2017/04/27 by Alexandra Koulouri, Koulouri, Alexandra
Engineering · Mathematics · Medicine · Physics and Astronomy · #Electrical and Bioimpedance Tomography #FOS: Biological sciences #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #Quantitative Methods (q-bio.QM) #Radioactive Decay and Measurement Techniques
paper · pdf · doi:10.48550/arxiv.1705.00708
openalex publication_date 2017/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Vector tomography methods intend to reconstruct and visualize vector fields\nin restricted domains by measuring line integrals of projections of these\nvector fields. Here, we deal with the reconstruction of irrotational vector\nfunctions from boundary measurements. As the majority of inverse problems,\nvector field recovery is an ill posed in the continuous domain and therefore\nfurther assumptions, measurements and constraints should be imposed for the\nfull vector field estimation. The reconstruction idea in the discrete domain\nrelies on solving a numerical system of linear equations which derives from the\napproximation of the line integrals along lines which trace the bounded domain.\nThis work presents an extensive description of a vector field recovery, the\nfundamental assumptions and the ill conditioning of this inverse problem. More\nimportantly we show that this inverse problem is regularized via the domain\ndiscretization, i.e. we show that the recovery of an irrotational vector field\nwithin a discrete grid employing a finite set of longitudinal line integrals,\nleads to a consistent linear system which has bounded solution errors. We\nelaborate on the estimation of the solution's error and we prove that this\nrelative error is finite and therefore a stable vector field reconstruction is\nensured. Such theoretical aspects are critical for future implementations of\nvector tomography in practical applications like the inverse bioelectric field\nproblem. We validate our theoretical results by performing simulations that\nreconstruct smooth irrotational fields based solely on a finite number of\nboundary measurements and without the need of any additional or prior\ninformation (e.g. transversal line integrals or source free assumption).\n