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

• Grant number: 16/24286-9
• Support Opportunities: 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 Investigator: Elias Salomão Helou Neto
• Grantee: Elias Salomão Helou Neto
• Host Investigator: Gabor Tamas Herman
• Host 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)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

DE LIMA, CAMILA; HELOU, ELIAS SALOMAO. Fast projection/backprojection and incremental methods applied to synchrotron light tomographic reconstruction. JOURNAL OF SYNCHROTRON RADIATION, v. 25, n. 1, p. 248-256, JAN 2018. (13/07375-0, 13/16762-7, 13/16508-3, 16/24286-9)

HELOU, ELIAS S.; ZIBETTI, MARCELO V. W.; HERMAN, GABOR T.. Fast Proximal Gradient Methods for Nonsmooth Convex Optimization for Tomographic Image Reconstruction. SENSING AND IMAGING, v. 21, n. 1, SEP 5 2020. (13/07375-0, 16/24286-9)

MIQUELES, EDUARDO; KOSHEV, NIKOLAY; HELOU, ELIAS S.. A Backprojection Slice Theorem for Tomographic Reconstruction. IEEE Transactions on Image Processing, v. 27, n. 2, p. 894-906, FEB 2018. (13/07375-0, 14/22040-7, 16/24286-9)

ZIBETTI, MARCELO VICTOR WUST; HELOU, ELIAS SALOMAO; REGATTE, RAVINDER R.; HERMAN, GABOR T.. Monotone FISTA With Variable Acceleration for Compressed Sensing Magnetic Resonance Imaging. IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING, v. 5, n. 1, p. 109-119, MAR 2019. (13/07375-0, 16/24286-9)