Generalized algebraic reconstruction techniques

Abstract

ABSTRACT We propose a family of new algorithms that can be viewed as a generalization of the Algebraic Reconstruction Techniques (ART). These algorithms can be tailored for trade-offs between convergence speed and memory requirement. They also can be made to include Gaussian a priori image models. A key advantage is that they can handle arbitrary data acquisition scheme. Approximations are required for practical sized image reconstruction. We discuss several approximations and demonstrate numerical simulation examples for computed tomography (CT) reconstructions. Keywords: Algebraic reconstruction techniques, recursive least squares, tomography. 1. INTRODUCTION Many computed imaging problems can be formulated as estimation of an unknown field from measurements that are modeled as linear functions of the field. For example, in x-ray computed tomography (CT), the measurements are line integrals of the unknown image; in magnetic resonance imaging (MRI), the measurements are the so called “k-space” data, which are Fourier transforms of the unknown image; in optical and acoustic diffraction tomography, under the weak scattering approximation, the Fourier diffraction theorem equates the Fourier transform of the measured scattered field to the Fourier transform of the unknown object along a semicircular arc; in synthetic aperture radar (SAR) imaging, the received signal is the coherent (complex linear) sum of the unknown field. Sometimes analytical closed form reconstruction formulae exist, such are filtered back projection for the reconstruction of fan beam x-ray CT, inverse Fourier transform for MRI with recta-linear grid data points, and again inverse Fourier transform for SAR in far field. More often we encounter problems where closed form solutions are not available. This is generally the case for two situations: (i) the data set is incomplete; (ii) the data acquisition configuration is not amenable to closed form analytical analysis. For example, in cone beam x-ray CT, practical data acquisition scheme only provides truncated cone beam projections on a helical orbit. As a second example, in cardio-vascular CT imaging, only a few projections can be measured before the cardiac phase change significantly a third example is MRI with the so called “fast pulse sequence”, where the k-space data aremeasured on a non-cartesian grid, such as a spiral. Similar problem exists in spotlight SAR, where the Fourier transforms of the scene are measured on a polar grid. In all of the above problems and many more, analytical reconstruction is not available, thus we must take a more numerical approach. After choosing appropriate basis functions for the unknown field and discretizing the measurement operation, we can generally formulate the reconstruction problem as solving a (large) linear system of equations: