I was trying to calculate the diagonal (bottom left to top right) Sobel operator but couldn't manage to do it. I know what it should look like, but getting there im not able to.
So thanks in advance if some kind soul can explain it.
I already tryed to synthesise it out of the horizontal and vertical Sobel operators, but got stuck in multiple wrong results.
The x and y derivatives together form the gradient vector. Using the Sobel operator you can compute these two derivatives.
Projecting the gradient vector onto a unit vector you can find the directional gradient. Your desired direction corresponds to the unit vector [sqrt(2), -sqrt(2)]. The projection is a dot product. Thus, your desired derivative image corresponds to:
dx = sobel(img)
dy = sobel(img)
out = sqrt(2)*dx - sqrt(2)*dy
Knowing that the Sobel operator is applied with a convolution, and that the convolution is associative and linear, we can see that the operations above correspond to a single convolution with a kernel formed by sqrt(2)*sobel_x - sqrt(2)*sobel_y. We can compute this kernel:
0 2.8284 2.8284
-2.8284 0 2.8284
-2.8284 -2.8284 0
Of course you could choose to normalize the kernel differently, the magnitude of the Sobel operator is wrong anyway. So you could instead use for example
0 1 1
-1 0 1
-1 -1 0