Trajectories in parallel optics

Abstract

In our previous work we showed the ability to improve the optical system's matrix condition by optical design, thereby improving its robustness to noise. It was shown that by using singular value decomposition, a target point-spread function (PSF) matrix can be defined for an auxiliary optical system, which works parallel to the original system to achieve such an improvement. In this paper, after briefly introducing the all optics implementation of the auxiliary system, we show a method to decompose the target PSF matrix. This is done through a series of shifted responses of auxiliary optics (named trajectories), where a complicated hardware filter is replaced by postprocessing. This process manipulates the pixel confined PSF response of simple auxiliary optics, which in turn creates an auxiliary system with the required PSF matrix. This method is simulated on two space variant systems and reduces their system condition number from 18,598 to 197 and from 87,640 to 5.75, respectively. We perform a study of the latter result and show significant improvement in image restoration performance, in comparison to a system without auxiliary optics and to other previously suggested hybrid solutions. Image restoration results show that in a range of low signal-to-noise ratio values, the trajectories method gives a significant advantage over alternative approaches. A third space invariant study case is explored only briefly, and we present a significant improvement in the matrix condition number from 1.9160e+013 to 34,526.

Experimental results for improving the matrix condition using a hybrid optical system

“…However, the required system is not always available or easy to design. In this section, we present the trajectories [15,16] and blurred trajectories [17] methods to realize matrix O with sufficient accuracy, by postprocessing manipulation on available auxiliary optics H aux . Here we introduce briefly these methods; detailed elaborations were given in [15][16][17].…”

“…In the trajectories method [15,16], we use the fact that, when we shift the image of the auxiliary optics (H aux ) relative to the image of the main system, we also change its matrix representation. In the case that H aux has a pixel confined sharp response, the matrix description of the shifted auxiliary optics is only a function of the shift (Δm; Δn) and is designated as O l i; j; Δm; Δn:…”

“…Previously we referred to O l i; j; Δm; Δn,Õ l i; j; Δm; Δn as trajectories [15,16] and blurred trajectories [17] PSF matrices, respectively. Since we use a translation operator to create a series of O l PSF matrices, the number of shifts is limited by the system FOV.…”

“…It can be seen that the shift of the image out of the FOV causes changes in the shape of the trajectories matrix. A detailed elaboration is given in [16]. In Fig.…”

“…It should be noted that the contribution of the auxiliary system signal can be significant, depending on the study case (main system and auxiliary optics). In simulations where we were more flexible in determining the study cases, the contribution of the auxiliary system signal to the overall signal was higher [16,17]. After estimating the signal and noise of each system, we calculate the system SNR (in decibels) [23]:…”