Last update: July. 18, 2001.

BACK

Publications by Petros Maragos et al. on Mathematical Morphology :

• Morphological Scale-Space Representation with Levelings, 1999.
• Multiscale Morphological Segmentations Based on Watershed, Flooding, and Eikonal PDE, 1999.
• Nonlinear PDEs and Numerical Algorithms for Modeling Levelings and Reconstruction Filters, 1999.
• Advances in Differential Morphology: Image Segmentation via Eikonal PDE & Curve Evolution & Reconstruction via Constrained Dilation Flow, 1998.
• Evolution Equations for Continuous-Scale Morphology, 1992 & 1994.
• Measuring the Fractal Dimension of Signals: Morphological Covers and Iterative Optimization, 1991.
• Morphological Systems for Multidimensional Signal Processing, 1990.
• A Representation Theory for Morphological Image and Signal Processing, 1989.
• Pattern Spectrum and Multiscale Shape Representation, 1989.
• Morphological Skeleton Representation and Coding of Binary Images, 1986.

BibTeX references

A morphological scale-space representation is presented based on a morphological strong filter, the levelings. The scale-properties are analysed and illustrated. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharp and perfectly localised. This paper is followed by a companion paper on PDE formulations of levelings.

The classical morphological segmentation paradigm is based on the watershed transform, constructed by flooding the gradient image seen as a topographic surface. For flooding a topographic surface, a topographic distance is defined from which a minimum distance algorithm is derived for the watershed. In a continuous formulation, this is modeled via the eikonal PDE, which can be solved using curve evolution algorithms. Various ultrametric distances between the catchment basins may then be associated to the flooding itself. To each ultrametric distance is associated a multiscale segmentation; each scale being the closed balls of the ultrametric distance.

In this paper we develop partial differential equations (PDEs) that model the generation of a large class of morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks ranging from enhancement, to geometric feature detection, to segmentation. The developed PDEs are nonlinear functions of the first spatial derivatives and model these nonlinear filters as the limit of a controlled growth starting from an initial seed signal. This growth is of the multiscale dilation or erosion type and the controlling mechanism is a switch that reverses the growth when the difference between the current evolution and a reference signal switches signs. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution and corresponding filter implementation, and provide insights via several experiments. Finally, we outline the use of these PDEs for improving the Gaussian scale-space by using the latter as initial seed to generate multiscale levelings that have a superior preservation of image edges and boundaries.

BACK

Page created & maintained by Frederic Leymarie, 2000.
Comments, suggestions, etc., mail to: leymarie@lems.brown.edu