Abstract
The two fundamental operations in morphological image processing are dilation and erosion. These processes are defined via structuring elements. It is of practical interest to consider a variety of structuring element shapes. The realisation of dilation/erosion for convex structuring elements by use of partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high degree of rotational invariance. We validate the FCT-scheme theoretically by proving consistency and stability. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvarez, L., Guichard, F., Lions, P. L., & Morel, J. M. (1993). Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis, 123, 199–257.
Arehart, A. B., Vincent, L., & Kimia, B. B. (1993). Mathematical morphology: The Hamilton–Jacobi connection. In Proc. fourth international conference on computer vision (pp. 215–219). Berlin: IEEE Computer Society Press.
Banon, G. J. F., Barrera, J., Braga-Neto, U. D. M., & Hirata, N. S. T. (Eds.) (2007). Proceedings of the 8th international symposium on mathematical morphology : Vol. 1. Full Papers. São José dos Campos: Instituto Nacional de Pesquisas Espaciais (INPE). http://urlib.net/dpi.inpe.br/ismm@80/2007/05.29.15.58.
Blum, H. (1973). Biological shape and visual science. Journal of Theoretical Biology, 38, 205–287.
Boris, J. P., & Book, D. L. (1973). Flux corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69.
Boris, J. P., & Book, D. L. (1976). Flux corrected transport. III. Minimal error FCT algorithms. Journal of Computational Physics, 20, 397–431.
Breuß, M., & Weickert, J. (2006). A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision, 25, 187–201.
Breuß, M., & Welk, M. (2007). Analysis of staircasing in semidiscrete stabilised inverse linear diffusion algorithms. Journal of Computational and Applied Mathematics, 206, 520–533.
Breuß, M., & Weickert, J. (2009). Highly accurate PDE-based morphology for general structuring elements. In X.-C. Tai et al. (Eds.), Lecture notes in computer science : Vol. 5567. Scale-space and variational methods in computer vision (pp. 758–769). Berlin: Springer.
Brockett, R. W., & Maragos, P. (1992). Evolution equations for continuous-scale morphology. In Proc. IEEE international conference on acoustics, speech and signal processing (Vol. 3, pp. 125–128). San Francisco, CA.
Butt, M. A., & Maragos, P. (1996). Comparison of multiscale morphology approaches: PDE implemented via curve evolution versus Chamfer distance transform. In P. Maragos, R. W. Schafer, & M. A. Butt (Eds.), Computational imaging and vision : Vol. 5. Mathematical morphology and its applications to image and signal processing (pp. 31–40). Dordrecht: Kluwer.
Farin, G. (2002). Curves and surfaces for CADG. San Mateo: Morgan-Kaufmann.
Gottlieb, S., & Shu, C.-W. (1998). Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67(221), 73–85.
Gottlieb, S., Shu, C.-W., & Tadmor, E. (2001). Strong stability-preserving high-order time discretisation methods. SIAM Review, 43(1), 89–112.
Goutsias, J., Vincent, L., & Bloomberg, D. S. (Eds.), (2000). Computational imaging and vision : Vol. 18. Mathematical morphology and its applications to image and signal processing. Kluwer: Dordrecht.
Hairer, E., Norsett, S., & Wanner, G. (1987). Springer series in computational mathematics : Vol. 8. Solving ordinary differential equations. I: Nonstiff problems. New York: Springer.
Heijmans, H. J. A. M., & Roerdink, J. B. T. M. (Eds.), (1998). Computational imaging and vision : Vol. 12. Mathematical morphology and its applications to image and signal processing. Dordrecht: Kluwer.
Kimia, B. B., Tannenbaum, A., & Zucker, S. W. (1995). Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space. International Journal of Computer Vision, 15, 189–224.
LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press.
Matheron, G. (1967). Eléments pour une théorie des milieux poreux. Paris: Masson.
Matheron, G. (1975). Random sets and integral geometry. New York: Wiley.
Osher, S., & Fedkiw, R. P. (2002). Applied mathematical sciences : Vol. 153. Level set methods and dynamic implicit surfaces. New York: Springer.
Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79, 12–49.
Pizarro, L., Burgeth, B., Breuß, M., & Weickert, J. (2009). A directional Rouy-Tourin scheme for adaptive matrix-valued morphology. In M. H. F. Wilkinson, & J. B. T. M. Roerdink (Eds.), Lecture notes in computer science : Vol. 5720. Proc. ninth international symposium on mathematical morphology (ISMM 2009) (pp. 250–260). Berlin: Springer.
Ronse, C., Najman, L., & Decencière, E. (Eds.), (2005). Computational imaging and vision : Vol. 30. Mathematical morphology: 40 years on. Dordrecht: Springer.
Rouy, E., & Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis, 29, 867–884.
Sapiro, G., Kimmel, R., Shaked, D., Kimia, B. B., & Bruckstein, A. M. (1993). Implementing continuous-scale morphology via curve evolution. Pattern Recognition, 26, 1363–1372.
Serra, J. (1967). Echantillonnage et estimation des phénomènes de transition minier. Ph.D. thesis, University of Nancy, France.
Serra, J. (1982). Image analysis and mathematical morphology (Vol. 1). London: Academic Press.
Serra, J. (1988). Image analysis and mathematical morphology (Vol. 2). London: Academic Press.
Sethian, J. A. (1999). Level set methods and fast marching methods (2nd ed.). Cambridge: Cambridge University Press. Paperback edition.
Siddiqi, K., Kimia, B. B., & Shu, C. W. (1997). Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution. Graphical Models and Image Processing, 59, 278–301.
Soille, P. (2003). Morphological image analysis (2nd ed.). Berlin: Springer.
Tadmor, E. (1984). Numerical viscosity and the entropy condition for conservative difference schemes. Mathematics of Computation, 43, 369–381.
Talbot, H., & Beare, R. (Eds.), (2002). Proc. sixth international symposium on mathematical morphology and its applications, Sydney, Australia. http://www.cmis.csiro.au/ismm2002/proceedings/.
van den Boomgaard, R. (1992). Mathematical morphology: Extensions towards computer vision. Ph.D. thesis, University of Amsterdam, The Netherlands.
van den Boomgaard, R. (1999). Numerical solution schemes for continuous-scale morphology. In M. Nielsen, P. Johansen, O. F. Olsen, & J. Weickert (Eds.), Lecture notes in computer science : Vol. 1682. Scale-space theories in computer vision (pp. 199–210). Berlin: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Breuß, M., Weickert, J. Highly Accurate Schemes for PDE-Based Morphology with General Convex Structuring Elements. Int J Comput Vis 92, 132–145 (2011). https://doi.org/10.1007/s11263-010-0366-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-010-0366-2