Decorrelating two image maps I am stuck on a research problem and do not know what kind of methods to use. I hope people in this forum can give me some good ideas, I've always had good brainstorming sessions here. 
My problem is that I have a laser and a stack of thin films made up of three different materials. If I shoot the laser at these thin films I get a circular image of the reflected light from the laser scattering on the films. However, the fourth layer is added and I want to measure the change in thickness of the fourth layer, similarly by shooting a laser at this stack. The other three bottom layers are "nuisance" parameters, but their thickness variations are also captured in the pupil image. How do I filter out the observed pupil image so that the information of the three base stacks are eliminated and only the variations of the fourth critical stack are captured? I have 1000 sets (1000 for 3-stack and 1000 for 4-stack) such that for each data point, the 3-stack and 4-stack films have gone through identical process conditions and there is an image for both these stacks. I've thought of maybe signal transforms? 
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Each image pixel of the pupil of the film stack is a nonlinear function of the thicknesses but the gradients at each parameter (as well as the image at a certain parameter value) point can be calculated via an optical simulator. 
$pix_i(i = 1...n) = f(h_1,h_2,h_3) + \epsilon_i$ 
where $pix_i$ is a pixel point of the acquired image and $h1,h2,h3$ are the film thicknesses of a 3-film stack. This can now be linearized around $h10,h20,h30$ to 
$pix_i(i=1...n) \sim f(h_{10},h_{20},h_{30}) + \frac{df}{dh_i}(h_i = h_{i0}) + \epsilon_i$
 A: I'm a little out of my domain here, but here's a thought at least. One method that might be effective is to create a smoothing spline model with a radial kernel that takes the coordinates of the pixel relative to the center and the thicknesses of the films as arguments. The resulting model would let you predict the changes in density of the image as you change the thickness of $h_4$ for arbitrary thicknesses of the first three films. But it won't give you any neat information about the impact of $h_4$ in the absence of the other three. The ssanova package in R is good for this. I'm assuming the images are essentially circular around some origin that you can calculate image-wise. 
Another thought is that you could subtract the matched pairs of images from each other and use the resulting images to estimate a function $f(d,h_4)$ where $d$ is the distance from the origin and $f$ gives the difference in light intensity. If you think the impact of $h_4$ is itself dependent on $h_1$-$h_3$ then this approach might be bad. 
Those are just a few ideas though. Some folks with signal processing experience might be more helpful for this. Also the dsp.stackexchange people may have other perspectives. 
