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How do you scale the intensity of a Fourier filtered image?

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Phillip on 16 Mar 2018
This seems so trivial, but I can't figure out the answer.
Let's say I have some image and zero out two quadrants of the FFT2 in order to focus on image information oriented in a certain direction. I then take the inverse FFT2. If I just add back the original DC term, I notice that the intensity of my objects is not what it should be (depending on the example... allow me to illustrate).
For example, imagine that the image is a sine wave oriented at 45 degrees with amplitude of 1 and DC of 2 (range of this function will be 1 to 3). This is superimposed on another sine wave at 135 degrees with amplitude of 3 and DC of 4 (range of this function will be 1 to 7). The overall image DC is 3 (average of 2 and 4).
If I IFFT2 in one direction and add back the 3 (DC), for the 45 degree direction I will have a range of 2 to 4 (i.e. the peaks are now 1 unit higher than they were in the original image), and if I did it for the 135 degree direction I would have a range of 0 to 6 (i.e. the peaks are now 1 unit lower than in the original image).
Clearly if I wanted to solve this problem I would need to adjust the DC to add to each image individually... but I can't seem to figure out a good way to do it. I've tried scaling by the amplitude and energy of the filtered data but appear to be missing something.
Any clues??
PS The reason I want to keep the image scales the same is that I have determined a threshold from the overall image, but need to apply the same threshold to the filtered images (if that helps at all...).


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