Tomographic Reconstruction of 3-D Irrotational Vector Fields via a Discretized Ray Transform

Abstract

In this paper, 3-D vector field tomography (3D-VFT) is employed to reconstruct three-dimensional, irrotational fields in a bounded cubic domain. A sampling process along the scanning lines that further assigns the derived points to preordained finite reconstruction points accomplishes data redundancy, lacking when the problem is formed in the continuous domain, and results in the formulation of an over-determined system of linear equations. The only precondition to the system solution, that corresponds to a discretized inversion of the Ray transform, is the known location and values of a limited number of boundary points. The method is accompanied by a theoretical analysis on the regularization achieved and the errors introduced. The effectiveness and robustness of the method are demonstrated by means of simulations of electric fields, a series of perturbation tests, and a comparison with two alternative baseline methodologies.

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Acknowledgments

This work is inspired by and dedicated to the memory of the late Professor Maria Petrou. The authors would like to thank Dr. Vasiliki Kosmidou and Ms. Alexandra Koulouri for their valuable contribution to this work. The latter was carried out as part of the GSRT Research Excellent Grant ARISTEIA, within the 4th Strategic Objective of the operational program “Education and Lifelong Learning” entitled ‘Supporting the Human Capital in order to Promote Research and Innovation’, under Grant agreement 440, Project CBP: Cognitive Brain signal Processing lab, coordinated by the Information Technologies Institute—Centre for Research & Technology—Hellas.

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Correspondence to Leontios J. Hadjileontiadis.

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Papadaniil, C.D., Hadjileontiadis, L.J. Tomographic Reconstruction of 3-D Irrotational Vector Fields via a Discretized Ray Transform. J Math Imaging Vis 52, 285–302 (2015). https://doi.org/10.1007/s10851-015-0559-y

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Keywords

  • 3-D vector field tomography
  • Ray transform
  • Inverse problems
  • Irrotational fields