Reconstruction from divergent ray projections

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Despite major advances in x-ray sources, detector arrays, gantry mechanical design and special computer performances, computed tomography (CT) enjoys the filtered back projection (FBP) algorithm as its first choice for the CT image reconstruction in the commercial scanners [1]. Over the years, a lot of fundamental work has been done in the area of finding the sophisticated solutions for the inverse problems using different kinds of optimization techniques. Recent literature in applied mathematics is being dominated by the compressive sensing techniques and/or sparse reconstruction techniques [2], [3]. Still there is a long way to go for translating these newly developed algorithms in the clinical environment. The reasons are not obvious and seldom discussed [1]. Knowing the fact that the filtered back projection is one of the most popular CT image reconstruction algorithms, one pursues research work to improve the different error estimates at different steps performed in the filtered back projection. In this paper, we present a back projection formula for the reconstruction of divergent beam tomography with unique convolution structure. Using such a proposed approximate convolution structure, the approximation error mathematically justifies that the reconstruction error is low for a suitable choice of parameters. In order to minimize the exposure time and possible distortions due to the motion of the patient, the fan beam method of collection of data is used. Rebinning [4] transformation is used to connect fan beam data into parallel beam data so that the well developed methods of image reconstruction for parallel beam geometry can be used. Due to the computational errors involved in the numerical process of rebinning, some degradation of image is inevitable. However, to date very little work has been done for the reconstruction of fan beam tomography. There have been some recent results [5], [6] on wavelet reconstruction of divergent beam tomography. In this paper, we propose a convolution algorithm for the reconstruction of divergent beam tomography, which is simpler than wavelet methods and provides small reconstruction error. As the formula is approximate in nature, we prove an estimate for the error associated with the formula. Using the estimate, we deduce the condition that minimizes the approximation error.