Two Enhanced Fourth Order Diffusion Models for Image Denoising

This paper presents two new higher order diffusion models for removing noise from images. The models employ fractional derivatives and are modifications of an existing fourth order partial differential equation (PDE) model which was developed by You and Kaveh as a generalization of the well-known second order Perona-Malik equation. The modifications serve to cure the ill-posedness of the You-Kaveh model without sacrificing performance. Also proposed in this paper is a simple smoothing technique which can be used in numerical experiments to improve denoising and reduce processing time. Numerical experiments are shown for comparison.

[1]  G. W. Wei,et al.  Generalized Perona-Malik equation for image restoration , 1999, IEEE Signal Processing Letters.

[2]  P. Guidotti.  A new nonlocal nonlinear diffusion of image processing , 2009 .

[3]  Joachim Weickert,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Properties of Higher Order Nonlinear Diffusion Filtering Properties of Higher Order Nonlinear Diffusion Filtering , 2022 .

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

[5]  Patrick Guidotti,et al.  Well-posedness for a class of fourth order diffusions for image processing , 2011 .

[6]  Luis Rueda,et al.  Advances in Image and Video Technology, Second Pacific Rim Symposium, PSIVT 2007, Santiago, Chile, December 17-19, 2007, Proceedings , 2007, PSIVT.

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

[8]  Atsushi Imiya,et al.  Regularity and Scale-Space Properties of Fractional High Order Linear Filtering , 2005, Scale-Space.

[9]  Herbert Amann,et al.  Time-delayed Perona-Malik type problems , 2007 .

[10]  Jian Bai,et al.  Fractional-Order Anisotropic Diffusion for Image Denoising , 2007, IEEE Transactions on Image Processing.

[11]  Yunmei Chen,et al.  On the Incorporation of Time-delay Regularization into Curvature-based Diffusion , 2001, Journal of Mathematical Imaging and Vision.

[12]  Joachim Weickert,et al.  Scale Space and PDE Methods in Computer Vision, 5th International Conference, Scale-Space 2005, Hofgeismar, Germany, April 7-9, 2005, Proceedings , 2005, Scale-Space.

[13]  Tony F. Chan,et al.  A fourth order dual method for staircase reduction in texture extraction and image restoration problems , 2010, 2010 IEEE International Conference on Image Processing.

[14]  Abdelmounim Belahmidi,et al.  Equations aux dérivées partielles appliquées à la restauration et à l'agrandissement des images , 2003 .

[15]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[16]  Joachim Weickert,et al.  Stability and Local Feature Enhancement of Higher Order Nonlinear Diffusion Filtering , 2005, DAGM-Symposium.

[17]  Mark Nitzberg,et al.  Nonlinear Image Filtering with Edge and Corner Enhancement , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Knut-Andreas Lie,et al.  Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings , 2009, SSVM.

[19]  Alain Oustaloup,et al.  Fractional differentiation for edge detection , 2003, Signal Process..

[20]  Wenyuan Xu,et al.  Behavioral analysis of anisotropic diffusion in image processing , 1996, IEEE Trans. Image Process..

[21]  Jean-Michel Morel,et al.  A Review of Image Denoising Algorithms, with a New One , 2005, Multiscale Model. Simul..

[22]  Mohammad Reza Hajiaboli.  An Anisotropic Fourth-Order Partial Differential Equation for Noise Removal , 2009, SSVM.

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

[24]  Chaomin Shen,et al.  Image restoration combining a total variational filter and a fourth-order filter , 2007, J. Vis. Commun. Image Represent..

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

[26]  Patrick Guidotti,et al.  A new well-posed nonlinear nonlocal diffusion , 2010 .

[27]  Mostafa Kaveh,et al.  Fourth-order partial differential equations for noise removal , 2000, IEEE Trans. Image Process..

[28]  Antonin Chambolle,et al.  Time-delay regularization of anisotropic diffusion and image processing , 2005 .

[29]  Satyanad Kichenassamy,et al.  The Perona-Malik Paradox , 1997, SIAM J. Appl. Math..

[30]  James V. Lambers,et al.  Enhancement of Krylov Subspace Spectral Methods by Block Lanczos Iteration , 2008 .

[31]  A. Bertozzi,et al.  Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes , 2004 .

[32]  James V. Lambers,et al.  Two New Nonlinear Nonlocal Diffusions for Noise Reduction , 2008, Journal of Mathematical Imaging and Vision.

[33]  Mohammad Reza Hajiaboli.  A Self-governing Hybrid Model for Noise Removal , 2009, PSIVT.

[34]  Greg Turk,et al.  LCIS: a boundary hierarchy for detail-preserving contrast reduction , 1999, SIGGRAPH.

[35]  Arvid Lundervold,et al.  Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time , 2003, IEEE Trans. Image Process..

[36]  Georges-Henri Cottet,et al.  A Volterra type model for image processing , 1998, IEEE Trans. Image Process..