Journal of Mathematical Imaging and Vision

, Volume 60, Issue 9, pp 1482–1502 | Cite as

Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging

  • Thomas VogtEmail author
  • Jan Lellmann


We develop a general mathematical framework for variational problems where the unknown function takes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation and provide an example where uniqueness fails to hold. Employing the Kantorovich–Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function-valued images, as commonly used in diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets.


Variational methods Total variation Measure theory Optimal transport Diffusion MRI Manifold-valued imaging 


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Authors and Affiliations

  1. 1.Institute of Mathematics and Image Computing (MIC)University of LübeckLübeckGermany

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