3D Spectral Imaging based on Compton scattering: data modelling and reconstruction strategies

• Applicant: Dr. Gael Rigaud
• Subject Area: Mathematics ; Medical Physics, Biomedical Technology
• Term: from 2019 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 419953439

• Final Report Year: 2022

Final Report Abstract

The current development of high-sensitive energy detectors, i.e. able to collect incoming photons in terms of energy, opens the way to new applications in imaging. Among them, Compton scattering imaging (CSI) is a nascent spectral imaging concept based on the Compton effect, i.e. the scattering of a photon by an electron, and exploiting the scattered radiation as a specimen of interest is illuminated by a gamma source. Indeed, the energy variable was unavailable in the past and so ignored in the standard imaging systems such as Computerized Tomography (CT) or Cone-beam CT. This project followed on from our previous DFG project in which we analyzed 2D/3D Compton scattering imaging in terms of modelling and reconstruction methods assuming only first-order scattering and a monochromatic external source. To go further, the new project strove to take into account a more realistic modelling for the spectral data and to develop suited and efficient reconstruction methods. In fact the associated inverse problem to CSI rises two challenges which are at the core of this project: (i) modelling the scattered spectral data since the measured photons can be scattered multiple times before to be detected, the photon beam suffers a significant attenuation along the traveling path and the stochastic nature of ionising sources leads to a significant noise; (ii) developing reconstruction techniques for the 2D/3D electron density map or at least features of the targeted object and able to handle model inexactness and limitations in the spectral CSI data. The first task of this project consisted in establishing a general formulation for the multiple scattering and an integral representation for the first-order and more importantly for the second-order scattering. Nonlinear, these integral operators were linearly approximated into Fourier integral operators and these latter were studied in terms of smoothness. This study showed that the first-order scattering carries a richer information than the rest of the spectrum and was validated by Monte-Carlo simulations. Therefore, a general reconstruction strategy stems from this observation on the smoothness properties and consisted in focusing on the first-order scattered radiation combined with differential operators. This strategy is reasonable as the computational complexity of the modelings of higher-order scattered radiation is way too expensive. This approach was successfully illustrated when we recovered the contours of the electron density from a filtered backprojection-type formula. Furthermore, we showed how an approximate of the electron density can be recovered from 2D CSI data with multiple source positions via a joint CT-CSI bimodality. Such an approximate is then built as the solution of a TV-constrained minimization problem based on this strategy. However, the approximation of the forward models as well as the multiple scattering in general leads to a subtstantial model inexactness. This is the reason why the second part of this project focused on the development of data-driven techniques able to handle model uncertainty and inexactness. We considered the recently developed RESESOP technique and proved some relationships (in terms of convergence and regularization properties) between a semi-discrete model and the fully-discrete model. The obtained reconstructions of the electron density were accurate and convincing. The second approach considered the DIP-architecture. Since the original loss function did not deliver satisfactory results, we enhanced it by incorporating the model uncertainty levels from the RESESOP. This improved vastly the reconstruction quality but did not compare with the RESESOP result. However, this opens the way to a larger class of DIP-architectures to tackle the model inexactness issue which we are currently working on. In conclusion, this project results in a successfull theoretical and algorithmic framework for CSI and has developed into further collaborations aiming to the construction of the first CSI-scanner.

Publications

• Joint fan-beam CT and Compton scattering tomography: analysis and reconstruction algorithm
  L. Kuger & G. Rigaud
  
• Reconstruction algorithm for 3D Compton scattering imaging with incomplete data. Inverse Problems in Science and Engineering, 29(7), 967-989.
  Rigaud, G. & Hahn, B. N.
  
• 3D Compton scattering imaging with multiple scattering: analysis by FIO and contour reconstruction. Inverse Problems, 37(6), 064001.
  Rigaud, Gaël
  
• Modeling and Reconstruction Strategy for Compton Scattering Tomography with Scintillation Crystals. Crystals, 11(6), 641.
  Kuger, Lorenz & Rigaud, Gael
  
• Imaging based on Compton scattering: model uncertainty and data-driven reconstruction methods. Inverse Problems, 39(3), 034004.
  Gödeke, Janek & Rigaud, Gaël