Theoretical foundations of spatially-variant mathematical morphology part I: Binary images

Nidhal Bouaynaya, Mohammed Charif-Chefchaouni, Dan Schonfeld

Research output: Contribution to journalArticlepeer-review

84 Scopus citations


We develop a general theory of spatially-variant (SV) mathematical morphology for binary images in the Euclidean space. The basic SV morphological operators (i.e., SV erosion, SV dilation, SV opening and SV closing) are defined. We demonstrate the ubiquity of SV morphological operators by providing a SV kernel representation of increasing operators. The latter representation is a generalization of Matheron's representation theorem of increasing and translation-invariant operators. The SV kernel representation is redundant, in the sense that a smaller subset of the SV kernel is sufficient for the representation of increasing operators. We provide sufficient conditions for the existence of the minimal basis representation in terms of upper-semi-continuity in the hit-or-miss topology. The latter minimal basis representation is a generalization of Maragos' minimal basis representation for increasing and translation-invariant operators. Moreover, we investigate the upper-semi-continuity property of the basic SV morphological operators. Several examples are used to demonstrate that the theory of spatially-variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatiallyvariant geometrical structuring elements (e.g., circular, affine and motion morphology). Simulation results illustrate the theory of the proposed spatially-variant morphological framework and show its potential power in various image processing applications.

Original languageEnglish (US)
Pages (from-to)823-836
Number of pages14
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number5
StatePublished - May 2008
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


Dive into the research topics of 'Theoretical foundations of spatially-variant mathematical morphology part I: Binary images'. Together they form a unique fingerprint.

Cite this