In computed tomography (CT), the fan-beam filtered backprojection (FFBP) algorithm is typically used for reconstruction of images directly from fan-beam data. In single- and multi-slice helical CT, the FFBP algorithm may also be used for reconstructing multiple slices of two-dimensional (2D) images from fan-beam data that are converted from measured helical data. The FDK algorithm may be used for cone-beam data. It has been known that the spatially variant weighting factor in the FFBP algorithm can significantly amplify data noise and sample aliasing in situations where the focal lengths are comparable to or smaller than the size of the field of view (FOV). One previously developed hybrid algorithm reconstructs images from parallel-beam data that are converted from the acquired fan-beam data. Because the previous hybrid algorithm involves no weighting factor similar to that in the FFBP algorithm, it is generally less susceptible to data noise and sample aliasing and is computationally more efficient than is the FFBP algorithm. However, because the previous hybrid algorithm invokes an explicit one-dimensional (1D) interpolation for converting fan-beam projections to parallel-beam projections along the radial direction, it can lead to reduced image resolution when fan-beam samples along the radial direction are sparse.
A new hybrid algorithm is described herein that retains the favorable resolution property of the FFBP algorithm and the favorable noise property of the previous hybrid algorithm while eliminating their shortcomings. Analytic formulas are also derived for evaluating variances in images reconstructed by use of the FFBP, the previous hybrid, and the new hybrid algorithms in their discrete forms. Such theoretical formulas not only may provide insights into the algorithms' precision in estimation tasks, but also may be used for assessing their performance by use of mathematical model observers in detection/classification tasks. Computer-simulation studies for quantitative evaluation of resolution and noise properties of these algorithms are provided. Using moduli of Fourier transforms of reconstructed images, image-resolution properties may be compared and possible distortions identified. From a large number of reconstructed noisy images, empirical image variances are computed and compared with the derived theoretical formulas. Numerical results of these simulation studies confirm that the proposed new hybrid algorithm combines the favorable resolution property of the FFBP algorithm and the noise property of the previous hybrid algorithm while eliminating their shortcomings. Empirical image variances also validate the novel theoretical formulas for image variances. The analysis contained herein has also be extended to half-scan CT, cone beam CT, asymmetric scanning and high resolution scanning techniques.