Complete Lattice Structure of Poincaré Upper-Half Plane and Mathematical Morphology for Hyperbolic-Valued Images

Jesus Angulo 1 Santiago Velasco-Forero 1
1 CMM - Centre de Morphologie Mathématique
Abstract : Mathematical morphology is a nonlinear image processing methodology based on the application of complete lattice theory to spatial structures. Let us consider an image model where at each pixel is given a univariate Gaussian distribution. This model is interesting to represent for each pixel the measured mean intensity as well as the variance (or uncertainty) for such measurement. The aim of this paper is to formulate morphological operators for these images by embedding Gaussian distribution pixel values on the Poincaré upper-half plane. More precisely, it is explored how to endow this classical hyperbolic space with partial orderings which lead to a complete lattice structure.
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First International Conference on Geometric Science of Information (GSI'2013), Aug 2013, Paris, France. Springer, Lecture Notes in Computer Science, vol 8085, pp.535 - 542, 2013, Nielsen F., Barbaresco F. (eds) Geometric Science of Information. 〈10.1007/978-3-642-40020-9_59〉
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Jesus Angulo, Santiago Velasco-Forero. Complete Lattice Structure of Poincaré Upper-Half Plane and Mathematical Morphology for Hyperbolic-Valued Images. First International Conference on Geometric Science of Information (GSI'2013), Aug 2013, Paris, France. Springer, Lecture Notes in Computer Science, vol 8085, pp.535 - 542, 2013, Nielsen F., Barbaresco F. (eds) Geometric Science of Information. 〈10.1007/978-3-642-40020-9_59〉. 〈hal-01536381〉

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