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I have two images:

$$ I_1 = w_{11}A + w_{12}B, \\ I_2 = w_{21}A + w_{22} \bar{B}. $$

$A$ and $B$ are unknown. $\bar{B}$ is ${B}$ rotated by 180 degree. For both images, $A$ has higher signal-to-noise ratio. What would be an effective way to extract $A$ out of these two images?

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Consider 2 points which are rotated version of each other, for these 2 point you could write 4 equation with 4 unknown and solve this system of linear equations (b=A*x, where b obtained from I1 and I2 at those 2 symmetric points and x are values of A and B at those points and the matrix A is obtained from Wij). Now you could solve this system of 4 linear equations for each pair of pixels, but by doing this you can't take into account the noise level unless you repeat your measurements!

To take into account the noise levels, you could use some measure of noise level like Total variation. You could build a cost function by adding the norm of (b-A*x) to a measure of noise level of your images then trying to find the unknowns by minimizing the cost function. In this case you cant use only a pair of points and you have to write the system of equations for all of points.

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