Wasserstein regularization of imaging problem

Abstract. We present a method to perform histogram matching between two color images based on the concept of harmonic mapping between Riemannian manifolds. The key idea is to associate the histogram of a color image to a Riemannian manifold. In this context, the energy of the matching between the two images is measured by the Dirichlet energy of the mapping between the Riemannian manifolds. Then, we assimilate optimal matchings to critical points of the Dirichlet energy. Such points are called harmonic maps. As there is no explicit expression for harmonic maps in general, we use a gradient descent flow with boundary condition to reach them, that we call harmonic flow. We present an application to color transfer, however many others applications can be envisaged using this general framework.

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“…They include most of the variational formulations of the color transfer literature (see e.g. [5], [9], [17], [19], [20], [25]). …”

Section: Color Transfermentioning

confidence: 99%

Harmonic Flow for Histogram Matching

Batard

Bertalmío

2014

Image and Video Technology – PSIVT 2013 Workshops

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“…Recent applications include astronomy [9, 18, 19], biomedical sciences [3, 25–27, 77, 81, 82, 88, 89], colour transfer [14, 17, 49, 62, 63], computer vision and graphics [7, 44, 60, 65, 68, 74, 75], imaging [36, 40, 64], information theory [78], machine learning [1, 15, 20, 34, 37, 48, 76], operational research [69] and signal processing [54, 58]. …”

Section: Introductionmentioning

confidence: 99%

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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“…The Wasserstein distance is also used in the context of contrast and colour modification in, e.g. [RP11,FPR+13]. In [ZHT03,HZTA04] the quadratic Wasserstein distance is considered to define a rigorous distance between images, applied to non-rigid image registration and warping.…”

Section: Introductionmentioning

confidence: 99%

A generalized model for optimal transport of images including dissipation and density modulation

et al. 2015

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In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.

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Wasserstein regularization of imaging problem

Cited by 39 publications

References 10 publications

Harmonic Flow for Histogram Matching

Harmonic Flow for Histogram Matching

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

A generalized model for optimal transport of images including dissipation and density modulation

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