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# Bar experiment. Evolution of the length of the upper level lines of the estimated density along time t: |ρ(t, x) > i/10|, for i = 1 · · · 9 . The left plot is the classic optimal transport of [4], the middle one is the proposed approach with an incompressible penalization and the right one corresponds to the proposed approach with a rigid penalization. Penalizing the norm of the velocity makes the level lines preserved along the computed path.

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Optimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier \cite{BB00} where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpo...

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... the example of Figure 7 that presents a rotating bar, the rigid penalization (last line) recovers a quasirotation, which better preserves the prior physics with respect to pure optimal transport (first line). As expected, it can also be observed in Figure 8 that the length of the level lines of the estimated density are preserved with the incompressible and rigid penalization approaches. We refer the reader to [11] for more synthetic examples involving such penalizations. ...

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## Citations

... Later, Kantorovich [36] introduced a convex relaxation of Monge's original formulation and applied it to economics. Since then, this topic has attracted more attentions and it has also played an increasing role in image processing [33,55,63], machine learning [3,24,35] and statistics [58,66]. The comprehensive theoretical investigations can be found in [68,69]. ...

This work provides an inexact primal-dual algorithm for a large class of optimal transport problems. It is based on the implicit Euler discretization of a proper dynamical system for linearly constrained convex optimization problems, and by using the tool of Lyapunov function, the global (super-)linear convergence rate is established for the objective residual and feasibility violation. The presented method contains an inner problem that possesses a strong semismoothness property, which motivates the use of the semismooth Newton iteration. In addition, by exploring the hidden structure of the problem itself, the linear equation arising from the Newton step is transferred equivalently into a graph Laplacian system, for which a robust algebraic multigrid method is proposed and analyzed via the famous Xu–Zikatanov identity. Finally, numerical tests are provided to validate the efficiency of our method.

... Recently, optimal transport has gained much interests in computational and applicable fields, such as machine learning [29,3] inverse problem [10,9,34], image processing [42,32,38,40,22], computer vision [46,48,30] and others. The original optimal transport problem was proposed by Monge in 1781 where the author considered a delivery task for transporting a certain amount of sand from one place to another with minimal cost [41]. ...

... The dynamic one provides us a view of the optimal flow of the mass movement at each time, which is valuable information for some specific applications. This, and its variants appearing subsequently, have also been applied for image processing and computer vision tasks such as image interpolation [43,17,32]. From the numerical point of view, the large scale of this problem will lead some troubles. ...

... Moreover, for the Wasserstein-1 distance problem, the multilevel primal-dual algorithm was discussed in [35]. The classical dynamic optimal transport problems, see [5,17,32,24] for instance, aim to find the sequential maps involving from a measure ρ 0 to another one ρ 1 under a least cost. Intuitively, these two measures are both known. ...

Optimal transport problem has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. In this paper, we incorporate optimal transport into linear inverse problems as a regularization technique. We establish a new variational model based on Benamou-Brenier energy to regularize the evolution path from a template to latent image dynamically. The initial state of the continuity equation can be regarded as a template, which can provide priors for the reconstructed images. Also, we analyze the existence of solutions of such variational problem in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a special grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation.

... Returning to optimal transport, we begin to review some literature. It has gained much interest in computational and applicable fields, such as machine learning [28,4], inverse problems [12,9,34], image processing [41,32,37,38,39,23], computer vision [45,47,29] and others. The original optimal transport problem was proposed by Monge in 1781 [40]. ...

... The well-known dynamic optimal transport was proposed in [5], equivalent to 2-Wasserstein distance as mentioned in (4). That and its variants appearing subsequently have been applied for image processing and computer vision tasks [43,19,32]. Also, the development of efficient algorithms for dynamic optimal transport has attracted much attention. ...

... As one knows, a centered-grid-only discretization strategy is not effective for the proposed problem due to the presence of the continuity equation, since it may result in chessboard oscillations. Therefore, we employ the staggered grids on variable m which can be found in [24,43,32], while others are on centered grids. In this paper, we focus on the case of d = 2 as the same dimension of the imaging problem. ...

... Later, Kantorovich [46] introduced a convex relaxation of Monge's original formulation and applied it to economics. Since then, this topic attracted more attentions and it also played an increasing role in imaging processing [42,67,77], machine learning [3,29,45] and statistics [71,80]. We refer the readers to [82,83] for comprehensive theoretical investigations. ...

This work is concerned with the efficient optimization method for solving a large class of optimal mass transport problems. An inexact primal-dual algorithm is presented from the time discretization of a proper dynamical system, and by using the tool of Lyapunov function, the global (super-)linear convergence rate is established for function residual and feasibility violation. The proposed algorithm contains an inner problem that possesses strong semismoothness property and motivates the use of the semismooth Newton iteration. By exploring the hidden structure of the problem itself, the linear system arising from the Newton iteration is transferred equivalently into a graph Laplacian system, for which a robust algebraic multigrid method is proposed and also analyzed via the famous Xu--Zikatanov identity. Finally, numerical experiments are provided to validate the efficiency of our method.

... Chewi et al. (2021) extend these works by providing computationally efficient algorithms for computing measure-valued splines. Hug et al. (2015) modify the usual displacement interpolation between probability measures by introducing anisotropy to the domain on which the measures are defined. This change corresponds to imposing preferred directions for the local displacement of probability mass. ...

... The kinetic energy in the integrand of (1) encourages probability mass to travel in straight lines. This assumption is occasionally undesirable, but it is straightforward to modify Eq. (1) to encourage v to point in specified directions (Hug et al., 2015): ...

... While works like (Hug et al., 2015;Zhang et al., 2022) investigate the modeling applications of anisotropic optimal transport, they only consider the case where the Riemannian metric A(x) is available a priori. This assumption is unrealistic for many problem domains, motivating our model, which learns a metric from cross-sectional samples from populations evolving over time. ...

We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using backpropagation. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.

... In this case, one is interested in reconstructing the evolution of a quantity of interest such as Sea Surface Temperature (SST) or Sea Surface Height (SSH) between two given observations. As highlighted in [19], for this type of applications one needs to include appropriate regularization terms to avoid the appearance of unphysical phenomena such as mass concentration in the reconstructed density evolution. ...

In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by Papadakis et al. (SIAM J Imaging Sci, 7(1):212–238, 2014). We solve the discrete problem using a proximal splitting approach and we show how to modify this in the presence of regularization terms which are relevant for physical data interpolation.

... Registrations between images or shapes can be built from velocity field-induced diffeomorphisms. Hug et al. [2015] regularize the velocity field in dynamical optimal transport and prove existence of minimizers with a velocity gradient regularizer. Eisenberger et al. [2019] compute static volume-preserving, velocity fields for mesh registration. ...

Much of computer-generated animation is created by manipulating meshes with rigs. While this approach works well for animating articulated objects like animals, it has limited flexibility for animating less structured creatures such as the Drunn in "Raya and the Last Dragon." We introduce Wassersplines, a novel trajectory inference method for animating unstructured densities based on recent advances in continuous normalizing flows and optimal transport. The key idea is to train a neurally-parameterized velocity field that represents the motion between keyframes. Trajectories are then computed by pushing keyframes through the velocity field. We solve an additional Wasserstein barycenter interpolation problem to guarantee strict adherence to keyframes. Our tool can stylize trajectories through a variety of PDE-based regularizers to create different visual effects. We demonstrate our tool on various keyframe interpolation problems to produce temporally-coherent animations without meshing or rigging.

... structing the evolution of a quantity of interest such as Sea Surface Temperature (SST) or Sea Surface Height (SSH) between two given observations. As highlighted in [69], for this type of applications one needs to include appropriate regularization terms to avoid the appearance of unphysical phenomena such as mass concentration in the reconstructed density evolution. In this work we propose a finite element approach to solve the dynamical formulation of optimal transport with quadratic cost on unstructured meshes (and therefore can be easily implemented on complex domains) and that can be easily modified to include different type of regularizations which are relevant for the dynamic reconstruction and interpolation of physical quantities. ...

This thesis is devoted to the design of locally conservative and structure preserving schemes for Wasserstein gradient flows, i.e. steepest descent curves in the Wasserstein space. The time discretization is based on variational approaches that mimic at the discrete in time level the behavior of steepest descent curves. These discretizations involve the computation of the Wasserstein distance, an instance of optimal transport problem. The space discretization is based on Two-Point Flux Approximation (TPFA) finite volumes, a well-known methodology particularly suited for the discretization of partial differential equations that present a conservative structure. In order to preserve the variational structure at the discrete level, we follow a first discretize then optimize approach. We start by presenting TPFA discretizations for the Wasserstein distance based on the Benamou-Brenier dynamical formulation. We expose some stability issues related to these discetizations, propose a possible solution to overcome them and derive quantitative estimate on the convergence of the discrete model. To solve the discrete optimization problem, we introduce an interior point strategy. Then, we propose first and second order accurate schemes for Wasserstein gradient flows. At this level, to reduce the computational complexity, we use an implicit linearization of the Wasserstein distance. By taking adavantage of the monotonicity of the upwind reconstruction, we propose a first order scheme which can be efficiently solved with a Newton method and show its convergence towards distributional solutions of the Fokker-Planck equation. In order to higher the accuracy in space, we use a centered reconstruction, which requires a different optimization technique. We use again the interior point strategy for this purpose. Finally, we propose a modified variational BDF2 time discretization and prove its convergence towards Wasserstein gradient flows. Thanks to these new discretizations, we design a second order accurate scheme in both time and space. All our approaches are validated with several numerical results.

... In the last 30 years, great advances in the understanding of the underlying theory have been achieved [3,45,48]. However, only recently these techniques are starting to be applied in order to solve computational problems in a great variety of fields, with logistic problems [8,[17][18][19], crowd dynamics [36,37], image processing [28,34,38,40,43,46,47], inverse problems [15,31] and machine learning [5,26,27,39,44,51] being a few examples. ...

... Moreover, it motivated recent developments in unbalanced optimal transport theory [20,21,32,33], that is, when the marginals are arbitrary positive measures. Finally, as the Benamou-Brenier energy provides a description of the optimal flow of the transported mass at each time t, which is a valuable information in applications, it was recently employed as a regularizer for variational inverse problems [13,15,28,34,49] (see also a forthcoming paper by Bredies, Carioni, Fanzon and Walter). The goal of this paper is to characterize the extremal points of the unit ball of the Benamou-Brenier energy B at (3), and of a coercive version of it, which is obtained by adding the total variation of ρ to B. Both functionals are constrained via the continuity equation (2). ...

... Note that Z is measurable, due to the measurability of the map L at (28). We claim that σ(Z) = 0. ...

In this paper, we characterize the extremal points of the unit ball of the Benamou–Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite-dimensional data and optimal-transport based regularization.

... Figure A.2: Representation of the well-known mass-splitting phenomenon with two gaussian distribution, from [54]. The two gaussian densities are initially vertical and we target a π/2-rotation. ...

... In the literature, we can nd generalized optimal transportation algorithms which impose physical constraints such as rigidity or incompressibility conditions [19,54]. The goal of such penalizations or ow constraints is to preserve the compactness and local properties of the mass densities. ...

... The goal of such penalizations or ow constraints is to preserve the compactness and local properties of the mass densities. For instance, standard optimal transportation validation tests such as the rotation of two gaussian distributions demonstrated a considerable improvement regarding the conservation of physics (see Fig. A.3). Figure A.3: Physic-like properties are imposed to recover proper rotation and mass movements, from [54]. The rotation of gaussians shows the impact of translation (left) and rigid (right) penalization: rigid constraints embed a rigid rotational motion. ...

Biologists use zebrafish as an animal model to study the effects of genetic or environmental factors related to human locomotor diseases in order to develop pharmacological treatments. The general objectives of the project were 1) to develop a numerical model based on real-world data capable of accurately simulating the escape swimming of the zebrafish eleuthero-embryo and 2) to provide, in addition to swimming kinematic parameters, a fine estimate of the energetic performance of locomotor behavior to enrich experimental studies on locomotion. Furthermore, an experiment-based numerical modeling might enhance the understanding of locomotor behavior. For this purpose, a computational fluid dynamics code describing the fluid flow around a moving and deforming immersed body was used to reproduce in silico the experimental escape response of a five-day post-fertilization eleuthero-embryo. The solution of the mechanistic model, governed by the incompressible Navier-Stokes equations and Newton's laws was approximated on a Cartesian mesh while the solid body represented by a level-set function, was described implicitly by a penalization method. As for the deformation kinematics, it was estimated directly from experimental locomotion videos by a Procrustes analysis. A first approach has been considered to extract the deformation velocity, in two dimensions, based on optimal transportation. In order to be faithful to three-dimensional (3D) physics, the morphology of the zebrafish eleuthero-embryo and the experimental escape kinematics were reconstructed in 3D, by tracking Lagrangian markers on the surface of the zebrafish body. Thus, a new approach has been developed to estimate the deformation velocity from experimental real data obtained by ultra-high-speed imaging after electric field pulse stimulation. Zebrafish eleuthero-embryo exhibits a highly stereotyped and complex escape behavior consisting of three swimming modules: C-bend, counter-bend and fast-swimming cyclic phase. The developed approach enables high-performance and realistic numerical simulations of real locomotion. After performing a numerical validation of the model based on each component, a study was conducted on the energetic performance of the zebrafish's escape response, challenged by a change in fluid viscosity. A linear response of the cost of transport, associated with a constant energy expenditure, regardless the fluid environment, was thus demonstrated. This energy study can be extended to any immersed, moving and deformable body and in particular, to any biological experiment such as exposure to a neuro-toxicant, which would alter the locomotor behavior of the eleuthero-embryo. Thus, numerical simulation may enrich the quantitative assessments of biological conditions and pharmacological treatments which lead to disturbing or recovering the locomotor behavior.