Modulo Radon Transform based Tomography (ModTomo)

• Applicant: Dr. Matthias Beckmann
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530863002

The term tomography refers to imaging an object under investigation based on indirect measurements. The most prominent example is X-ray computerized tomography, where X-radiation is used to produce multiple projectional radiographs from different views. At the conceptual core of the mathematical model for tomography is the Radon transform, which maps a function to its line integral values that can be linked to the observed measurements, or a generalization thereof. The internal structure of the object is then algorithmically reconstructed by applying sophisticated inversion schemes tailored to the specific application and measurement process. All reconstruction approaches, however, heavily rely on the quality of the hardware data as any loss of information in the hardware pipeline will degrade the image reconstruction quality. A particular fundamental bottleneck is that all physical sensors operate with a fixed dynamic range. Signal amplitudes that are higher than the dynamic threshold of the sensor result in saturation and this in turn yields a permanent loss of information. Tackling this problem of dynamic range limitations in the context of tomography has recently led to the novel field of high dynamic range (HDR) tomography. While conventional approaches for HDR tomography are based on the fusion of multiple low dynamic range measurements, our recent line of work proposes a radically new approach based on the so-called modulo Radon transform (MRT), a novel nonlinear generalization of the classical Radon transform. In its implementation, the MRT is analogous to the conventional Radon transform in that, at each angle, it computes the line integral values of a function in Euclidean space. The unconventional aspect of the MRT is that instead of encoding these pointwise Radon projections that may potentially saturate, the MRT incorporates a modulo operation that performs a reset before the saturation level is reached. This computational imaging approach can be realized in hardware by incorporating so-called modulo analog-to-digital converters, which have been recently developed, and leads to a nonlinear inverse problem that requires new mathematical solution strategies. Our previous work focused on an end-to-end development of a one-shot approach for high dynamic range tomography and only first steps towards a mathematically sound theory have been made. The main objective of this project is to develop a profound and in-depth theory of the modulo Radon transform as a mathematical model for one-shot HDR tomography and to verify the theoretical advances by practical experiments and hardware validations. The overall perspective is to develop a mathematical framework required for the model-based reconstruction in high dynamic range tomography in an acquisition protocol suitable for practical applications such as non-destructive testing and medical imaging.

• DFG Programme: Research Grants