Background subtraction

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I have a histogram of counts which is made from ion fragmentation and noise superimposed on top of it. I also have an image of just the noise. What I want to do is to subtract the noise of the total image in order to only be left with the ion counts. The problem that I am encounter is that if I simply subtract the background, I get negative numbers of counts at certain pixels (because the noise is statistical). This would be fine, except that I then want to do an abel inversion which requires all the counts to be positive. Also, just setting all of the negative values equal to zero biases the data. Does anyone know of a method for background subtraction that will not allow any of the numbers to be negative but doesn't bias the data? Maybe something involving a poisson error bar. The help would be very appreciated!

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Try adding to all the values the larger negative noise value +1. A simple shift. If there is some problem with the result, then scale to the maximum.