I've taken PCAs of in-focus centered simulated images in the spectrograph with 30 images at varying FRD (6,12,18,24,30) with varying Poisson+read noise and varying signal-to-noise ratio. Each image was label according to the FRD added, and each had a base at 60000 counts in the maximum pixel. The PCA result is shown in PCA2comp.png. Due to the high number of counts, the 1st component of the PCA is dominated by the bright, in-focus signal, but the 2nd component already shows signs of separation in FRD. This investigation is continuing to quantify the FRD extractability and precision, as well as variation with position and higher-order components. 

The effect of the second component of the PCA was primarily due to a dipole shift between images, corresponding to a shift of centroids. As indicated in PIPE2D-563, such a centroid shift must be subtracted out of the image analysis. Therefore, the next few components of the FRD were found and analyzed to determine if they could be used to extract FRD values from the spectrograph images.

The 3rd component of the PCA images is compared to the 2nd component (which nicely separates images at different FRDs) in figure 2nd3rd_Components.png (third component along y axis). Of course, for distinguishing FRD location in a higher dimensional space would be used; here, however, the 2d plots are used for visualization purposes. Notably the images do separate within the third component, although surprisingly 6 mrad FRD and 30 mrad FRD have similar values. Between the range of 20-30 mrad, however, it appears that these images would be uniquely distinguishable from the third component.

The 4th component of the PCA images is compared to the 2nd component in figure 2nd4th_Components.png (fourth component along y axis). The images do not separate out in this component.

The visualization of each of the first four components of the images are shown in figure 4componentPCA_2D.png. Notably while component 1 describes the amplitude and component 2 describes the centroid shift, the next few components are harder to distinguish. 

4 components of a PCA using the same procedure as above was applied to centroid-corrected images at the center of the camera. Align_2nd3rd_Components.png and Align_3rd4th_Components.png demonstrate the structure of the PCA components in these orders, which is remarkably similar to the nonaligned images.  However, Align_4componentPCA_2D.png demonstrates that especially the third component of the PCA demonstrates a more symmetric count decrease in the center and increase toward the edge of the PSF. The magnitude of the second component appears to have a slight dipole component still (alignment still has variations on the order of 1/1000 of a pixel) but is more nearly azimuthally symmetric. 

The PCA for images at varying FRDs was calculated and indicates promise in distinguishing images at different FRDs to high precision over noise (with stacking). Future tickets may utilize PCA to help determine structures in the PSFs that are most correlated with FRD, but the initial investigation, which is the scope of this ticket, has been completed.