In an influential paper, Tadmor, Nezzar and Vese (2004) 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 Calderón 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.
A multiscale theory for image registration and nonlinear inverse problems / K. Modin, A. Nachman, L. Rondi. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 346(2019 Apr), pp. 1009-1066.
|Titolo:||A multiscale theory for image registration and nonlinear inverse problems|
|Parole Chiave:||Multiscale decomposition; Image registration; Diffeomorphisms; LDDMM; Inverse problems; Calderón problem|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||apr-2019|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.aim.2019.02.014|
|Appare nelle tipologie:||01 - Articolo su periodico|