Multi-physics optimal transportation and image interpolation

Hug, Romain; Maitre, Emmanuel; Papadakis, Nicolas

doi:10.1051/m2an/2015038

Cited by 20 publications

(31 citation statements)

References 25 publications

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“…For example in the colour transfer application we could have considered regularization terms/constraints which would have improved the performance, e.g. [17, 28, 51, 56, 62, 63]. It was not the aim to propose a state-of-the-art method for each application, indeed each application would constitute a paper within its own right.…”

Section: Discussionmentioning

confidence: 99%

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A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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Transport based distances, such as the Wasserstein distance and earth mover'sdistance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the TLp distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled non-positive one-dimensional signals and two-dimensional images, and color transfer.

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Section: Discussionmentioning

confidence: 99%

“…Such choices of regularization on the transport map include penalizing the gradients [17, 62, 63], sparsity [63], average transport [56] and rigidity [28]. One could apply any of the above regularizations to spatially correlated histogram specification.…”

Section: Tlp In Multivariate Signal and Image Processingmentioning

confidence: 99%

A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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“…Being expressed in terms of fluid mechanics quantities, it makes the model very flexible, and allows generalizations to non balanced problems [22,9] which are relevant for practical applications. The introduction of new physical constraints (anisotropy of the domain, free divergence or rigidity of the velocity field) in the dynamic problem has been a subject of study in [18]. The problem is reformulated as the minimization of a convex, proper, lower semi-continuous (l.s.c) energy and can be solved using convex optimization tools [3,24].…”

Section: Introductionmentioning

confidence: 99%

On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities

Hug

Maitre²,

Papadakis

2020

Journal of Mathematical Analysis and Applications

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The dynamical formulation of the optimal transport problem, introduced by J. D. Benamou and Y. Brenier [4], corresponds to the time-space search of a density and a momentum minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate a saddle point of a Lagrangian. We study the convergence of this algorithm in the most general case where initial and final densities vanish on some areas of the transportation domain. Under these conditions, the main difficulty of our study is the proof of existence of a saddle point and of uniqueness of the densitymomentum component, as it leads to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required.

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“…This is different from the case of applications to PDEs, where OT is a tool to prove existence or uniqueness results, or to provide interpretations of the corresponding evolutions, and also from the case of other real-life applications where OT and its variants are mainly a modeling tool. The reader will find many different applications to different issues of interest for the image community in [2,5,8,[10][11][12].…”

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confidence: 99%

“…Think that one of the most spectacular applications of optimal transport, the reconstruction of the early universe 1 , required days of computational time for a set of 10 000 target points, while the methods presented, for instance, in [9] allow to deal with data sets 100 times larger in some minutes. Different methodologies are addressed in this volume, including non-smooth methods [5,9], and methods based on, or extending, the Benamou-Brenier approach [2,8,10,11].…”

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confidence: 99%

Preface

Maury¹,

Santambrogio²

2015

ESAIM: M2AN

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The history of Optimal Transportation is not a long, quiet river. Its birth is commonly dated back to the end of the 18th century, with the problem set by G. Monge in his Mémoire sur la théorie des déblais et des remblais, where the motivation for the problem was the optimal way of moving sand piles for building or military applications. For many modern mathematicians, Monge is the father of OT theory, but his work did not answer to all the questions that we would consider as important issues nowadays, and in particular Monge did not even wonder about a rigorous proof of the existence of an optimizer.A long period of sleep followed, until Kantorovich, who, facing optimization problems related to railway supply chain during World War II, proposed an alternative, relaxed formulation of the transport problem set by Monge. In particular, his work allowed to give both existence and duality results, and paved the way to a modern mathematical treatise of the problem.Initially motivated by those two engineering issues, the problem turned out to be incredibly rich from the mathematical point of view, thereby triggering a huge amount of theoretical questions, some of which are still open. Its connections with many other branches of pure mathematics, and in particular the geometry of manifolds and metric spaces are a very important feature of the theory. In the invited preface by Cédric Villani, the reader will find the point of view on the various topics that are developed in this volume of a mathematician who has been highly involved in the research on these applications to "pure" mathematics.However, in parallel to a huge activity around existence, regularity of transport maps, geometric properties of the Wasserstein space, etc. . . many connections with real world application have surfaced in the last decades. Those links have taken different forms, that we aim at highlighting in the present volume.Firstly, OT has been identified as a natural framework to various existing models (in particular in physics or life sciences), allowing to establish new mathematical properties, or to suggest numerical strategies.In particular, OT allows to study several types of evolution PDEs describing the motion of a density of particles, as it is the case for many (typically parabolic) equations which can be interpreted as gradient flows of suitable energy functionals with respect to optimal transport distances. In [1] the authors study the parabolicparabolic Keller Segel model in chemotaxis via these methods, while in [3] a higher-order optimal transport problem involving gradients is proposed in connection with combinatorics, and a gradient-flow equation is derived from it. Among other models from physics which fit in an OT framework, a very important role is played by the multielectron problems of the Density Functional Theory, which can be interpreted as a multi-marginal Kantorovich problem. In [6] the reader will find new duality results on this model, and in [12] a general introduction to multi-marginal problems and their applicat...

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Multi-physics optimal transportation and image interpolation

Cited by 20 publications

References 25 publications

A Transportation $$L^p$$ Distance for Signal Analysis

A Transportation $$L^p$$ Distance for Signal Analysis

On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities

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