We present novel microlocal results for generalized ellipsoid and hyperboloid Radon transforms in Euclidean Space and we apply our results to Ultrasound Reflection Tomography (URT). We introduce a new Radon transform, $R$, which integrates compactly supported distributions over ellipsoids and hyperboloids with centers on a smooth nypersurface, $S$ in $\mathbb{R}^n$. $R$ is shown to be a Fourier Integral Operator and in our main theorem we prove that $R$ satisfies the Bolker condition if and only if the support of the function is in a connected set that is not intersected by any plane tangent to $S$. In this case, backprojection type reconstruction operators such as the normal operator $R^* R$ do not add artifacts to the reconstruction.

We apply our results to a cylindrical geometry that could be used in URT. We prove injectivity results and investigate the visible singularities in this modality. In addition, we present reconstructions of image phantoms in two dimensions that illustrate our microlocal theory.

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. Once the time-dependent motion parameters are known, the particle’s scattering potential can be reconstructed based on the Fourier diffraction theorem, considering it is compatible with making the first order Born or Rytov approximation.

The aim of this presentation is twofold: We first need to detect the motion parameters from the tomographic data by detecting common circles in the Fourier-transformed data. This can be seen as analogue to method of common lines from cryogenic electron microscopy (cryo-EM), which is based on the assumption that the assumption that the light travels along straight lines. Then we can reconstruct the scattering potential of the object utilizing non-uniform Fourier methods.

[1] M. Quellmalz, P. Elbau, O. Scherzer, G. Steidl. Motion Detection in Diffraction Tomography by Common Circle Methods 2022. https://arxiv.org/abs/2209.08086

Modelling the Compton scattering effect leads to many challenges such as non-linearity of the forward model, multiple scattering and high level of noise for moving targets. While the non-linearity is addressed by a necessary linear approximation of the first-order scattering with respect to the sought-for electron density, the multiple-order scattering stands for a substantial and unavoidable part of the spectral data which is difficult to handle due to highly complex forward models. However, the smoothness properties of the operators modelling the different scattering orders suggests that differential operators can be used to reduce the level of multiple scattering. Last but not least, the stochastic nature of the Compton effect may involve a large measurement noise, in particular when the object under study is subject to motion, and therefore time must be taken into account. To tackle these different issues, we discuss in this talk a Bayesian method based on the generalized Golub-Kahan bidiagonalization and explore the possibilities to mimic and improve the stochastic approach with deep neural networks.

As shown by the rise of data-driven and learning techniques, the use of specific features in datasets is essential to build satisfactory solutions to ill-posed inverse problems suffering limitations, sparsity and large level of noise. The well-known total-variation regularization has become a standard approach due to producing good results without any a priori knowledge. Providing additional information, it is possible to improve the reconstruction using forward/backward diffusions. An example is the so-called Perona-Malik functional which is based on a priori on the global contrast. Such a construction of regularizers is particularly relevant with machine learning techniques in which a database can provide natural features and informations such as contrast and edges. We propose to study this approach and to validate its potential on multi-energy CT (computerized tomography) subject to limitations such as sparsitiy and limited angles.