Tropical and Morphological Operators for Signals on Graphs

Blusseau, Samy; Velasco-Forero, Santiago; Angulo, Jesús; Bloch, Isabelle

doi:10.1109/icip.2018.8451395

Cited by 9 publications

(12 citation statements)

References 20 publications

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“…It would also be interesting to compare these models with the invariance obtained from data augmentation and possibly combine the two approaches in a complementary way. Additionally, future works will explore the use of the proposed morphological scale-spaces on other types of data like 3D point clouds [1], graphs [2], and high dimensional images.…”

Section: Discussionmentioning

confidence: 99%

Scale Equivariant Neural Networks with Morphological Scale-Spaces

Sangalli,

Blusseau,

Velasco-Forero

et al. 2021

Preprint

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The translation equivariance of convolutions can make convolutional neural networks translation equivariant or invariant. Equivariance to other transformations (e.g. rotations, affine transformations, scalings) may also be desirable as soon as we know a priori that transformed versions of the same objects appear in the data. The semigroup cross-correlation, which is a linear operator equivariant to semigroup actions, was recently proposed and applied in conjunction with the Gaussian scale-space to create architectures which are equivariant to discrete scalings. In this paper, a generalization using a broad class of liftings, including morphological scale-spaces, is proposed. The architectures obtained from different scale-spaces are tested and compared in supervised classification and semantic segmentation tasks where objects in test images appear at different scales compared to training images. In both classification and segmentation tasks, the scale-equivariant architectures improve dramatically the generalization to unseen scales compared to a convolutional baseline. Besides, in our experiments morphological scalespaces outperformed the Gaussian scale-space in geometrical tasks.

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Section: Discussionmentioning

confidence: 99%

Scale Equivariant Neural Networks with Morphological Scale-Spaces

Sangalli,

Blusseau,

Velasco-Forero

et al. 2021

Preprint

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“…In the present paper, we focus on complete lattices of the type [a, b] n , where a and b represent the minimal and maximal possible signal values (typically a = 0 and b = 255 for 8-bits images), and n is an integer representing the size of the signal (typically, the number of pixels of an image, reshaped as a column vector). This is a theoretical contribution that can be viewed as a companion paper to previous studies where this framework has been successfully applied to adaptive anisotropic filtering [2,3] 3 . In Section 2 we introduce the matrix-based morphological setting and prove simple but fundamental results: in particular, we characterise the adjunctions that can be represented by matrices and show that these matrices need to be doubly-0-astic.…”

Section: Introductionmentioning

confidence: 92%

“…Proposition 2 Let W ∈ M n and ε W be the function defined by (2). Then ε W is an erosion mapping L to L if and only if W is column-0-astic.…”

Section: Erosions and Adjunctionsmentioning

confidence: 99%

“…While the matrix point of view is not the most appropriate for the implementation of these operators, especially translation-invariant ones, it is a valuable insight for their theoretical understanding. In particular, it can help predict and control complex behaviours such as those of iterated operators based on adjunctions with non-flat, spatially variant and input-adapted structuring functions [2,3]. Indeed, it is a flexible and general framework which embraces a very broad part of morphological literature, and it is supported by the rich theory of Minimax algebra [1,6].…”

Section: Introductionmentioning

confidence: 99%

“…In the abundant literature on spatially-variant morphological image processing, only a few approaches explicitly used the matrix formulation [2,3,13,14], whereas most contributions were limited to flat structuring elements and focused on the local effects of the adaptive strategy. On the theoretical side, the representation of morphological adjunctions by matrices was studied in a setting that does not directly apply to digital signal and image processing, as the co-domain is usually an unbounded lattice, stable under any vertical translation [11].…”

Section: Introductionmentioning

confidence: 99%

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

Blusseau,

Velasco-Forero,

Angulo

et al. 2022

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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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