Unbalanced Multi-marginal Optimal Transport

Beier, Florian; Lindheim, Johannes von; Neumayer, Sebastian; Steidl, Gabriele

doi:10.1007/s10851-022-01126-7

Cited by 12 publications

(6 citation statements)

References 32 publications

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“…We can obtain an optimal matching between more than two particle types by multi-marginal OT (Kim and Pass, 2014; Pass, 2015) and at the same time account for the not necessarily equal numbers of support points per channel by utilizing an unbalanced OT formulation (Chizat et al, 2018). A combination of both OT generalizations, i.e., multi-marginal optimal unbalanced transport problems, have been recently discussed in the literature (Friesecke et al, 2021; Heinemann et al, 2022; Beier et al, 2022; Le et al, 2022).…”

Section: Resultsmentioning

confidence: 99%

MultiMatch: Geometry-Informed Colocalization in Multi-Color Super-Resolution Microscopy

Naas,

Nies,

et al. 2024

Preprint

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With recent advances in multi-color super-resolution light microscopy it has become possible to simultaneously visualize multiple subunits within complex biological structures at nanometer resolution. To optimally evaluate and interpret spatial proximity of stainings on such an image, colocalization analysis tools have to be able to integrate prior knowledge on the local geometry of the recorded biological complex. Here, we present MultiMatch to analyze the abundance and location of chain-like particle arrangements in multi-color microscopy based on multi-marginal optimal unbalanced transport methodology. Our object-based colocalization model statistically addresses the effect of incomplete labeling efficiencies enabling inference on existent, but not fully observable particle chains. We showcase that MultiMatch is able to consistently recover all existing chain structures in three-color STED images of DNA origami nanorulers and outperforms established geometry-uninformed triplet colocalization methods in this task in a simulation study. Furthermore, MultiMatch also excels in the evaluation of simulated four-color STED images and generalizations to even more color channels can be immediately derived from our analysis. MultiMatch is provided as a user-friendly Python package comprising intuitive colocalization visualizations and a computationally efficient network flow implementation.

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Section: Resultsmentioning

confidence: 99%

MultiMatch: Geometry-Informed Colocalization in Multi-Color Super-Resolution Microscopy

Naas,

Nies,

et al. 2024

Preprint

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“…Beier et al proposed a new framework for the introduction of imbalanced optimal transportation by combining the balanced multi marginal optimal transportation problem and its dual with the Sinkhorn algorithm, and then examining the imbalanced methods with multiple edges and settings. The results indicate that the algorithm framework has good convergence 18 . The Dagger team converted the multi‐edge optimal transport problem to a tensor scaling problem using entropy regularization.…”

Section: Related Workmentioning

confidence: 95%

“…The results indicate that the algorithm framework has good convergence. 18 The Dagger team converted the multi-edge optimal transport problem to a tensor scaling problem using entropy regularization. They solved this mathematically with the multi-edge Sinkhorn algorithm and further sped up computation by combining the tensor network duality of the graphical model with additional low-rank approximations.…”

Section: Related Workmentioning

confidence: 99%

Local feature matching and tracking optimization of machine VSLAM based on AG Sinkhorn

Pan

2024

Concurrency and Computation

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SummaryAt present, the visual Simultaneous localization and mapping method has a wide range of applications in mobile robots and unmanned driving. However, when the environment changes, the accuracy of the system drops sharply. Aiming at the low matching accuracy of local feature matching and tracking optimization in current visual simultaneous localization and mapping methods, the attention map neural network and Sinkhorn algorithm are used to extract and match the image features. Then, it improves the real‐time performance of the visual simultaneous localization and mapping system by local optimization of feature point matching, and proposes an optimization method for visual simultaneous localization and mapping in view of an end‐to‐end algorithm. The experiment illustrates that in the experimental comparison of HPatches View dataset, the Homography estimation was improved by 0.092, the matching accuracy was improved by 0.087, and the matching recall was improved by 0.038. In the experimental comparison of HPatches Illum dataset, the Homography estimation was improved by 0.015, the matching accuracy is improved by 0.076, and the matching recall was improved by 0.036. The local feature matching and tracking optimization method of the visual Simultaneous localization and mapping system through the attention map neural network and Sinkhorn algorithm can enhance the matching accuracy of local feature matching and tracking optimization in dynamic environment and under changing lighting conditions.

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“…The idea behind sliced optimal transport has been generalized and transferred to many related problems. There exists sliced variants [8,16] of partial optimal transport [19,27], where only a fraction of mass is transported, and a sliced version [20] of multi-marginal optimal transport [10,12,30], considering the transport between several measures instead of only two. For optimal transport on Riemannian manifolds, sliced Wasserstein distances based on the push-forward of the eigenfunctions of the Laplacian have been proposed in [77].…”

Section: Introductionmentioning

confidence: 99%

Sliced optimal transport on the sphere

Quellmalz,

Beinert,

Steidl

2023

Inverse Problems

Self Cite

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Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback-Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.

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Unbalanced Multi-marginal Optimal Transport

Cited by 12 publications

References 32 publications

MultiMatch: Geometry-Informed Colocalization in Multi-Color Super-Resolution Microscopy

MultiMatch: Geometry-Informed Colocalization in Multi-Color Super-Resolution Microscopy

Local feature matching and tracking optimization of machine VSLAM based on AG Sinkhorn

Sliced optimal transport on the sphere

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