Abstract
In this chapter, we study the mathematical imaging problem of diffraction tomography (DT), which is an inverse scattering technique used to find material properties of an object by illuminating it with probing waves and recording the scattered waves. Conventional DT relies on the Fourier diffraction theorem, which is applicable under the condition of weak scattering. However, if the object has high contrasts or is too large compared to the wavelength, it tends to produce multiple scattering, which complicates the reconstruction. In this chapter, we give a survey on diffraction tomography and compare the reconstruction of low- and high-contrast objects. We also implement and compare the reconstruction using the full waveform inversion method which, contrary to the Born and Rytov approximations, works with the total field and is more robust to multiple scattering.
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Acknowledgements
We thank the anonymous reviewer for carefully reading the manuscript and making various suggestions for its improvement. This work is supported by the Austrian Science Fund (FWF) within SFB F68 (“Tomography across the Scales”), Projects F68-06 and F68-07. FF is funded by the Austrian Science Fund (FWF) under the Lise Meitner fellowship M 2791-N. Funding by the DFG under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, Projektnummer: 390685689) as well as by the DFG project STE 571/19 (Projektnummer: 495365311) is gratefully acknowledged. For the numerical experiments, we acknowledge the use of the Vienna Scientific Cluster VSC-4 (https://vsc.ac.at/).
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Faucher, F., Kirisits, C., Quellmalz, M., Scherzer, O., Setterqvist, E. (2022). Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_115-1
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