An Alternative Variational Framework for Image Denoising

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method.

[1]  L. Vese.  Problemes variationnels et edp pour l'analyse d'images et l'evolution de courbes , 1996 .

[2]  Tony F. Chan,et al.  Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diiusion in Image Processing , 1996 .

[3]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[4]  Stanley Osher,et al.  Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal , 2000, SIAM J. Sci. Comput..

[5]  B. Dacorogna.  Introduction to the calculus of variations , 2004 .

[6]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[7]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[8]  Tony F. Chan,et al.  Aspects of Total Variation Regularized L[sup 1] Function Approximation , 2005, SIAM J. Appl. Math..

[9]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[10]  Qiang Liu,et al.  On a reaction-diffusion system applied to image decomposition and restoration , 2011, Math. Comput. Model..

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

[12]  Boying Wu,et al.  Adaptive Perona–Malik Model Based on the Variable Exponent for Image Denoising , 2012, IEEE Transactions on Image Processing.

[13]  J. Koenderink.  The structure of images , 2004, Biological Cybernetics.

[14]  Scott T. Acton,et al.  Speckle reducing anisotropic diffusion , 2002, IEEE Trans. Image Process..

[15]  Mohamed-Jalal Fadili,et al.  Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients , 2008, Journal of Mathematical Imaging and Vision.

[16]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[17]  L. Vese.  A Study in the BV Space of a Denoising—Deblurring Variational Problem , 2001 .

[18]  Hore,et al.  [IEEE 2010 20th International Conference on Pattern Recognition (ICPR) - Istanbul, Turkey (2010.08.23-2010.08.26)] 2010 20th International Conference on Pattern Recognition - Image Quality Metrics: PSNR vs. SSIM , 2010 .

[19]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[20]  Tony F. Chan,et al.  High-Order Total Variation-Based Image Restoration , 2000, SIAM J. Sci. Comput..

[21]  Yunmei Chen,et al.  Variable Exponent, Linear Growth Functionals in Image Restoration , 2006, SIAM J. Appl. Math..

[22]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[23]  Oscar Firschein,et al.  Readings in computer vision: issues, problems, principles, and paradigms , 1987 .

[24]  Djemel Ziou,et al.  Image Quality Metrics: PSNR vs. SSIM , 2010, 2010 20th International Conference on Pattern Recognition.

[25]  L. Evans,et al.  Various Properties of Solutions of the Infinity-Laplacian Equation , 2005 .

[26]  Liang Xiao,et al.  Image Variational Denoising Using Gradient Fidelity on Curvelet Shrinkage , 2010, EURASIP J. Adv. Signal Process..

[27]  Tony F. Chan,et al.  Extensions to total variation denoising , 1997, Optics & Photonics.

[28]  Zhuoqun Wu,et al.  Nonlinear Diffusion Equations , 2002 .

[29]  Existence and uniqueness of weak solutions for a generalized thin film equation , 2005 .

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

[31]  M. Fuchs,et al.  A Variational Approach to the Denoising of Images Based on Different Variants of the TV-Regularization , 2012 .

[32]  Yunmei Chen,et al.  Adaptive total variation for image restoration in BV space , 2002 .

[33]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[34]  Qiang Liu,et al.  Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal , 2007 .

[35]  Zhuoqun Wu,et al.  Elliptic & parabolic equations , 2006 .

[36]  Tony F. Chan,et al.  Mathematical Models for Local Nontexture Inpaintings , 2002, SIAM J. Appl. Math..

[37]  Xiaodong Zhou.  An evolution problem for plastic antiplanar shear , 1992 .

[38]  J. McLeod.  Nonlinear Diffusion Equations. , 1985 .

[39]  Jesús Ildefonso Díaz Díaz,et al.  Some qualitative properties for the total variation flow , 2002 .

[40]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[41]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .