8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"Fourier Reconstruction in Diffraction Tomography"

Quellmalz, Michael

We study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a rigid particle rotating in a trap created by acoustic or optical forces. Under the influence of the inhomogeneous forces, the particle carries out a time-dependent smooth, but irregular motion. The rotation axis is not fixed, but continuously undergoes some variations, and the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, and that the particle’s scattering potential is compatible with making the first order Born or Rytov approximation. By the Fourier diffraction theorem, the measurements of the scattered wave are related to the Fourier transform of the scattering potential on an irregular grid. We derive novel backpropagation formulae for the reconstruction of the scattering potential, which depends on the refractive index inside the object, taking its complicated motion into account. This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform. Furthermore, we consider the case of missing phase information, since often only intensity measurements are available in practice. We propose a new reconstruction approach for ODT with unknown phase information, utilizing a hybrid input-output scheme with TV regularization, and show numerical results.

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