil- problem in signal processing. Common power- paintings when applied to images that contain ful techniques for image denoising are based on complex textures and shading. wavelets as well as on total variation (TV) and To improve upon these methods, combinations diffusion. of these routines have been proposed (e.g. [9], By relying on certain smoothness assumptions, [10], [11], [12], [13]). The main goal of these wavelet theory can be used to provide an effective methods was to reduce the the Gibbs-type ring- way to denoise image. For example, if the image ing by adding a constraint on the non-retained 2 R2 is assumed to be a function of class C ( ) away coefficients. In an opposite approach, wavelet- 2 2 from a C edge (namely, a composite of a C inspired concepts were used in [14] to improve function plus an indicator function of a set whose the performance and computational efficiency of 2 boundary is C ), then the nonlinear approxima- TV-based methods. Other PDE-based methods tion of f consisting of the N largest wavelet co- 1 influenced by concepts from wavelet theory have efficients has error rate O(N − ). Thus, a good been developed in [15], [16], and [17]. approximation can be obtained from some of the In this article, we propose a method for de- System Planning Corporation 1000 Wilson Boulevard, noising images based on combining the new tight Arlington, VA 22209, USA ([email protected]) frame of shearlets with TV techniques. A key North Carolina State University, Campus Box 8205, feature is that the discrete shearlet transform Raleigh, NC 27695, USA ([email protected]) has many flexible attributes that lead to bet- George Mason University, 4400 University Drive, Fair- fax, VA 22030, USA ([email protected]) ter stability and reduced Gibbs-type artifacts. A EDICS: 2-REST, 2-WAVP. closely related approach in [19] suggested com- 2 IEEE TRANSACTIONS IMAGE PROCESSING ∂u bining the tight frame of curvelets with nonlinear with the boundary condition ∂n = 0 on ∂Ω, anisotropic diffusion. The results given in [19] where t is interpreted as an artificial time- indicate that this technique is highly effective. marching parameter [20]. Other techniques for We shall demonstrate that our method based finding the solution to the Euler-Lagrange equa- on combining shearlets with TV performs better tion include duality-based methods [21], [22], than this curvelet-based technique. Furthermore, [23]. the number of iterations is significantly reduced. The TV method described above is a special In some cases, the reduction in the number of case of the method based on minimizing the func- iterations is nearly six-fold. tional In Section II, we give a brief overview of the λ 2 TV and the nonlinear diffusion methods. The φ( u ) dxdy + (u u0) dxdy, ZΩ k∇ k 2 ZΩ − shearlet transform and its implementation are where φ C2(R) is an even regularization func- described in Section III. In Section IV, we present ∈ a new method which exploits the best features tion [24]. The solution is obtained by solving of shearlets and TV to obtain superior denois- ∂u φ0( u ) = k∇ k u λ(u u ) (1) ing capabilities. In Section V, we discuss the ex- ∂t ∇· u ∇ − − 0 perimental results of the comparison among dif- k∇ k ferent state-of-the-art techniques, and show that subjected to the Neumann boundary condition. For λ =0 and lim φ0(x)/x = 0, equation (1) is the method we propose yields significantly better x outcomes. The concluding remarks are given in a special case of→∞ the Perona and Malik diffusion Section VI. equation ∂u II. Total Variation and Diffusion = (ρ( u ) u) , ∂t ∇· k∇ k ∇ Let Ω be a bounded region in R2. The total where ρ(x) = φ (x)/x [25]. variation of a function u C1(Ω) is defined as 0 ∈ In diffusion, the auxiliary function ρ is used to control the amount of smoothing. In regions T V (u) = u dA, where the gradient u is small, which may corre- ZΩ k∇ k spond to noise or the∇ lack of an edge, the diffusion where u = ∂u , ∂u and is the standard process is strong. On the other hand, in regions ∂x1 ∂x2 ∇ k k where u is large, which are likely to correspond Euclidean norm. ∇ A common TV technique for the purpose of to the location of an edge, the diffusion process denoising is based on minimizing the functional is weak or non-existent. In this context, as we shall see, shearlets be- λ 2 have like the gradient. Indeed, shearlets can be F (u) = u dA + (u u0) dA, ZΩ k∇ k 2 ZΩ − used to detect the presence of an edge [26]. where u is the estimated image, u0 is the noisy III. Shearlet Transform image, and λ R is a penalty parameter (see + The continuous wavelet transform W pro- [20]). The associated∈ Euler-Lagrange equation is ψ vides a decomposition of a signal over dilated and u translated versions of a fixed waveform ψ. More ∇ + λ(u u )=0, 2 n −∇ · u − 0 precisely, for a fixed ψ L (R ) (n N), this is ∈ ∈ 2 n k∇ k defined as the mapping Wψ with domain L (R ) ∂u 2 Rn with the Neumann boundary condition ∂n = 0 such that for f L ( ) on ∂Ω. To improve stability, the term u is ∈ 2 k∇ k replaced by u α = u + α, where α is a Wψf(a,t) = f(x) ψa,t(x) dx, (2) k∇ k k∇ k Rn positive parameter. p Z n/2 1 A method for finding the minimizer of the where ψa,t(x) = a− ψ(a− (x t)), a > 0 and functional F is based on looking for the steady t Rn. If the function ψ satisfies− the admissibil- state solution of ity∈ condition ∂u u ∞ 2 da n = ∇ λ(u u0) ψˆ(aξ) =1 fora.e. ξ R , ∂t ∇· u − − Z | | an ∈ k∇ k 0 Shearlet-Based Total Variation for Denoising 3 then ψ is called a wavelet, and any f L2(Rn) ξ can be recovered via the reproducing formula:∈ 2 1 (a, s)=( 4 , 0) ∞ da HYH f = f, ψa,t ψa,t dt a2n . @ Z0 ZRn h i @ 1 @R (a, s) = ( 32 , 1) One of the most remarkable properties of the wavelet transform is its ability to identify the sin- ξ1 gularities of a signal. In fact, if f is a function 6 which is smooth apart from a discontinuity at a 1 point x0, the transform Wψf(a,t) will signal the (a, s) = ( 32 , 0) location of the singularity by its asymptotic de- cay at fine scales. More precisely, provided ψ is a “nice” wavelet, then Wψf(a,t) decays rapidly Fig. 1. Frequency support of the horizontal shearlets as a 0, unless t is near x0 [27]. This property (left) and vertical shearlets (right) for different values of allows→ one to resolve the singular support of f, a and s. that is, to identify the set of points where f is not regular. The generating function ψ is well localized and However, the continuous wavelet transform is satisfies appropriate admissibility conditions [28], unable to provide additional information about [29], so that for each f L2(R2), we have the geometry of the set of singularities of f.

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