Nonlinear Operators on Graphs via Stacks

Velasco-Forero, Santiago; Angulo, Jesús

doi:10.1007/978-3-319-25040-3_70

Cited by 4 publications

(4 citation statements)

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“…The dilation δ W , already introduced in [2,11,15], can be viewed as a generalisation of the non-local and adaptive mathematical morphology [13,14] on signals and images. Each column W :,j of W represents the structuring function corresponding to pixel (or instant) j.…”

Section: Erosions and Adjunctionsmentioning

confidence: 99%

Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing

Blusseau,

Velasco-Forero,

Angulo

et al. 2022

Preprint

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In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.

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Section: Erosions and Adjunctionsmentioning

confidence: 99%

Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing

Blusseau,

Velasco-Forero,

Angulo

et al. 2022

Preprint

Self Cite

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show abstract

“…Lately, signal processing on graphs has gained interest to address this kind of issues [1,2,3]. This effort includes the extension of non-linear operators, and in particular mathematical morphology, to signals on graphs [4].…”

Section: Introductionmentioning

confidence: 99%

“…In [4] the authors introduce a formalism for MM for signals on graphs that generalises a wide range of mathematical morphology operators usually defined for images, including flat, non-flat and adaptive erosions and dilations. This formalism is clearly inspired by the idea of non-local morphology [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…This formalism is clearly inspired by the idea of non-local morphology [9,10]. Interestingly, although in [4] the authors do not emphasise this aspect, they write dilations and erosions as max-plus and min-plus "linear" functions, in other words as the product of a matrix by a vector in the "max-plus" and "min-plus" algebras, also called idempotent or tropical alge-bras [11]. This is just a hint of how much the field of idempotent mathematics can be suited to the processing of signals on graphs from a mathematical morphology perspective.…”

Section: Introductionmentioning

confidence: 99%

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