Adding images for wiki, showing dual convolution with various methods for keeping the kernels compact.
In the presence of noise, it is difficult for the least-squares solution to provide kernels that are compact (since convolution of one image by a large kernel can be matched with the convolution of the other image by a similarly large kernel). One solution is to add a penalty term into the least-squares problem (c.f. Yuan & Akerlof, 2008ApJ...677..808Y). Another solution is to solve the equation once, and then compare the corresponding terms in each of the kernels, masking the smaller (or both if small). Below we test these solutions using the cross-directed Gaussian and PS1_V1 PSFs (5.2pix, A/R = 1.2, Theta = +30 vs 5.2pix, A/R = 1.2, Theta = -30).
- NONE: No attempt was made to force the kernels to be compact.
- MASK: We compared corresponding terms in each of the kernels, masking the smaller; we also masked any with a value less than 10-3 of the derived normalisation.
- PENALTY: We added a penalty function following Yuan & Akerlof, scaled to match the background diagonal term in the least-squares matrix.
- BOTH: We applied both the MASK and PENALTY methods.
The below images are a montage of the residuals and the convolution kernels for the above methods, laid out:
