Trajectories by a blurred auxiliary system

Simple optical imaging systems tend to suffer from a space variant (SV) blur and ill matrix condition and, thus, tend to amplify additive noise in the essential stage of image restoration. Previously, we showed that the matrix condition of systems with SV blur can be improved by adding a system-tailored parallel auxiliary system. We introduced such a solution with the "trajectories" method, where we used auxiliary optics with a pixel confined point spread function to decompose the required auxiliary system. In this paper, by removing the pixel confined requirement, we extend the trajectories to a "blurred trajectories" method, which relies on the more common case of auxiliary optics with blurred response. The method is simulated and shown to be effective in two cases. First, we show that the matrix condition is significantly improved. In one case the condition number of a space variant system is reduced from 87640 down to 1212. In a second case of a highly defocused system, the matrix condition number is reduced from 6412.5 to 238.7. We then investigate the influence of the improvement in the matrix condition on image restoration by regularization with and without the auxiliary system. Blurred trajectories with regularization yields better restoration than regularization only. The new (to our knowledge) system is compared to other previously suggested optical designs. The method's flexibility is demonstrated when applied as postprocessing on a system that includes the original ill-conditioned system and a quartic phase filter and yields an improvement in the overall matrix condition.