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Sparsely Connected Autoassociative Lattice Memories with an Application for the Reconstruction of Color Images

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Abstract

Sparsely connected autoassociative lattice memories (SCALMs) are very general models defined on complete lattices, a mathematical structure which is obtained by imposing some ordering on a set. They are computationally cheaper and mathematically simpler than “traditional” models and other memories such as the original autoassociative morphological memories (AMMs) of Ritter and Sussner because they only compute maximums and minimums. This paper provides theoretical results on SCALMs defined on a general complete lattice as well as an application of these memories for the storage and recall of color images. Precisely, we characterize the recall phase of SCALMs in terms of their fixed points. Then, we show that any endomorphic lattice polynomial—a concept that generalizes the notion of lattice polynomial of Birkhoff—on the fundamental memory set represents a fixed point of the SCALMs. Also, we discuss the relationship between SCALMs and the original AMMs. Finally, we provide some experimental results on the performance of SCALMs, defined on different color lattices, for the reconstruction of color images corrupted by either Gaussian or impulsive noise.

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  1. Department of Mathematics, Center for Exact Sciences, University of Londrina, CEP 86051-990, Londrina, PR, Brazil

    Marcos Eduardo Valle

  2. Center for Exact Sciences and Technology, State University of West Paraná, CEP 85819-110, Cascavel, PR, Brazil

    Daniela Maria Grande Vicente

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  1. Marcos Eduardo Valle
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Valle, M.E., Grande Vicente, D.M. Sparsely Connected Autoassociative Lattice Memories with an Application for the Reconstruction of Color Images. J Math Imaging Vis 44, 195–222 (2012). https://doi.org/10.1007/s10851-011-0322-y

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