# A Linear Optimal Transportation Framework for Quantifying and Visualizing Variations in Sets of Images

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## Abstract

Transportation-based metrics for comparing images have long been applied to analyze images, especially where one can interpret the pixel intensities (or derived quantities) as a distribution of ‘mass’ that can be transported without strict geometric constraints. Here we describe a new transportation-based framework for analyzing sets of images. More specifically, we describe a new transportation-related distance between pairs of images, which we denote as linear optimal transportation (LOT). The LOT can be used directly on pixel intensities, and is based on a linearized version of the Kantorovich-Wasserstein metric (an optimal transportation distance, as is the earth mover’s distance). The new framework is especially well suited for computing all pairwise distances for a large database of images efficiently, and thus it can be used for pattern recognition in sets of images. In addition, the new LOT framework also allows for an isometric linear embedding, greatly facilitating the ability to visualize discriminant information in different classes of images. We demonstrate the application of the framework to several tasks such as discriminating nuclear chromatin patterns in cancer cells, decoding differences in facial expressions, galaxy morphologies, as well as sub cellular protein distributions.

## Keywords

Optimal transportation Linear embedding## Notes

### Acknowledgements

The authors wish to thank the anonymous reviewers for helping significantly improve this paper. W. Wang, S. Basu, and G.K. Rohde acknowledge support from NIH grants GM088816 and GM090033 (PI GKR) for supporting portions of this work. D. Slepčev was also supported by NIH grant GM088816, as well as NSF grant DMS-0908415. He is also grateful to the Center for Nonlinear Analysis (NSF grant DMS-0635983 and NSF PIRE grant OISE-0967140) for its support.

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