Wasserstein Barycenter and Its Application to Texture Mixing

Rabin, Julien; Peyré, Gabriel; Delon, Julie; Bernot, Marc

doi:10.1007/978-3-642-24785-9_37

Cited by 383 publications

(457 citation statements)

References 33 publications

(16 reference statements)

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“…Some of these results have been extended by one of us [32] to the case where the support of Ω ∈ P (R n ) is parameterised by a 1-dimensional continuum. Let us note that in addition to economics [8], Wasserstein barycenters in the multi-measure setting have appeared in the literature with applications in image processing [34] and statistics [4].…”

Section: Definition 11 (Wasserstein Barycenter Measure)mentioning

confidence: 99%

Wasserstein barycenters over Riemannian manifolds

Kim

Pass

2017

Advances in Mathematics

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We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω ∈ P (P (M )) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger [9] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

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Section: Definition 11 (Wasserstein Barycenter Measure)mentioning

confidence: 99%

Wasserstein barycenters over Riemannian manifolds

Kim

Pass

2017

Advances in Mathematics

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“…It is also used for other image processing applications such as warping [23] and texture synthesis [24]. This metric is related to the assignment problem [25].…”

Section: Previous Workmentioning

confidence: 99%

“…We also make use of the L p Wasserstein distance using an assignment formulation of the metric, while previous works restrict their attention to p = 1. To speed up the evaluation of the Wasserstein distance, we approximate it using a series of 1-D projections, which is a method introduced in [24] for histogram equalization.…”

Section: Comparison With Previous Workmentioning

confidence: 99%

“…The numerical complexity of solving (9) in dimension d > 1 is prohibitive for imaging applications such as our segmentation problem. To obtain a fast numerical scheme, we follow the work of Rabin et al [24] that introduces a sliced Wasserstein distance. It is defined as an aggregation of 1-D Wasserstein distances of projected distributions…”

Section: -D Wasserstein Distancementioning

confidence: 99%

“…Evaluating this sliced distance (11) has a complexity of O(|Θ|N log(N )) operations which is advantageous over the original Wasserstein distance (9) if Θ is not too large. Although there is no mathematical proof of the quality of the approximation of W using SW , numerical observations suggest that SW is a good approximation to solve minimization problems involving the Wasserstein metric, see [24].…”

Section: -D Wasserstein Distancementioning

confidence: 99%

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Texture Segmentation via Non-local Non-parametric Active Contours

Jung

Peyré

Cohen

2011

Lecture Notes in Computer Science

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Abstract. This article introduces a novel active contour model that makes use of non-parametric estimators over patches for the segmentation of textured images. It is based on an energy that enforces the homogeneity of these statistics. This smoothness is measured using Wasserstein distances among discretized probability distributions that can handle features in arbitrary dimension. It is thus usable for the segmentation of color images or other high dimensional features. The Wasserstein distance is more robust than traditional pointwise statistical metrics (such as the Kullback-Leibler divergence) because it takes into account the relative distances between modes in the distributions. This makes the corresponding energy robust and does not require any smoothing of the statistical estimators. To speed-up the computational time, we propose an alternative metric that retains the main qualities of the Wasserstein distance, while being faster to compute. It aggregates 1-D Wasserstein distances over a set of directions, and thus benefits from the simplicity of 1-D statistical metrics while being able to discriminate high dimensional features. We show numerical results that instantiate this novel framework using grayscale and color values distributions. This allows us to segment regions with smoothly varying intensities or colors as well as complicated textures.

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Differentiation of Bolus Texture During Deglutition via High‐Density Surface Electromyography: A Pilot Study

et al. 2023

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ObjectiveSwallowing is a complex neuromuscular task. There is limited spatiotemporal data on normative surface electromyographic signal during swallow, particularly across standard textures. We hypothesize the pattern of electromyographic signal of the anterior neck varies cranio‐caudally, that laterality can be evaluated, and categorization of bolus texture can be differentiated by high‐density surface electromyography (HDsEMG) through signal analysis.MethodsAn HDsEMG grid of 20 electrodes captured electromyographic activity in eight healthy adult subjects across 240 total swallows. Participants swallowed five standard textures: saliva, thin liquid, puree, mixed consistency, and dry solid. Data were bandpass filtered, underwent functional alignment of signal, and then placed into binary classifier receiver operating characteristic (ROC) curves. Muscular activity was visualized by creating two‐dimensional EMG heat maps.ResultsSignal analysis results demonstrated a positive correlation between signal amplitude and bolus texture. Greater differences of amplitude in the cranial most region of the array when compared to the caudal most region were noted in all subjects. Lateral comparison of the array revealed symmetric power levels across all subjects and textures. ROC curves demonstrated the ability to correctly classify textures within subjects in 6 of 10 texture comparisons.ConclusionThis pilot study suggests that utilizing HDsEMG during deglutition can noninvasively differentiate swallows of varying texture noninvasively. This may prove useful in future diagnostic and behavioral swallow applications.Level of Evidence4 Laryngoscope, 133:2695–2703, 2023

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Wasserstein Barycenter and Its Application to Texture Mixing

Cited by 383 publications

References 33 publications

Wasserstein barycenters over Riemannian manifolds

Wasserstein barycenters over Riemannian manifolds

Texture Segmentation via Non-local Non-parametric Active Contours

Differentiation of Bolus Texture During Deglutition via High‐Density Surface Electromyography: A Pilot Study

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