Abstract

We present an analysis of two well-known iterative reconstruction-from-projections algorithms, ART (algebraic reconstruction technique) and SIRT (simultaneous iterative reconstruction technique), that demonstrates how individual spatial-frequency components in the image converge at different rates to their respective object components. The analysis proceeds by considering the continuous versions of the ART and SIRT algorithms in the limit of continuous sampling along the projections and in angle. Explicit convergence formulas are derived that show that the continuous ART and SIRT algorithms converge to the correct solutions, that the convergence is geometric, and how the rate of convergence depends on spatial frequency. Moreover, it is shown how the continuous ART and SIRT algorithms can be expressed as a multiplication of the object spectrum by a spatial-frequency transfer function that varies in a simple way with the iteration number. The transfer-function formulation also makes it easy to compare the continuous ART and SIRT algorithms with convolution backprojection, a well-known non-iterative technique. The continuous ART and SIRT convergence formulas may help in establishing meaningful stopping criteria for the discrete ART and SIRT algorithms, particularly when the intrinsic bandwidth of the projection data is known in advance. A numerical example is given, which agrees with the predicted behavior of the continuous ART and SIRT algorithms.

© 1985 Optical Society of America

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