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Mathematics of Super-Resolution Biomedical Imaging

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Professor Prof. Dr. Habib Ammari
Time Mo 10-12, Th 13-15
Coordinator
Francisco Romero Room HG E 22

Aims of the Course

Super-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other. In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold: (i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence of small anomalies; (ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies; (iii) To design efficient inversion algorithms in multi-wave modalities; (iv) to develop inversion techniques using multi-frequency measurements; (v) to develop a mathematical and numerical framework for nanoparticle imaging.

Applications of the anomaly detection and multi-wave approaches in medical imaging are described in some detail. In particular, the use of asymptotic analysis to improve a multitude of emerging imaging techniques is highlighted. These imaging modalities include both single-wave and multi-wave approaches. They can be divided into three groups: (i) Those using boundary or scattering measurements such as electrical impedance tomography, ultrasound, and infrared tomographies; (ii) Those using internal measurements such as magnetic resonance elastography; (iii) Those using boundary measurements obtained from internal perturbations of the medium such as photoacoustic tomography, impediography, and magnetoacoustic imaging.

This course is offered every year in the Spring semester.

Course Material

Introductory Lesson

Lecture Notes

Tutorial Notes

Matlab Codes

SVD regularization code

Random medium generation codes

Spherical means Radon transform inversion

Neumann Poincaré Operator

Electrical Impedance Tomography

Anomaly detection algorithms: MUSIC for Conductivity and MUSIC, Backpropagation and Kirchhoff migration for Helmholtz

Inversion of the spherical radon transform with total variation regularization

Gradient descent for magneto acoustic imaging

OCT Elastography

 

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© 2016 Mathematics Department | Imprint | Disclaimer | 30 June 2016
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