Singular Feature Extraction and Artefact Reduction in Dynamic Imaging
Final Report Abstract
Imaging modalities are concerned with the non-invasive recovery of some characteristic functions of an object under investigation. From a mathematical point of view, this process corresponds to the solution of an inverse problem and requires a careful modeling and analysis of the problem as well as the development of appropriate numerical solution schemes. Most methods are based on the assumption that the searched-for quantity is independent of time. However, this assumption is violated in many medical and industrial applications, e.g. due to patient and organ motion or while imaging engines at working stage. In this case, the standard reconstruction techniques lead to motion artifacts in the computed images which can significantly impede a reliable diagnostics. To compensate for the motion implies to incorporate the time-dependency of the investigated object in the inverse problem associated to the static case. Adding the time dimension to the searched-for quantity does not only lead to an underdetermined problem, it also alters the nature of the static problem such as the influence of data errors on the solution, the spatial resolution or leads to limited data issues. The goal of this project was to develop a suitable regularization theory for dynamic imaging, including in particular the modeling of motion information, the formulation and analysis of the underlying inverse problems, and the development of efficient reconstruction algorithms. The starting point was a general modeling approach using Fourier integral operators (FIO), which covers a wide range of imaging applications. In the first part of the project, dynamic problems with known motion were analyzed and we developed solution methods of filtered backprojection type. These allow the characteristic features of the object, which are encoded in the dynamic data, to be extracted from the measured data stably and efficiently without motion artifacts. For the case that the exact motion is unknown and only an inaccurate estimate is available, we derived an explicit characterization of the resulting artifacts arising in the reconstruction. Furthermore, local deformations pose a particular challenge, since individual measurement points usually combine information from regions with different dynamic behavior and consequently also induce artifacts in the reconstruction step. Based on our explicit artifact characterization, we were able to develop a numerical method that can significantly reduce these artifacts in a post-processing step. In the second part of the project, the task was devoted to motion estimation by studying two different approaches. Many dynamic processes can be described by partial differential equations, i.e. their (numerical) solution could provide the required motion fields. The potential of this approach was demonstrated together with our cooperation partners at the University of Wurzburg ¨ using the example of an elastic motion model described by the Navier-Cauchy equations. Our second approach circumvents the need for an explicit motion estimation: instead of the exact but unknown dynamic forward operator, a simplified model is used, e.g., the operator of the static case, together with a regularization procedure that takes into account the respective model inexactness. We developed such a procedure based on the sequential subspace optimization method which even holds for general linear inverse problems with inexact forward operator. Overall, we succeeded within this project to develop solution methods that significantly improve the quality of reconstruction in tomographic applications affected by object related motion and, in the long term, enable the non-invasive visualization of dynamic processes in a wide range of applications.
Publications
- A motion artefact study and locally deforming objects in computerized tomography. Inverse Problems, Vol. 33, (2017) 114001
B. N. Hahn
(See online at https://doi.org/10.1088/1361-6420/aa8d7b) - 3D Compton scattering imaging and contour recon-struction for a class of Radon transforms. Inverse Problems, Vol. 34, (2018) 075004
G. Rigaud and B. N. Hahn
(See online at https://doi.org/10.1088/1361-6420/aabf0b) - An efficient reconstruction approach for a class of dynamic imaging operators. Inverse Problems, Vol. 35 (2019) 094005
B. N. Hahn and M.-L. Kienle Garrido
(See online at https://doi.org/10.1088/1361-6420/ab178b) - Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging. Inverse Problems, Vol. 36 (2020) 124001
S. E. Blanke, B. N. Hahn and A. Wald
(See online at https://doi.org/10.1088/1361-6420/abb5e1) - Microlocal properties of dynamic Fourier integral operators in Time-Dependent Problems in Imaging and Parameter Identification, ed. by B. Kaltenbacher, T. Schuster, A. Wald (Springer, Cham, 2021), 85-120
B. N. Hahn, M. L. Kienle-Garrido and E. T. Quinto
(See online at https://doi.org/10.1007/978-3-030-57784-1_4) - Motion compensation strategies in tomography in Time-Dependent Problems in Imaging and Parameter Identification, ed. by B. Kaltenbacher, T. Schuster, A. Wald (Springer, Cham, 2021), 51-83
B. N. Hahn
(See online at https://doi.org/10.1007/978-3-030-57784-1_3) - Reconstruction algorithm for 3D Compton scattering imaging with incomplete data, Inverse Problems in Science and Engineering 29 (2021), 967-989
G. Rigaud and B. N. Hahn
(See online at https://doi.org/10.1080/17415977.2020.1815723) - Using the Navier-Cauchy equation for motion estimation in dynamic imaging, Inverse Problems and Imaging
B. N. Hahn, M. L. Kienle-Garrido, C. Klingenberg and S. Warnecke
(See online at https://doi.org/10.3934/ipi.2022018)
