We propose a direct nonlinear reconstruction algorithm for Computed Tomography (CT), designed to handle low-dose measurements. It involves the filtered back-projection and adaptive nonlinear filtering in both the projection and the image domains. The filter is an extension of the learned shrinkage method by Hel-Or and Shaked to the case of indirect observations. The shrinkage functions are learned using a training set of reference CT images. The optimization is performed with respect to an error functional in the image domain that combines the mean square error with a gradient-based penalty, promoting image sharpness. Our numerical simulations indicate that the proposed algorithm can manage well with noisy measurements, allowing a dose reduction by a factor of 4, while reducing noise and streak artifacts in the FBP reconstruction, comparable to the performance of a statistically based iterative algorithm. 1. Introduction 1.1. Problem Statement Computed tomography (CT) imaging produces a 3D map of the scanned object, where the different materials are distinguished by their X-ray attenuation properties. In medicine, such a map has a great diagnostic value, making the CT scan one of the most frequent noninvasive exploration procedures practiced in almost every hospital. The attenuation of biological tissues is measured by comparing the intensity of the X-rays entering and leaving the body. The main problem precluding pervasive use of the CT scan for diagnostics and monitoring is the damage caused to the tissues by the X-ray radiation. CT manufacturers make great efforts to reduce the X-ray dose required for images of diagnostic quality. In this work we propose an algorithm that enables a high-quality reconstruction from low-dose (and thus noisy) measurements. In ideal conditions, the information obtained in the scan suffices to build an exact attenuation map, called the CT image. In practice, the measurements are degraded by a number of physical phenomena. The main factors are off-focal radiation, afterglow and crosstalk in the detectors, beam hardening, and Compton scattering (see [1] for a detailed overview). These introduce a structured error into the measurements, mostly the type that is modeled by a convolution with some kernel. Another source of deterioration, dominant in the low-dose scenario, is the stochastic noise. One type of such noise stems from the low photon counts, which occur when the X-rays pass through high-attenuation areas. This phenomenon is similar to the shot noise, encountered in photo cameras in poor lighting conditions.
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