Question

I calculated the histogram(a simple 1d array) for an 3D grayscale Image.
Now I would like to calculate the gradient for the this histogram at each point. So this would actually mean I have to calculate the gradient for a 1D function at certain points. However I do not have a function. So how can I calculate it with concrete x and y values?

For the sake of simplicity could you probably explain this to me on an example histogram – for example with the following values (x is the intensity, and y the frequency of this intensity):

x1 = 1; y1 = 3

x2 = 2; y2 = 6

x3 = 3; y3 = 8

x4 = 4; y4 = 5

x5 = 5; y5 = 9

x6 = 6; y6 = 12

x7 = 7; y7 = 5

x8 = 8; y8 = 3

x9 = 9; y9 = 5

x10 = 10; y10 = 2

I know that this is also a math problem, but since I need to solve it in c++ I though you could help me here.

Thank you for your advice
Marc

Answer 1

I think you can calculate your gradient using the same approach used in image border detection (which is a gradient calculus). If your histogram is in a vector you can calculate an approximation of the gradient as*:

for each point in the histogram compute 
     gradient[x] = (hist[x+1] - hist[x])

This is a very simple way to do it, but I’m not sure if is the most accurate.

• approximation because you are working with discrete data instead of continuous

Edited:

Other operators will may emphasize small differences (small gradients will became more emphasized). Roberts algorithm derives from the derivative calculus:

lim delta -> 0 = f(x + delta) - f(x) / delta

delta tends infinitely to 0 (in order to avoid 0 division) but is never zero. As in computer’s memory this is impossible, the smallest we can get of delta is 1 (because 1 is the smallest distance from to points in an image (or histogram)).

Substituting

lim delta -> 0 to lim delta -> 1

we get

f(x + 1) - f(x) / 1 = f(x + 1) - f(x) => vet[x+1] - vet[x]