Model-based reconstruction for magnetic particle imaging in 2D and 3D

Thomas  März; Andreas  Weinmann

doi:10.3934/ipi.2016033

Article Contents

2016, Volume 10, Issue 4: 1087-1110. Doi: 10.3934/ipi.2016033

This issue Previous Article A globally convergent numerical method for a 1-d inverse medium problem with experimental data Next Article Location of eigenvalues for the wave equation with dissipative boundary conditions

Model-based reconstruction for magnetic particle imaging in 2D and 3D

Thomas März^1, and
Andreas Weinmann^2,

1.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quater, Woodstock Road, Oxford OX2 6GG
2.
Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Schöfferstraße 3, 64295 Darmstadt

Received: September 2015

Revised: July 2016

Published: November 2016

Abstract / Introduction Related Papers Cited by

Abstract

We contribute to the mathematical modeling and analysis of magnetic particle imaging which is a promising new in-vivo imaging modality. Concerning modeling, we develop a structured decomposition of the imaging process and extract its core part which we reveal to be common to all previous contributions in this context. The central contribution of this paper is the development of reconstruction formulae for MPI in 2D and 3D. Until now, in the multivariate setup, only time consuming measurement-based approaches are available, whereas reconstruction formulae are only available in 1D. The 2D and the 3D (describing the real world) reconstruction formulae which we derive here are significantly different from the 1D situation -- in particular there is no Dirac property in dimensions greater than one in the high resolution limit. As a further result of our analysis, we conclude that the reconstruction problem in MPI is severely ill-posed. Finally, we obtain a model-based reconstruction algorithm.

Keywords:

Mathematics Subject Classification: Primary: 92C55, 94A12, 94A08; Secondary: 44A35, 65R32.

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