Summary

The aim of this project is to reduce the number of x-ray exposures and enhance the spatial resolution of the reconstructed object in x-ray computed tomography.

Introduction

X-Ray computed tomography (CT) has many applications in biomedical imaging, medical and material sciences, and microscopy. The aim of x-ray CT is to reconstruct certain characteristics of an object using a series of images taken along different angles. Two important characteristics are the absorption coefficient and the distribution of the refractive index of the object.

Conventional x-ray CT exploits differences in the absorption of radiation of the materials in the object. Many of the materials present in biological and medical samples ( e.g. , soft tissues) produce only weak absorption contrasts; meanwhile, they still result in perceptible phase shifts in the x-ray beam. Therefore, the information encoded in the phase of an x-ray beam is a suitable alternative to the absorption coefficient for characterizing such objects. The purpose of phase-contrast and differential phase-contrast x-ray CT is to extract the distribution of the refractive index of the object using the phase.

The linear mathematical models of conventional, phase-contrast, and differential phase-contrast x-ray computed tomography are all based on either the Radon transform or on the first and second derivative of the Radon transform. We have proposed discrete models for the Radon transform and its derivatives and designed their fast and effective implementation.

In practical applications, a challenging issue is that the number of imaging directions is severely limited. To overcome this problem, we formulate the reconstruction as an inverse problem and develop iterative algorithms using prior information to enhance the spatial resolution.

Main Contribution

Our preliminary results suggest that iterative methods provide reconstructions that are more faithful than FBP-like approaches. Specifically, we have proposed a preconditioned conjugate-gradient (PCG) iterative method that reaches the same reconstruction quality as the FBP approach while using half the number of projections. We want now to find a fast TV regularization method for our reconstruction problem. We also investigate accurate but nonlinear models of phase-contrast tomography to develop iterative methods to solve nonlinear inverse problems.