Space Domain Models for Optical Imaging Systems
keyword: ECE 637, digital image processing
were gonna use a lot of linear space invariant system theory to describe imaging systems so the reason we can do that is that they are well approximated by that. as i described, when you have a lens, and there's an aperture here, and there's a focal plane, the lens is the fourier domain and the image plane is the image domain. so in the image domain the image here you should have formed f(x,y) will be convoled with some psf that we'll call h(xy) to from the actual image g(xy).
h(xy) = inv CTFT {H(uv)} where H(u,v) is the aperture the things i'm saying here are not precisely true because i'm mixing a few metaphors. what's happening here is that this interpretation is precisely true ig g(xy) was a function for the electromagnetic field but the images that are formed here are from the energy in the signal but using the approximations of non coherent integration of photons, it all works out pretty much the same way. what happens is that the image you form on the focal plane array (g(xy))is given by the convolution of f(xy)(the image you should have formed) with h(xy) (the psf of the system).
insert equation
so if you take away the magnification factor, the resulting image is likt the convolution of what the image should have been with the psf of the system
$ g(x,y) = \frac{1}{M^{2}} h(x,y)*f(x,y) $
