This Article 
 Bibliographic References 
 Add to: 
Image Enhancement and Denoising by Complex Diffusion Processes
August 2004 (vol. 26 no. 8)
pp. 1020-1036

Abstract—The linear and nonlinear scale spaces, generated by the inherently real-valued diffusion equation, are generalized to complex diffusion processes, by incorporating the free Schrödinger equation. A fundamental solution for the linear case of the complex diffusion equation is developed. Analysis of its behavior shows that the generalized diffusion process combines properties of both forward and inverse diffusion. We prove that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, we develop two examples of nonlinear complex processes, useful in image processing: a regularized shock filter for image enhancement and a ramp preserving denoising process.

[1] L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, Axioms and Fundamental Equations of Image Processing, Archives of Rational and Mechanical Analysis, vol. 123, no. 3, pp. 199-257, 1993.
[2] L. Alvarez and L. Mazorra, Signal and Image Restoration Using Shock Filters and Anisotropic Diffusion SIAM J. Numerical Analysis, vol. 31, no. 2, pp. 590-605, 1994
[3] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing Applied Math. Sciences 147. New York: Springer Verlag, 2002.
[4] F. Barbaresco, Calcul des Variations et Analyse Spectrale: Equations de Fourier et de Burgers pour Modeles Autoregressifs Regularises Traitement du Signal, vol. 17, nos. 5/6, 2000.
[5] P.J. Burt and E.H. Adelson, “The Laplacian Pyramid as a Compact Image Code,” IEEE Trans. Comm., vol. 31, no. 4, pp. 532-540, 1983.
[6] O. Coulon and S.R. Arridge, Dual Echo MR Image Processing Using Multi-Spectral Probabilistic Diffusion Coupled with Shock Filters Proc. MIUA '2000, British Conf. Medical Image Understanding and Analysis, 2000.
[7] M.C. Cross and P.C. Hohenberg, Pattern Formation Outside of Equilibrium Rev. of Modern Physics, vol. 65, pp. 854-1090, 1993.
[8] D. Gabor, Theory of Communication J. Inst. of Electrical Eng., vol. 93, no. 3, pp. 429-457, 1946.
[9] G. Gilboa, Y.Y. Zeevi, and N. Sochen, Complex Diffusion Processes for Image Filtering Proc. Conf. Scale-Space 2001, pp. 299-307, 2001.
[10] G. Gilboa, Y.Y. Zeevi, and N. Sochen, Image Enhancement and Denoising by Complex Diffusion Processes CCIT Report 404, Technion, Israel, Nov. 2002.
[11] G. Gilboa, N. Sochen, and Y.Y. Zeevi, Regularized Shock Filters and Complex Diffusion Proc. European Conf. Computer Vision (ECCV '02), pp. 399-313, 2002.
[12] G. Gilboa, N. Sochen, and Y.Y. Zeevi, Image Enhancement Segmentation and Denoising by Time Dependent Nonlinear Diffusion Processes Proc. Int'l Conf. Image Processing (ICIP) 2001, vol. 3, pp. 134-137, 2001.
[13] G. Gilboa, N. Sochen, and Y.Y. Zeevi, A Forward-and-Backward Diffusion Process for Adaptive Image Enhancement and Denoising IEEE Trans. Image Processing, vol. 11, no. 7, pp. 689-703, 2002.
[14] G. Gilboa, filtering.html , 2003.
[15] V.L. Ginzburg and L.D. Landau,, Zh. Eksp. Teor. Fiz. 20, 1064, 1950; English translation: see Men of Physics: Landau, vol. II, edited by D. ter Haar, Pergamon, New York, pp. 546-568, 1965.
[16] P. Goupillaud, A. Grossmann, and J. Morlet, Cycle-Octave and Related Transforms in Seismic Signal Analysis Geoexploration, vol. 23 pp. 85-102, 1984-1985.
[17] A. Grossmann and J. Morlet, Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape SIAM J. Math. Analysis, vol. 15, pp. 723-736, 1984.
[18] F. Guichard and J-M. Morel, Image Analysis and PDE's, (new book in preparation).
[19] R. Kimmel, R. Malladi, and N. Sochen, Images as Embedding Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images Int'l J. Computer Vision, vol. 39, no. 2, pp. 111-129, Sept. 2000.
[20] J.J. Koenderink, The Structure of Images Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[21] P. Kornprobst, R. Deriche, and G. Aubert, Image Coupling, Restoration and Enhancement via PDE's Proc. Int'l Conf. Image Processing, pp. 458-461, 1997.
[22] T. Lindeberg and B. ter Haar Romeny, Linear Scale-Space: (I) Basic Theory and (II) Early Visual Operations Geometry-Driven Diffusion, pp. 1-77, 1994.
[23] P. Maragos and F. Meyer, Nonlinear PDEs and Numerical Algorithms for Modeling Levelings and Reconstruction Filters Space-Scale Theories in Computer Vision, vol. 1682, pp. 363-374, 1999.
[24] D. Marr, Vision. Freeman&Co., 1982.
[25] M. Nagasawa, Schrödinger Equations and Diffusion Theory Monographs in Math., vol. 86, 1993.
[26] A.C. Newell, Envelope Equations Lect. in Applied Math. 15, pp. 157-163, 1974.
[27] S.J. Osher and L.I. Rudin, Feature-Oriented Image Enhancement Using Shock Filters SIAM J. Numerical Analysis, vol. 27, pp. 919-940, 1990.
[28] P. Perona and J. Malik, "Scale-Space and Edge Detection Using Anisotropic Diffusion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629639, July 1990.
[29] A.P. Prudnikov, Y.A. Brychkov, and O.I. Marichev, Integrals and Series (English translation by N.M. Queen), vol. 1, 1986.
[30] B.M. ter Haar Romeny, Geometry Driven Diffusion in Computer Vision. Kluwer Academic Publishers, 1994.
[31] F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina. Holden-Day, 1965.
[32] P. Rosenau, Free Energy Functionals at the High Gradient Limit Physical Rev. A, vol. 41, pp. 2227-2230, 1990.
[33] N. Rougon and F. Preteux, Controlled Anisotropic Diffusion Proc. SPIE Conf. Nonlinear Image Processing VI IS&T/ SPIE Symp. Electronic Imaging, Science, and Technology '95, vol. 2424, pp. 329-340, 1995.
[34] L. Rudin, S. Osher, and E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms Physica D, vol. 60, pp. 259-268, 1992.
[35] R. Whitaker and G. Gerig, Vector-Valued Diffusion Geometry-Driven Diffusion, pp. 93-134, 1994.
[36] A.P. Witkin, Scale Space Filtering Proc. Int'l Joint Conf. Artificial Intelligence, pp. 1,019-1,023, 1983.
[37] M. Zibulski and Y.Y. Zeevi, Analysis of Multi-Window Gabor-Type Schemes by Frame Methods J. Applied and Computational Harmonic Analysis, vol. 4, pp. 188-221, 1997.
[38] A. Spira, R. Kimmel, and N. Sochen, Efficient Beltrami Flow Using a Short-Time Kernel Proc. Fourth Int'l Conf. Scale-Space Methods in Computer Vision 2003, vol. 2695, pp. 511-522, 2003.

Index Terms:
Scale-space, image filtering, image denoising, image enhancement, nonlinear diffusion, complex diffusion, edge detection, shock filters.
Guy Gilboa, Nir Sochen, Yehoshua Y. Zeevi, "Image Enhancement and Denoising by Complex Diffusion Processes," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 8, pp. 1020-1036, Aug. 2004, doi:10.1109/TPAMI.2004.47
Usage of this product signifies your acceptance of the Terms of Use.