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Mathematics of Super-Resolution Biomedical Imaging

by H. Ammari and J. Garnier and H. Kang and L. H. Nguyen and L. Seppecher

(Report number 2016-31)

Abstract
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 these lecture notes 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.

Keywords: Hybrid imaging; Super-resolution; Mathematical Imaging.

@Techreport{AGKNS16_668,
  author = {H. Ammari and J. Garnier and H. Kang and L. H. Nguyen and L. Seppecher},
  title = {Mathematics of Super-Resolution Biomedical Imaging},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-31},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-31.pdf },
  year = {2016}
}

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