Grain Filters

Motivated by operators simplifying the topographic map of a function, we study the theoretical properties of two kinds of “grain” filters. The first category, discovered by L. Vincent, defines grains as connected components of level sets and removes those of small area. This category is composed of two filters, the maxima filter and the minima filter. However, they do not commute. The second kind of filter, introduced by Masnou, works on “shapes”, which are based on connected components of level sets. This filter has the additional property that it acts in the same manner on upper and lower level sets, that is, it commutes with an inversion of contrast. We discuss the relations of Masnou's filter with other classes of connected operators introduced in the literature. We display some experiments to show the main properties of the filters discussed above and compare them.

[1]  Pascal Monasse,et al.  Fast computation of a contrast-invariant image representation , 2000, IEEE Trans. Image Process..

[2]  Petros Maragos,et al.  Morphological filters-Part I: Their set-theoretic analysis and relations to linear shift-invariant filters , 1987, IEEE Trans. Acoust. Speech Signal Process..

[3]  Lionel Moisan,et al.  Affine plane curve evolution: a fully consistent scheme , 1998, IEEE Trans. Image Process..

[4]  Henk J. A. M. Heijmans.  Connected Morphological Operators for Binary Images , 1999, Comput. Vis. Image Underst..

[5]  Guillermo Sapiro,et al.  Affine invariant scale-space , 1993, International Journal of Computer Vision.

[6]  Montse Pardàs,et al.  Morphological operators for image and video compression , 1996, IEEE Trans. Image Process..

[7]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[8]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Jean Paul Frédéric Serra.  Connections for sets and functions , 2000, Fundam. Informaticae.

[10]  C. Lantuejoul.  Geodesic methods in image anlysis , 1984 .

[11]  Henk J. A. M. Heijmans,et al.  Fundamenta Morphologicae Mathematicae , 2000, Fundam. Informaticae.

[12]  Pascal Monasse,et al.  Scale-Space from a Level Lines Tree , 2000, J. Vis. Commun. Image Represent..

[13]  H. Heijmans.  Morphological image operators , 1994 .

[14]  Philippe Salembier,et al.  Morphological operators for very low bit rate video coding , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[15]  Jacques Froment,et al.  A Compact and Multiscale Image Model Based on Level Sets , 1999, Scale-Space.

[16]  C. Lantuéjoul,et al.  On the use of the geodesic metric in image analysis , 1981 .

[17]  Philippe Salembier,et al.  Morphological multiscale segmentation for image coding , 1994, Signal Process..

[18]  V. Caselles,et al.  The M-components of level sets of continuous functions in WBV , 2001 .

[19]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[20]  Luc Vincent,et al.  Morphological Area Openings and Closings for Grey-scale Images , 1994 .

[21]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[22]  V. Caselles,et al.  The Tree of Shapes of an Image , 2003 .

[23]  M. H. A. Newman,et al.  Topology . Elements of the topology of plane sets of points , 1939 .

[24]  Henk J. A. M. Heijmans.  Self-dual morphological operators and filters , 2004, Journal of Mathematical Imaging and Vision.

[25]  Philippe Salembier,et al.  Antiextensive connected operators for image and sequence processing , 1998, IEEE Trans. Image Process..

[26]  J. Morel,et al.  Connected components of sets of finite perimeter and applications to image processing , 2001 .

[27]  Jean-Michel Morel,et al.  Image restoration involving connectedness , 1998, Other Conferences.

[28]  J. Serra.  Introduction to mathematical morphology , 1986 .

[29]  L. Vincent.  Grayscale area openings and closings, their efficient implementation and applications , 1993 .

[30]  J. W. Modestino,et al.  Flat Zones Filtering, Connected Operators, and Filters by Reconstruction , 1995 .

[31]  F. Guichard.  Image iterative smoothing and P.D.E.'s , 2000 .

[32]  Philippe Salembier,et al.  Connected operators and pyramids , 1993, Optics & Photonics.

[33]  S. Beucher,et al.  Morphological segmentation , 1990, J. Vis. Commun. Image Represent..

[34]  Petros Maragos,et al.  Morphological filters-Part II: Their relations to median, order-statistic, and stack filters , 1987, IEEE Trans. Acoust. Speech Signal Process..