2020
DOI: 10.4171/rlm/908
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The equivalence of Fourier-based and Wasserstein metrics on imaging problems

Abstract: We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At di¤erence with the original one, the new Fourier-based metrics are well-defined also for probability distributions with di¤erent centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fo… Show more

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Cited by 7 publications
(6 citation statements)
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“…Generally, there is a large literature on algorithms to calculate optimal transport, of which we cite only a recent few. Among popular cutting-edge algorithms are fast approximations in the Fourier space [35] and in the wavelet space [36]. Also popular is entropic regularization [37], which is known as Sinkhorn distance.…”
Section: Methodsmentioning
confidence: 99%
See 1 more Smart Citation
“…Generally, there is a large literature on algorithms to calculate optimal transport, of which we cite only a recent few. Among popular cutting-edge algorithms are fast approximations in the Fourier space [35] and in the wavelet space [36]. Also popular is entropic regularization [37], which is known as Sinkhorn distance.…”
Section: Methodsmentioning
confidence: 99%
“…One interesting and straightforward future improvement is to reduce the number of transports needed by setting d i,j = ∞ if |i − j| > c for some threshold c. Generally, there is a large literature on algorithms to calculate optimal transport, of which we cite only a recent few. Among popular cutting-edge algorithms are fast approximations in the Fourier space [35] and in the wavelet space [36]. Also popular is entropic regularization [37], which is known as Sinkhorn distance.…”
Section: Materials and Methods (A) Wasserstein Distancementioning
confidence: 99%
“…The Fourier-based metric, also known as the Tanaka metric, was first introduced in [8] based on the work of [19,20] to study the large-time asymptotics of Boltzmann equation for Maxwellian molecules. In recent years, several investigations were performed to relate the optimal transport theory to Fourier-based metric [2,1,16]. In this work, we generalize the Fourier-based metric and present its connection with the weighted 2 -norm for FWI applications.…”
Section: Theoretical Framework 21 the Fourier-based Metricmentioning
confidence: 99%
“…One interesting and straightforward future improvement is to reduce the number of transports needed by setting d i,j " 8 if |i ´j| ą c for some threshold c. Generally, there is a large literature on algorithms to calculate optimal transport, of which we cite only a recent few. Among popular cutting-edge algorithms are fast approximations in the Fourier space [3] and in the wavelet space [48]. Also popular is entropic regularization [11], which is known as Sinkhorn distance.…”
Section: Wasserstein Distancementioning
confidence: 99%