Start date
Betweenand

# Computational and theoretical advances in inverse problems with applications to tomographic image reconstruction

 Grant number: 16/24286-9 Support type: Scholarships abroad - Research Effective date (Start): July 01, 2017 Effective date (End): June 30, 2018 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics Principal researcher: Elias Salomão Helou Neto Grantee: Elias Salomão Helou Neto Host: Gabor Tamas Herman Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil Research place: City University of New York, New York (CUNY), United States Abstract The present project approaches two relevant current themes in applied mathematics. The first branch aims at studying Stein-like estimators that are exact, in average and except for a constant term, replacements for non-computable mean squared errors in denoising and inverse problems. We propose to extend current techniques to operate with more general Bregman-divergences and, more importantly, to study theoretical concentration properties of such estimators, which we have observed in practice. A practical application for these techniques is selecting parameters in denoising/regularization methods. The second branch we propose relates to iterative image reconstruction in tomographic imaging. High-resolution tomographic image reconstruction is a very computationally intensive procedure, usually taking roughly $O( n^3 )$ flops for the reconstruction of a $n \times n$ image. Iterative algorithms perform such amount of operations on each iteration, making things even worse computation-wise, despite a noticeable improvement in image quality. The main goal is to use techniques for $O( n^2 \log n )$ projection/backprojection during the iterations of fast incremental methods, and to run these computations on a GPGPU. This would result in the threefold benefit of (i) incremental algorithms (ii) using $O( n^2 \log n )$ flops per iteration (iii) running on parallel architectures. (AU)