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 the total variation (TV) method for denoising im- ages. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Com- mon approaches in combining wavelet-like repre- sentations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts af- ter obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive to- tal variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demon- strate that these schemes are highly effective at denoising complex images and outperform a re- lated method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competi- tors. Index Terms—Shearlets, curvelets, total varia- tion, diffusion, regularization, denoising. I. Introduction Restoring images contaminated by measure- ment errors that cause noise is an important problem in signal processing. Common power- ful techniques for image denoising are based on wavelets as well as on total variation (TV) and diffusion. By relying on certain smoothness assumptions, wavelet theory can be used to provide an effective way to denoise image. For example, if the image is assumed to be a function of class C 2 (R 2 ) away from a C 2 edge (namely, a composite of a C 2 function plus an indicator function of a set whose boundary is C 2 ), then the nonlinear approxima- tion of f consisting of the N largest wavelet co- efficients has error rate O(N −1 ). Thus, a good approximation can be obtained from some of the System Planning Corporation 1000 Wilson Boulevard, Arlington, VA 22209, USA (geasley@sysplan.com) North Carolina State University, Campus Box 8205, Raleigh, NC 27695, USA (dlabate@scsu.edu) George Mason University, 4400 University Drive, Fair- fax, VA 22030, USA (fcolonna@gmu.edu) EDICS: 2-REST, 2-WAVP. largest wavelet coefficients and a denoised esti- mate of the image can be made by removing the wavelet coefficients whose absolute value is below a specified noise level [1]. This approach, how- ever, often leads to the formation of Gibbs-type (or ringing) artifacts around sharp discontinu- ities, due to the elimination of small wavelet coef- ficients that should have been retained. In addi- tion, this technique as well as other sophisticated wavelet coefficient reduction schemes (e.g. [2]) do not necessarily remove all high-noise values (outliers). Although new wavelet extensions such as curvelets [3], [4], [5], [6] (which inspired the source of many of these extensions) and shear- lets [7], [8] have a better approximation rate, they may also suffer from the same types of effects. TV and diffusion-based methods are other powerful tools for denoising and greatly reduce these ringing effects. It is generally understood that they have superior denoising performance when applied to simple classes of images with no textures, such as images of conic shapes with flat colors. These methods, however, often pro- duce approximations that are reminiscent of oil- paintings when applied to images that contain complex textures and shading. To improve upon these methods, combinations of these routines have been proposed (e.g. [9], [10], [11], [12], [13]). The main goal of these methods was to reduce the the Gibbs-type ring- ing by adding a constraint on the non-retained coefficients. In an opposite approach, wavelet- inspired concepts were used in [14] to improve the performance and computational efficiency of TV-based methods. Other PDE-based methods influenced by concepts from wavelet theory have been developed in [15], [16], and [17]. In this article, we propose a method for de- noising images based on combining the new tight frame of shearlets with TV techniques. A key feature is that the discrete shearlet transform has many flexible attributes that lead to bet- ter stability and reduced Gibbs-type artifacts. A closely related approach in [19] suggested com-