Counter examples for unmatched projector/backprojector in an iterative algorithm.

It is rather controversial whether it is justified to use an unmatched projector/backprojector pair in an iterative image reconstruction algorithm. One common concern of using an unmatched projector/backprojector pair is that the optimal solution cannot be reached. This concern is misleading and must be clarified. We define a figure-of-merit in the image domain as the distance between the reconstructed image and the true image, as the normalized mean-squared-error (NMSE). The NMSE is used to determine whether an unmatched matched projector/backprojector pair can provide a better image than a matched projector/backprojector pair. Hot and cold lesion's contrast-to-noise ratio is also used as an alternative secondary figure-of-merit for algorithm comparison. Computer-generated counterexamples are used to test the performance for matched and unmatched projection/backprojection pairs for different reconstruction algorithms. The projectors are ray-driven, and the backprojectors are ray-driven and pixel-driven. For the attenuation-free data examples, the unmatched pixel-driven backprojector outperforms the matched ray-driven backprojector. For the attenuated data example, the matched ray-driven backprojector performs better. The ray-driven backprojector can be slightly improved by using an attenuation coefficient that is larger than the true one; in this case the backprojector becomes unmatched. Unmatched projector/backprojector pairs are fairly flexible. If the backprojector is properly chosen, good results can be obtained. However, we have not found a general rule to select a good backprojector.