Representations for Morphological Image Operators and Analogies with Linear Operators

Maragos, Petros

doi:10.1016/b978-0-12-407702-7.00002-4

Cited by 24 publications

(26 citation statements)

References 112 publications

(201 reference statements)

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“…In particular, given A ∈ R m×n max , x ∈ R n max , b ∈ R m , it is given by the following formula 2 : n j=1 (A ij + x j ) = b i , i = 1, · · · , m 1 An alternative notation that has been used in the literature is ⊕ for maximum (max-plus "addition") and ⊗ for addition (max-plus "multiplication")-see Cuninghame-Green (1979) or Baccelli et al (1992). Here, we follow the notation of lattice theory-see Birkhoff (1967), Maragos (2013), Maragos (2017), where the symbol ∨/∧ is used for max/min operations. We also use the classic symbol "+" for real addition, without obscuring the addition with the less intuitive symbol ⊗.…”

Section: Max-plus Linear Equation and Exact Solutionmentioning

confidence: 99%

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Sparsity in max-plus algebra and systems

Tsiamis

Maragos

2019

Discrete Event Dyn Syst

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We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the ℓ 1 residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations. * The paper was published in the Discrete Event (2008). A unification of max-type algebras and their duals using weighted lattices with applications to nonlinear dynamical systems was presented in Maragos (2017). Meanwhile, in the last decade, we have experienced an increase of interest in sparsity in linear equations and linear systems. A solution of a linear equation is sparse when it has many zero elements. The reason we are interested in such solutions, is that they need less elements to describe the same information. They provide us a way of compressing the available data, throwing away those that are unnecessary (Donoho, 2006). They also reveal the structure of partially known signals (Candès et al, 2006) or systems (Chen et al, 2009). In control systems, sparsity has been sought in the sense of minimizing the number of sensors or actuators, subject to energy (Tzoumas et al, 2016;Summers et al, 2016) or observability-controllability constraints (Pequito et al, 2016).Although sparsity has been extensively studied in the linear setting (Elad, 2010), it is still not much developed in more general nonlinear settings. In this work, we aim to define and study sparsity in the max-plus algebraic setting. A sparse solution of a max-plus equation is a solution with many non-informative elements, i.e. the infinite elements. As in the linear case, such solutions use the least number of elements to describe the same information, thus yielding compressed data. But there are many other applications where sparsity could be relevant. For example, in max-plus systems (either static or dynamical), finding sparse inputs implies that we are activating fewer actuators/machines, thus, saving resources. Similarly, the problem of selecting few sensors to observe a max-plus system could be expressed in terms of designing sparse output matrices. Another application co...

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Section: Max-plus Linear Equation and Exact Solutionmentioning

confidence: 99%

“…The notion of residuated and residual maps is also related to the notion of adjunctions in lattice theory, e.g. seeMaragos (2013),Maragos (2017), as well as the notion of Galois Connections, e.g seeAkian et al (2005).…”

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confidence: 99%

Sparsity in max-plus algebra and systems

Tsiamis

Maragos

2019

Discrete Event Dyn Syst

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“…Further, while previous work focused mainly on the (max, +) or (min, +) formalism, we join both using CWLs and generalize them by replacing + with any operation ⋆ that distributes over ∨ and a dual operation ⋆ ′ that distributes over ∧. The corresponding generalized scalar arithmetic is governed by a rich algebraic structure, called clodum, which we developed in previous work [45,46] and further refine herein. This clodum serves as the 'field of scalars' for the CWLs and binds together a pair of dual 'additions' with a pair of dual 'multiplications'; as opposed to max-plus, in some cases the 'multiplications' do not have inverses.…”

Section: Contributions Of Our Workmentioning

confidence: 99%

“…To the above definitions we add the word complete if M is a complete lattice and the distributivities involved are infinite. We call the resulting algebra a complete lattice-ordered double monoid, in short clodum [45,46]. Previous works on minimax or max-plus algebra and their applications have used alternative names 2 for algebraic structures similar to the above definitions which emphasize semigroups and semirings instead of lattices.…”

Section: Lattice-ordered Monoids and Clodummentioning

confidence: 99%

“…Other cases are much less studied or relatively unknown. Our theoretical analysis is based on a relatively new type of nonlinear space we have developed in recent work [46] and further refine herein, which we call complete weighted lattice (CWL). This combines two vector or signal generalized 'additions' of the supremum (∨) or the infimum (∧) type and two generalized scalar multiplications, ⋆ and its dual ⋆ ′ , which distribute over ∨ and ∧ respectively.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical systems on weighted lattices: general theory

Maragos

2017

Math. Control Signals Syst.

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In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra. Their algebraic structure has a polygonal geometry. Some of the special cases unified include max-plus, maxproduct, and probabilistic dynamical systems. We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using lattice adjunctions, stability and controllability. We outline applications in state-space modeling of nonlinear filtering; dynamic programming (Viterbi algorithm) and shortest paths (distance maps); fuzzy Markov chains; and tracking audio-visual salient events in multimodal information streams using generalized hidden Markov models with control inputs.

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Tropical Geometry, Mathematical Morphology and Weighted Lattices

Maragos

2019

Lecture Notes in Computer Science

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Representations for Morphological Image Operators and Analogies with Linear Operators

Cited by 24 publications

References 112 publications

Sparsity in max-plus algebra and systems

Sparsity in max-plus algebra and systems

Dynamical systems on weighted lattices: general theory

Tropical Geometry, Mathematical Morphology and Weighted Lattices

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