Bellman processes

Akian, Marianne; Quadrat, Jean-Pierre; Viot, M.

doi:10.1007/bfb0033561

Cited by 39 publications

(57 citation statements)

References 4 publications

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“…Moreover, they described this relation by means of the Cramér transform. In the context of decision theory, this connection has been analysed by Akian et al [2]. However, since their formulation would lead to divergent integrals we have to persue a different strategy.…”

Section: Introductionmentioning

confidence: 99%

“…On the one hand, this framework extends the results of Burgeth and Weickert [12] to differential and pseudodifferential operators. On the other hand, it allows to transfer the results of Akian et al [2] to a scale-space and image processing setting. In particular, it enables us to derive the morphological counterparts of the Poisson scale-space, α-scale-spaces, summed α-scale-spaces, relativistic scalespaces, and anisotropic variants of them.…”

Section: Introductionmentioning

confidence: 99%

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Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Schmidt

Weickert

2016

J Math Imaging Vis

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It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramér-Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton-Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, α-scale-spaces, summed α-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Schmidt

Weickert

2016

J Math Imaging Vis

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“…Here the homomorphism between dilation/erosion and diffusion/inverse diffusion is given by the Cramer transform C = F • log •L , [4], [1], which is a concatenation of the multivariate Laplace transform, logarithm and Fenchel transform. The Fenchel transform maps a convex function c :…”

Section: Phase Invariant Convection On Gabor Transformsmentioning

confidence: 99%

Left Invariant Evolution Equations on Gabor Transforms

Duits

Führ

Janssen

2011

Computational Imaging and Vision

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By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H r . By using the left invariant vector fields on H r and the corresponding left-invariant vector fields on on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive left-invariant convection (reassignment), while maintaining the original signal.

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“…We have first to define convergence of sequences of decision variables. We have defined counterparts of each of the four classical kinds of convergence used in probability in previous papers (see [3]). Let us recall the definition of the two most important ones.…”

Section: Limit Theorems For Decision Variablesmentioning

confidence: 99%

“…The analogues of Markov chains, continuous time Markov processes, Brownian and diffusion processes have also been given in [3].…”

Section: Limit Theorems For Decision Variablesmentioning

confidence: 99%

Geometric and Dynamical Aspects of Real Submanifolds of Complex Space

Webster

1995

Proceedings of the International Congress of Mathematicians

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In the modeling of human activities, in contrast to natural phenomena, quite frequently only the operations max (or min) and + are needed. A typical example is the performance evaluation of synchronized processes such as those encountered in manufacturing (dynamic systems made up of storage and queuing networks). Another typical example is the computation of a path of maximum weight in a graph and more generally of the optimal control of dynamical systems. We give examples of such situations. The max-plus algebra which is a mathematical framework well suited to handle such situations. We present results on i) linear algebra, ii) system theory, iii) duality between probability and optimization based on this algebra.1 Max-Plus Linear Algebra Definition 1. 1. A abelian monoid K is a set endowed with one operation which is associative, commutative and has a zero element ". 2.A semiring is an abelian monoid endowed with a second operation which is associative and distributive with respect to which has an identity element denoted e, with " absorbing (that is " a = a " = ").3. A dioid is a semiring which is idempotent (that is a a = a, 8a 2 K ). 4. A semifield is a semiring having its second operation invertible on K ? = K n f " g . 5.A semifield which is also a dioid is called an idempotent semifield. 6. We will say that these structures are commutative when the product is also commutative.7. We call R max [resp. R min ]the set R[ f 1g [resp. R[ f + 1g] endowed with the two operations = max [resp = min] and = + .8. We call R n n max and analogously R n n min the set of n n matrices with entries belonging to R max endowed with denoting the max entry by entry and defined by [AB]

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Bellman processes

Cited by 39 publications

References 4 publications

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Left Invariant Evolution Equations on Gabor Transforms

Geometric and Dynamical Aspects of Real Submanifolds of Complex Space

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