Fast variance image predictions for quadratically regularized statistical image reconstruction in fan-beam tomography

Accurate predictions of variance can be useful for algorithm analysis and for the design of regularization methods. Computing predicted variances at every pixel using matrix-based approximations is impractical. Even the recently adopted methods that are based on local discrete Fourier approximations are impractical since they would require two 2D FFT calculations for every pixel, particularly for shift-variant systems like fan-beam tomography. This paper describes a new analytical approach to predict the approximate variance maps of images reconstructed by penalized likelihood estimation with quadratic regularization in a fan-beam geometry. This analytical approach requires computation equivalent to one backprojection and some simple summations, so it is computationally practical even for the data sizes in X-ray CT. Simulation results show that it gives accurate predictions of the variance maps. The parallel-beam geometry is a simple special case of the fan-beam analysis.