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Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

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Abstract

This paper explores some aspects of the algebraic theory of mathematical morphology from the viewpoints of minimax algebra and translation-invariant systems and extends them to a more general algebraic structure that includes generalized Minkowski operators and lattice fuzzy image operators. This algebraic structure is based on signal spaces that combine the sup-inf lattice structure with a scalar semi-ring arithmetic that possesses generalized ‘additions’ and ✶-‘multiplications’. A unified analysis is developed for: (i) representations of translation-invariant operators compatible with these generalized algebraic structures as nonlinear sup-✶ convolutions, and (ii) kernel representations of increasing translation-invariant operators as suprema of erosion-like nonlinear convolutions by kernel elements. The theoretical results of this paper develop foundations for unifying large classes of nonlinear translation-invariant image and signal processing systems of the max or min type. The envisioned applications lie in the broad intersection of mathematical morphology, minimax signal algebra and fuzzy logic.

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Authors and Affiliations

  1. National Technical University of Athens, School of Electrical & Computer Engineering, Athens, 15773, Greece

    Petros Maragos

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  1. Petros Maragos
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Correspondence to Petros Maragos.

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Petros Maragos received the Diploma degree in electrical engineering from the National Technical University of Athens in 1980, and the M.Sc.E.E. and Ph.D. degrees from Georgia Tech, Atlanta, USA, in 1982 and 1985.

In 1985 he joined the faculty of the Division of Applied Sciences at Harvard University, Cambridge, Massachusetts, where heworked for 8 years as professor of electrical engineering, affiliated with the interdisciplinary Harvard Robotics Lab. He has also been a consultant to several industry research groups including Xerox’s research on document image analysis. In 1993, he joined the faculty of the School of Electrical and Computer Engineering at Georgia Tech. During parts of 1996-98 he was on academic leave working as a senior researcher at the Institute for Language and Speech Processing in Athens. In 1998, he joined the faculty of the National Technical University of Athens where he is currently working as professor of electrical and computer engineering. His current research and teaching interests include the general areas of signal processing, systems theory, pattern recognition, and their applications to image processing and computer vision, and computer speech processing and recognition.

He has served as associate editor for the IEEE Trans. on Acoustics, Speech, and Signal Processing, editorial board member for the Journal of Visual Communications and Image Representation, and guest editor for the IEEE Trans. on Image Processing; general chairman for the 1992 SPIE Conference on Visual Communications and Image Processing, and co-chairman for the 1996 International Symposium on Mathematical Morphology; member of two IEEE DSP committees; and president of the International Society for Mathematical Morphology.

Dr. Maragos’ research work has received several awards, including: a 1987 US National Science Foundation Presidential Young Investigator Award; the 1988 IEEE Signal Processing Society’s Paper Award for the paper ‘Morphological Filters’; the 1994 IEEE Signal Processing Society’s Senior Award and the 1995 IEEE Baker Award for the paper ‘Energy Separation in Signal Modulations with Application to Speech Analysis’; and the 1996 Pattern Recognition Society’s Honorable Mention Award for the paper ‘Min-Max Classifiers’. In 1995, he was elected Fellow of IEEE for his contributions to the theory and applications of nonlinear signal processing systems.

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Maragos, P. Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems. J Math Imaging Vis 22, 333–353 (2005). https://doi.org/10.1007/s10851-005-4897-z

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