Efficient and reliable schemes for nonlinear diffusion filtering

IEEE Transactions on Image Processing

Volumn 7, Issue 3, 1998, Pages 398-410

  Weickert, Joachim ^a   Ter Haar Romeny, Bart M ^b,c   Viergever, Max A ^b  

^a UNIVERSITY OF COPENHAGEN   (Denmark)

^b UNIVERSITY MEDICAL CENTER UTRECHT   (Netherlands)

^c IEEE

Author keywords

Absolute stability; Nonlinear diffusion; Recursive filters

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; LYAPUNOV METHODS; MATHEMATICAL MODELS; RECURSIVE FUNCTIONS;

GAUSSIAN ALGORITHM; NONLINEAR DIFFUSION FILTERING; THOMAS ALGORITHM;

IMAGE PROCESSING;

EID: 0032026465     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/83.661190     Document Type: Article

References (52)

1. S. T. Acton, A. C. Bovik, and M. M. Crawford, "Anisotropic diffusion pyramids for image segmentation," in Proc. IEEE Int. Conf. Image Processing, Austin, TX, Nov. 13-16, 1994, vol. 3, pp. 478-482.
2. L. Alvarez, "Images and PDE's," in ICAOS'96: Images, Wavelets and PDE's, vol. 219., M.-O. Berger, R. Deriche, I. Herlin, J. Jaffré, and J.-M. Morel, Eds. New York: Springer, 1996, pp. 3-14.
3. L. Alvarez and L. Mazorra, "Signal and image restoration using shock filters and anisotropic diffusion," SIAM J. Numer. Anal., vol. 31, pp. 590-605, 1994.
4. K. Åström, and A. Heyden, "Stochastic analysis of image acquisition and scale-space smoothing," in Gaussian Scale-Space Theory, J. Sporring, M. Nielsen, L. Florack, and P. Johansen, Eds. Boston, MA: Kluwer, 1997, pp. 129-136.
5. E. Bänsch and K. Mikula, "A coarsening finite element strategy in image selective smoothing," Faculty Math., Univ. Freiburg, Freiburg, Germany, preprint no. 18/1996, 1996; to be published in Computation and Visualization in Science.
6. B. Benhamouda, "Parameter adaptation for nonlinear diffusion in image processing," Master's thesis, Dept. Math., Univ. Kaiserslautern, Kaiserslautern, Germany, 1994.
7. W. L. Briggs, A Multigrid Tutorial. Philadelphia, PA: SIAM, 1987.
8. A. M. Bruckstein, G. Sapiro, and D. Shaked, "Evolution of planar polygons," Int. J. Pattern Recognit. Artif. Intell., vol. 9, pp. 991-1014, 1995.
9. L. D. Cai, "Some notes on repeated averaging smoothing," in Pattern Recognition, vol. 301, J. Kittler, Ed. New York: Springer-Verlag, 1988, pp. 597-605.
10. F. Catté, "Convergence of iterated affine and morphological filters by nonlinear semi-group theory," in ICAOS'96: Images, Wavelets and PDE's, vol. 219, M.-O. Berger, R. Deriche, I. Herlin, J. Jaffré, and J.-M. Morel, Eds. New York: Springer-Verlag, 1996, pp. 125-133.
11. F. Catté, F. Dibos, and G. Koepfler, "A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets," SIAM J. Numer. Anal., vol. 32, pp. 1895-1909, 1995.
12. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM J. Numer. Anal., vol. 29, 182-193, 1992.
13. T. Cohignac, F. Eve, F. Guichard, C. Lopez, and J.-M. Morel, "Numerical analysis of the fundamental equation of image processing," Preprint no. 9254, CEREMADE, Univ. Paris IX-Dauphine, Paris, France, 1992.
14. R. Deriche, "Fast algorithms for low-level vision," IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 78-87, 1990.
15. _, "Recursively implementing the Gaussian and its derivatives," in Proc. 2nd Int. Singapore Conf. on Image Processing, V. Srinivasan, S. H. Ong, and Y. H. Ang, Eds., Singapore, Sept. 7-11, 1992, pp. 263-267.
16. R. Deriche and O. Faugeras, "Les EDP en traitement des images et vision par ordinateur," Traite. Signal, vol. 13, no. 6, 1996.
17. J. Douglas and J. E. Gunn, "A general formulation of alternating direction methods. Part I. Parabolic and hyperbolic problems," Numer. Math., vol. 6, pp. 428-453, 1964.
18. J. Fröhlich and J. Weickert, "Image processing using a wavelet algorithm for nonlinear diffusion," Rep. no. 104, Lab. Technomath., Univ. Kaiserslautern, Kaiserslautern, Germany, 1994.
19. T. Gijbels et al., "A VLSI-architecture for parallel nonlinear diffusion with applications in vision," in Proc. IEEE Workshop on VLSI Signal Processing, 1994, vol. 7, pp. 398-707.
20. A. R. Gourlay, "Implicit convolution," Image Vis. Comput., vol. 3, pp. 15-23, 1985.
21. B. M. ter Haar Romeny, Ed., Geometry-Driven Diffusion in Computer Vision. Boston, MA: Kluwer, 1994.
22. R. A. Hummel, "Representations based on zero-crossings in scale space," in Proc. IEEE Comput. Soc. Conf. Computer Vision and Pattern Recognition, Miami Beach, FL, June 22-26, 1986; pp. 204-209.
23. T. Iijima, "Basic theory of pattern normalization (for the case of a typical one-dimensional pattern)," Bull. Electrotech. Lab., vol. 26, pp. 368-388, 1962 (in Japanese).
24. C. Lamberti, M. Sitta, and F. Sgallari, "Improvements to the anisotropic diffusion model for 2-D echo image processing," in Proc. Annual Int. Conf. IEEE Engineering in Medicine and Biology Soc., 1992, vol. 14, pp. 1872-1873.
25. X. Li and T. Chen, "Nonlinear diffusion with multiple edginess thresholds," Pattern Recognit., vol. 27, pp. 1029-1037, 1994.
26. T. Lindeberg, "Scale-space for discrete signals," IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 234-254, 1990.
27. D.-S. Luo, M. A. King, and S. Glick, "Local geometry variable conductance diffusion for post-reconstruction filtering," IEEE Trans. Nucl. Sci., vol. 41, pp. 2800-2806, 1994.
28. G. I. Marchuk, "Splitting and alternating direction methods," Handbook of Numerical Analysis, vol. I, P. G. Ciarlet and J.-L. Lions, Eds. Amsterdam, The Netherlands: North Holland, 1990, pp. 197-462.
29. T. Meis and U. Marcowitz, Numerische Behandlung Partieller Differentialgleichungen. New York: Springer-Verlag, 1978.
30. A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations. New York: Wiley, 1980.
31. E. Nicolet and M. Spühler, "Charakterisierung von unscharfen blobförmigen 3-D strukturen," Dipl. thesis, Commun. Technol. Lab., Image Sci. Div., ETH-Zentrum, Zürich, Switzerland, 1996.
32. M. Nielsen, L. Florack, and R. Deriche, "Regularization, scale-space and edge detection filters," J. Math. Imag. Vis., vol. 7, pp. 291-307, 1997.
33. W. J. Niessen, K. L. Vincken, J. Weickert, and M. A. Viergever, "Nonlinear multiscale representations for image segmentation," Comput. Vis. Image Understand., vol. 66, pp. 233-245, 1997.
34. E. Norman, "A discrete analogue of the Weierstrass transform," Proc. Amer. Math. Soc., vol. 11, 596-604, 1960.
35. J. M. Ortega, Matrix Theory. New York: Plenum, 1987.
36. P. Perona and J. Malik, "A network for multiscale image segmentation," in Proc. IEEE Int. Symp. Circuits and Systems, Espoo, Finland, June 7-9, 1988, pp. 2565-2568.
37. _, "Scale space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 629-639, 1990.
38. A. H. Salden, B. M. ter Haar Romeny, and M. A. Viergever, "Linear scale-space theory from physical principles," J. Math. Imag. Vis., to appear.
39. H. R. Schwarz, Numerische Mathematik. Stuttgart, Germany: Teubner, 1988.
40. J. A. Sethian, Level Set Methods. Cambridge, U.K.: Cambridge Univ. Press, 1996.
41. K. Siddiqi, B. B. Kimia, and C.-W. Shu, "Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution," Graph. Models Image Process., vol. 59, pp. 278-301, 1997.
42. R. A. Varga, Matrix Iterative Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1962.
43. J. Weickert, "Zwischenbericht zum projekt 'nichtlineare diffusionsfilter'," Ber. über wissenschaftliche Tätigkeit, Ctr. Appl. Math., Darmstadt-Kaiserslautern, Germany, pp. 133-142, 1992.
44. _, "Theoretical foundations of anisotropic diffusion in image processing," Computing, suppl. 11, pp. 221-236, 1996.
45. _, "Nonlinear diffusion scale-spaces: From the continuous to the discrete setting," in Proc. ICAOS'96: Images, Wavelets and PDE's, vol. 219, M.-O. Berger et al., Eds. New York: Springer, 1996, pp. 111-118.
46. _, Anisotropic Diffusion in Image Processing. Stuttgart, Germany: Teubner, 1998.
47. _, "Recursive separable schemes for nonlinear diffusion filters," Scale-Space Theory in Computer Vision, vol. 1252, B. ter Haar Romeny, et al., Eds. New York:, Springer-Verlag: 1997, pp. 260-271.
48. J. Weickert, S. Ishikawa, and A. Imiya, "On the history of Gaussian scale-space axiomatics," Gaussian Scale-Space Theory, J. Spurring et al., Eds. Boston, MA: Kluwer, 1997, pp. 45-59.
49. J. Weickert, K. J. Zuiderveld, B. M. ter Haar Romeny, and W. J. Niessen, "Parallel implementations of AOS schemes: A fast way of nonlinear diffusion filtering," in Proc. 1997 IEEE Int. Conf. Image Processing, Santa Barbara, CA, Oct. 26-29, 1997.
50. R. T. Whitaker and S. M. Pizer, "A multi-scale approach to nonuniform diffusion," CVGIP: Image Understand., vol. 57, pp. 99-110, 1993.
51. N. N. Yanenko, The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables. New York: Springer-Verlag, 1971.
52. D. Zhao and B. Li, "A new implementation of discrete multiscale filtering," in Proc. IEEE Int. Conf. Image Processing, Lausanne, Switzerland, Sept. 16-19, 1996, vol. 1, pp. 383-386.