Shearlet-Based Total Variation Diffusion for Denoising

We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after 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 total variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competitors.

[1]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[2]  Wotao Yin,et al.  Image Cartoon-Texture Decomposition and Feature Selection Using the Total Variation Regularized L1 Functional , 2005, VLSM.

[3]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[4]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[5]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2002, IEEE Trans. Image Process..

[6]  Jianwei Ma,et al.  Combined Complex Ridgelet Shrinkage and Total Variation Minimization , 2006, SIAM J. Sci. Comput..

[7]  I. Selesnick,et al.  Bivariate shrinkage with local variance estimation , 2002, IEEE Signal Processing Letters.

[8]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[9]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Jacques Froment,et al.  Reconstruction of Wavelet Coefficients Using Total Variation Minimization , 2002, SIAM J. Sci. Comput..

[11]  Wang-Q Lim,et al.  Optimally Sparse Image Representations using Shearlets , 2006, 2006 Fortieth Asilomar Conference on Signals, Systems and Computers.

[12]  Hamid Krim,et al.  A Shearlet Approach to Edge Analysis and Detection , 2009, IEEE Transactions on Image Processing.

[13]  Minh N. Do,et al.  Ieee Transactions on Image Processing the Contourlet Transform: an Efficient Directional Multiresolution Image Representation , 2022 .

[14]  D. Labate,et al.  Resolution of the wavefront set using continuous shearlets , 2006, math/0605375.

[15]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[16]  Wang-Q Lim,et al.  Sparse multidimensional representation using shearlets , 2005, SPIE Optics + Photonics.

[17]  N. Sochen,et al.  Texture Preserving Variational Denoising Using an Adaptive Fidelity Term , 2003 .

[18]  Jianwei Ma,et al.  Curvelets for surface characterization , 2007 .

[19]  Ronald R. Coifman,et al.  Combining the Calculus of Variations and Wavelets for Image Enhancement , 2000 .

[20]  Stphane Mallat,et al.  A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way , 2008 .

[21]  Demetrio Labate,et al.  Optimally Sparse Multidimensional Representation Using Shearlets , 2007, SIAM J. Math. Anal..

[22]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[23]  Joachim Weickert,et al.  Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising , 2005, International Journal of Computer Vision.

[24]  G. Easley,et al.  Sparse directional image representations using the discrete shearlet transform , 2008 .

[25]  Minh N. Do,et al.  The Nonsubsampled Contourlet Transform: Theory, Design, and Applications , 2006, IEEE Transactions on Image Processing.

[26]  Gerlind Plonka-Hoch,et al.  Nonlinear Regularized Reaction-Diffusion Filters for Denoising of Images With Textures , 2008, IEEE Transactions on Image Processing.

[27]  Michel Barlaud,et al.  Image coding using wavelet transform , 1992, IEEE Trans. Image Process..

[28]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[29]  Los Angeles,et al.  Dual Methods for Total Variation-Based Image , 2001 .

[30]  Eitan Tadmor,et al.  A Multiscale Image Representation Using Hierarchical (BV, L2 ) Decompositions , 2004, Multiscale Model. Simul..

[31]  Wang-Q Lim,et al.  Edge analysis and identification using the continuous shearlet transform , 2009 .

[32]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[33]  Jianwei Ma.  Characterization of textural surfaces using wave atoms , 2007 .

[34]  Y. Meyer.  Wavelets and Operators , 1993 .

[35]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[36]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[37]  M. Barlaud,et al.  Nonlinear image processing: modeling and fast algorithm for regularization with edge detection , 1995, Proceedings., International Conference on Image Processing.

[38]  Gerlind Plonka-Hoch,et al.  Combined Curvelet Shrinkage and Nonlinear Anisotropic Diffusion , 2007, IEEE Transactions on Image Processing.

[39]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .