Abstract
Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Euler-forward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semi-implicit schemes is studied, so-called additive operator splitting (AOS) methods. They share the advantages of explicit and (semi-)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scale-spaces. Their efficiency is due to the fact that they can be separated into one-dimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes.
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References
S.T. Acton, A.C. Bovik, M.M. Crawford, Anisotropic diffusion pyramids for image segmentation, Proc. IEEE Int. Conf. Image Processing (ICIP-94, Austin, Nov. 13–16, 1994), Vol. 3, IEEE Computer Society Press, Los Alamitos, 478–482, 1994.
L. Alvarez, Images and PDE's, M.-O. Berger, R. Deriche, I. Herlin, J. Jaffré, J.-M. Morel (Eds.), ICAOS '96: Images, wavelets and PDEs, Lecture Notes in Control and Information Sciences, Vol. 219, Springer, London, 3–14, 1996.
L.D. Cai, Some notes on repeated averaging smoothing, J. Kittler (Ed.), Pattern recognition, Lecture Notes in Comp. Science, Vol. 301, Springer, Berlin, 597–605, 1988.
F. Catté, P.-L. Lions, J.-M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., Vol. 29, 182–193, 1992.
R. Deriche, Fast algorithms for low-level vision, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 12, 78–87, 1990.
J. Fröhlich, J. Weickert, Image processing using a wavelet algorithm for nonlinear diffusion, Report No. 104, Laboratory of Technomathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany, 1994.
T. Gijbels, P. Six, L. Van Gool, F. Catthoor, H. De Man, A. Oosterlinck, A VLSI-architecture for parallel non-linear diffusion with applications in vision, Proc. IEEE Workshop on VLSI Signal Processing, Vol. 7, 398–707, 1994.
A.R. Gourlay, Implicit convolution, Image Vision Comput., Vol. 3, 15–23, 1985.
B.M. ter Haar Romeny (Ed.), Geometry-driven diffusion in computer vision, Kluwer, Dordrecht, 1994.
R.A. Hummel, Representations based on zero-crossings in scale space, Proc. IEEE Comp. Soc. Conf. Computer Vision and Pattern Recognition (CVPR '86, Miami Beach, June 22–26, 1986), IEEE Computer Society Press, Washington, 204–209, 1986.
T. Iijima, Basic theory of pattern normalization (for the case of a typical one-dimensional pattern), Bulletin of the Electrotechnical Laboratory, Vol. 26, 368–388, 1962 (in Japanese).
J. Kačur, K. Mikula, Solution of nonlinear diffusion appearing in image smoothing and edge detection, Appl. Num. Math., Vol. 17, 47–59, 1995.
T. Lindeberg, Scale-space theory in computer vision, Kluwer, Boston, 1994.
G.I. Marchuk, Splitting and alternating direction methods, P.G. Ciarlet, J.-L. Lions (Eds.), Handbook of numerical analysis, Vol. I, 197–462, 1990.
M. Nielsen, L. Florack, R. Deriche, Regularization, scale-space, and edge detection filters, B. Baxton, R. Cipolla (Eds.), Computer vision — ECCV '96, Volume II, Lecture Notes in Comp. Science, Vol. 1065, Springer, Berlin, 70–81, 1996.
W.J. Niessen, K.L. Vincken, J.A. Weickert, M.A. Viergever, Nonlinear multiscale representations for image segmentation, Computer Vision and Image Understanding, 1997, in press.
P. Perona, J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 12, 629–639, 1990.
J. Weickert, Theoretical foundations of anisotropic diffusion in image processing, W. Kropatsch, R. Klette, F. Solina (Eds.), Theoretical foundations of computer vision, Computing Suppl. 11, Springer, Wien, 221–236, 1996.
J. Weickert, Nonlinear diffusion scale-spaces: From the continuous to the discrete setting, M.-O. Berger, R. Deriche, I. Herlin, J. Jaffré, J.-M. Morel (Eds.), ICAOS '96: Images, wavelets and PDEs, Lecture Notes in Control and Information Sciences, Vol. 219, Springer, London, 111–118, 1996.
J. Weickert, Anisotropic diffusion in image processing, Teubner Verlag, Stuttgart, 1997, to appear.
J. Weickert, B.M. ter Haar Romeny, M.A. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Proc., 1998, to appear.
J. Weickert, S. Ishikawa, A. Imiya, On the history of Gaussian scale-space axiomatics, J. Sporring, M. Nielsen, L. Florack, P. Johansen (Eds.), Gaussian scale-space theory, Kluwer, Dordrecht, 1997, in press.
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Weickert, J. (1997). Recursive separable schemes for nonlinear diffusion filters. In: ter Haar Romeny, B., Florack, L., Koenderink, J., Viergever, M. (eds) Scale-Space Theory in Computer Vision. Scale-Space 1997. Lecture Notes in Computer Science, vol 1252. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63167-4_56
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DOI: https://doi.org/10.1007/3-540-63167-4_56
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