The effect of image arithmetic on histograms

Question

There are two images f(x,y) and g(x,y) and their histograms hf and hg .

If I do image arithmetic and name the result z(x,y) and its histogram hz , how can I determine hz as a function of hf and hg ?

As far as I have found :

z(x,y)=f(x,y)+g(x,y) => hz = hf *(convolution) hg (it needs to be scaled)

How about the other arithmetic calculations? How can I determine it on Matlab?

Answer 1

The short answer is you cannot. The histogram hz of a combined image

z(x,y)=f(x,y)+g(x,y)

cannot be uniquely determined from the histograms hf and hg of the individual images.

Example why not:

Suppose f is binary and zero whereever g (also binary) is one. The combined histogram will most likely have a smaller variance than the single histograms. If however now the pixels in g are rearranged so that its histogram hg stays the same but correlation between f and g is maximal then the variance of the combined histogram will be larger.

The result depends on the exact images f(x,y) and g(x,y) and the histograms alone do not contain enough information.