Reconstruction of Voronoi diagrams in electrical impedance tomography

• Grant number: 22/16733-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2023
• End date: March 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Ernesto Julián Goldberg Birgin
• Grantee: Danilo Rodrigues de Souza
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 18/24293-0 - Computational methods in optimization, AP.TEM

Abstract

The inverse problem of electrical impedance tomography (EIT), also known as the inverse conductivity or Calderón's problem, is an active field of research with an extensive literature. The goal of electrical impedance tomography is to reconstruct the electrical conductivity inside a medium, using measurements of the electrical potential on the boundary obtained by applying boundary currents. EIT is a low cost, noninvasive, radiation free and portable imaging modality with various applications in medical imaging, geophysics, civil engineering and nondestructive testing. In the EIT literature, the conductivity is often supposed to be a relatively smooth, continuous function. However, the case where the conductivity presents discontinuities is important for applications, in particular in geophysics and civil engineering, but also in medicine. In this project we consider the particular case where the conductivity is a piecewise constant function. The domain of definition of the conductivity can then be partitioned into cells such that the conductivity is constant in each cell. In this work we also suppose that the set of cells is given by a Voronoi diagram.The research plan includes a variety of theoretical and numerical challenges that range from proving existence of solutions, computing and evaluating derivatives, solving nonlinear programming problems that requires computing Voronoi diagrams and solving PDE equations at each iteration. This project will be developed together with Prof. Antoine Laurain.