Random Sets pp 47-71 | Cite as

On Estimating Granulometric Discrete Size Distributions of Random Sets

  • Krishnamoorthy Sivakumar
  • John Goutsias
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 97)


Morphological granulometries, and the associated size distributions and densities, are important shape/size summaries for random sets. They have been successfully employed in a number of image processing and analysis tasks, including shape analysis, multiscale shape representation, texture classification, and noise filtering. For most random set models however it is not possible to analytically compute the size distribution. In this contribution, we investigate the problem of estimating the granulometric (discrete) size distribution and size density of a discrete random set. We propose a Monte Carlo estimator and compare its properties with that of an empirical estimator. Theoretical and experimental results demonstrate superiority of the Monte Carlo estimation approach. The Monte Carlo estimator is then used to demonstrate existence of phase transitions in a popular discrete random set model known as a binary Markov random field, as well as a tool for designing “optimal” filters for binary image restoration.

Key words

Granulometries Image Restoration Markov Random Fields Mathematical Morphology Monte Carlo Estimation Random Sets Size Distribution Phase Transition 


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Krishnamoorthy Sivakumar
    • 1
  • John Goutsias
    • 2
  1. 1.Texas Center for Applied Technology and Department of Electrical EngineeringTexas A&M University, College StationUSA
  2. 2.Department of Electrical and Computer Engineering, Image Analysis and Communications LaboratoryThe Johns Hopkins UniversityBaltimoreUSA

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