A shock-capturing algorithm for the differential equations of dilation and erosion

Journal of Mathematical Imaging and Vision

Volumn 25, Issue 2, 2006, Pages 187-201

  Breuß, Michael ^a   Weickert, Joachim ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

Finite difference methods; Flux corrected transport; Morphological dilation; Morphological erosion

Indexed keywords

FINITE DIFFERENCE METHODS; FLUX CORRECTED TRANSPORT; MORPHOLOGICAL DILATION; MORPHOLOGICAL EROSION;

ALGORITHMS; EROSION; IMAGE ANALYSIS; ROTATION; VISCOSITY;

DIFFERENTIAL EQUATIONS;

EID: 33749570584     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-006-9696-7     Document Type: Article

References (43)

1. L. Alvarez, R Guichard, P.-L. Lions, and J.-M. Morel, "Axioms and fundamental equations in image processing," Archive for Rational Mechanics and Analysis,Vol. 123, pp. 199-257, 1993.
2. A.B. Arehart, L. Vincent, and B.B. Kimia, "Mathematical morphology: The Hamilton-Jacobi connection," in Proc. Fourth International Conference on Computer Vision, Berlin. IEEE Computer Society Press, May 1993, pp. 215-219.
3. J.P. Boris and D.L. Book, "Flux corrected transport. I. SHASTA, a fluid transport algorithm that works," Journal of Computational Physics, Vol. 11, No. 1, pp. 38-69, 1973.
4. J.P. Boris and D.L. Book. "Flux corrected transport. III. Minimal error FCT algorithms," Journal of Computational Physics, Vol. 20, pp. 397-431, 1976.
5. J.P. Boris, D.L. Book, and K. Hain, "Flux corrected transport. II. Generalizations of the method," Journal of Computational Physics, Vol. 18, pp. 248-283, 1975.
6. M. Breuß, "An analysis of discretisations of inverse diffusion equations," To appear in Computing Letters.
7. M. Breuß, T. Brox, T. Sonar, and J. Weickert, "Stabilized nonlinear inverse diffusion for approximating hyperbolic PDEs," in Scale-Space and PDE Methods in Computer Vision, R. Kimmel, N. Sochen, and J. Weickert (Eds.), Vol. 3459 of Lecture Notes in Computer Science, Springer, Berlin, 2005, pp. 536-547.
8. R.W. Brockett and P. Maragos, "Evolution equations for continuous-scale morphology," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Vol. 3, San Francisco, CA, Mar. 1992, pp. 125-128.
9. R.W. Brockelt and P. Maragos, "Evolution equations for continuous-scale morphological filtering," IEEE Transactions on Signal Processing, Vol. 42, No. 12, pp. 3377-3386, 1994.
10. M. A. Butt and P. Maragos, "Comparison of multiscale morphology approaches: PDE implemented via curve evolution versus Chamfer distance transform," in Mathematical Morphology and its Applications to Image and Signal Processing, P. Maragos, R.W. Schafer, and M.A. Butt (Eds.), Vol. 5 of Computational Imaging and Vision, Kluwer: Dordrecht, 1996, pp. 31-40.
11. R. Courant, K. Friedrichs, and H. Lewy, "Über die partiellen differenzengleichungen der mathematischen Physik," Mathematische Annalen, Vol. 100, pp. 32-74, 1928.
12. E.R. Dougherty, Mathematical Morphology in Image Processing. Marcel Dekker: New York, 1993.
13. L.C. Evans, Partial Differential Equations, Vol. 19 of Graduate. Studies in Mathematics. American Mathematical Society, Providence, 1998.
14. S.J. Farlow, Partial Differential Equations for Scientists and Engineers. Dover: New York, 1993.
15. E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Edition Marketing, 1991.
16. J. Goutsias, L. Vincent, and D.S. Bloomberg (Eds.), Mathematical Morphology and its Applications to Image and Signal Processing, Vol. 18 of Computational Imaging and Vision. Kluwer: Dordrecht, 2000.
17. H.J.A.M. Heijmans, Morphological Image Operators. Academic Press: Boston, 1994.
18. H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.), Mathematical Morphology and its Applications to Image and Signal Processing, Vol. 12 of Computational Imaging and Vision. Kluwer: Dordrecht, 1998.
19. D. Kuzmin, R. Löhner, and S. Turek (Eds.), Flux-Corrected Transport. Springer: Berlin, 2005.
20. D. Kuzmin and S. Turek, "Flux correction tools for finite elements," Journal of Computational Physics, Vol. 175, pp. 525-558, 2002.
21. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, UK, 2002.
22. P. Maragos, "Partial differential equations for morphological scale-spaces and eikonal applications," in The Image and Video Compression Handbook, 2nd edn., A.C. Bovik, Elsevier Academic Press, 2005, pp. 587-612.
23. P. Maragos, R.W. Schafer, and M.A. Butt (Eds.), Mathematical Morphology and its Applications to Image and Signal. Processing, Vol. 5 of Computational Imaging and Vision. Kluwer: Dordrecht, 1996.
24. G. Matheron, Random Sets and Integral Geometry. Wiley: New York, 1975.
25. S. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Vol. 153 of Applied Mathematical Sciences. Springer: New York, 2002.
26. S. Osher and J.A. Sethian, "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations," Journal of Computational Physics, 79:12-49, 1988.
27. C. Ronse, L. Najman, and E. Decencière (Eds.), Mathematical Morphology: 40 Years On, Vol. 30 of Computational Imaging and Vision. Springer: Dordrecht, 2005.
28. E. Rouy and A. Tourin, "A viscosity solutions approach to shape-from-shading," SIAM Journal on Numerical Analysis, Vol. 29, pp. 867-884, 1992.
29. G. Sapiro, R. Kimmel, D. Shaked, B.B. Kimia, and A.M. Bruckstein, "Implementing continuous-scale morphology via curve evolution," Pattern Recognition, Vol. 26, pp. 1363-1372, 1993.
30. M. Schmitt and J. Mattioli, Morphologie mathématique. Masson: Paris, 1993.
31. J. Serra, Image Analysis and Mathematical Morphology, Vol. 1. Academic Press: London, 1982.
32. J. Serra. Image Analysis and Mathematical Morphology, Vol. 2. Academic Press: London, 1988.
33. J. Serra and P. Salembier (Eds.), Proc. Internationanal Workshop on Mathematical Morphology and its Applications to Signal Processing. Barcelona, Spain, May 1993.
34. J. Serra and P. Soille (Eds.), Mathematical Morphology and its Applications to Image Processing, Vol. 2 of Computational Imaging and Vision. Kluwer: Dordrecht, 1994.
35. J.A. Sethian, Level Set Methods and Fast Marching Methods 2nd edn. Cambridge University Press: Cambridge, UK, 1999. aperback edition.
36. K. Siddiqi, B.B. Kimia, and C.W. Shu, "Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution," Graphical Models and Image Processing, Vol. 59, pp. 278-301, 1997.
37. P. Soille, Morphological Image Analysis 2nd edn. Springer, Berlin, 2003.
38. P. Stoll, C. Shu, and B.B. Kimia, "Shock-capturing numerical methods for viscosity solutions of certain PDEs in computer vision: The Godunov, Osher-Sethian and ENO schemes," Technical Report LEMS-132, Division of Engineering, Brown University, Providence, RI, 1994.
39. H. Talbot and R. Beare (Eds.), in Proc. Sixth International Symposium on Mathematical Morphology and its Applications. Sydney, Australia, April 2002. http://www.cmis.csiro.au/ismm2002/proceedings/.
40. R. van den Boomgaard, "Mathematical morphology: extensions towards computer vision," PhD thesis, University of Amsterdam, The Netherlands, 1992.
41. R. van den Boomgaard, "Numerical solution schemes for continuous-scale morphology," in Scale-Space Theories in Computer Vision, M. Nielsen, P. Johansen, O.F. Olsen, and J. Weickert (Eds.), Vol. 1682 of Lecture Notes in Computer Science, Springer: Berlin, 1999, pp. 199-210.
42. S.T. Zalesak, "Fully multidimensional flux-corrected transport algorithms for fluids," Journal of Computational Physics, Vol. 31, pp. 335-362, 1979.
43. S.T. Zalesak, "Introduction to "Flux corrected transport. I. SHASTA, a fluid transport algorithm that works," Journal of Computational Physics, Vol. 135, pp. 170-171, 1997.