Journal of Mathematical Imaging and Vision

, Volume 59, Issue 2, pp 187–210 | Cite as

A Transportation \(L^p\) Distance for Signal Analysis

  • Matthew ThorpeEmail author
  • Serim Park
  • Soheil Kolouri
  • Gustavo K. Rohde
  • Dejan Slepčev


Transport-based distances, such as the Wasserstein distance and earth mover’s distance, 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 non-negative and normalised. 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 \(TL^p\) distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multichannelled 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 multichannelled non-positive one-dimensional signals and two-dimensional images, and colour transfer.


Optimal transport Signal comparison Colour transfer Histogram specification Image Registration 



Authors gratefully acknowledge funding from the NSF (CCF 1421502) and the NIH (GM090033, CA188938) in contributing to a portion of this work. DS also acknowledges funding by NSF (DMS-1516677). The authors are grateful to the Center for Nonlinear Analysis at CMU for its support. In addition, the authors would like to thank the referees for their valuable comments that lead to significant improvements in the paper.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.University of VirginiaCharlottesvilleUSA
  3. 3.HRL LaboratoriesMalibuUSA

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