A multiscale theory for image registration and nonlinear inverse problems
Journal article, 2019

In an influential paper, Tadmor et al. (2004) [42] introduced a hierarchical decomposition of an image as a sum of constituents of different scales. Here we construct analogous hierarchical expansions for diffeomorphisms, in the context of image registration, with the sum replaced by composition of maps. We treat this as a special case of a general framework for multiscale decompositions, applicable to a wide range of imaging and nonlinear inverse problems. As a paradigmatic example of the latter, we consider the Calderon inverse conductivity problem. We prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity, driven by the inverse problem itself. We provide novel convergence proofs which work in the general abstract settings, yet are sharp enough to prove that the hierarchical decomposition of Tadmor, Nezzar and Vese converges for arbitrary functions in L-2, a problem left open in their paper. We also give counterexamples that show the optimality of our general results.

Image registration

Inverse problems

Calderon problem


Multiscale decomposition



Klas Modin

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Adrian Nachman

University of Toronto

Luca Rondi

University of Milan

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 346 1009-1066

Subject Categories

Computational Mathematics

Computer Vision and Robotics (Autonomous Systems)

Mathematical Analysis


Basic sciences



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