How do derivative masks work for finding edges in image?

Question

As I know, masks like follow are derivative mask,

but I don't get it why they can detect edges and which one of them can find vertical edges and which one can find harizontal edges?

Answer 1

Basically, the triplet 1,0,-1 (and similarly 2,0,-2) is a discrete, 3-point derivative operator. It can be applied column- (left) or row-wise (right). Let us try first with a row: a flat one, f = 0,0,0,0,0,0,0,0, a step-like s = 0,0,0,1,1,1,1,1 and one hat-like 0,0,0,1,1,0,0,0. 0 here denotes black, and 1 white.

The results will be, on the $6$ central points for each:

• f: 0,0,0,0,0,0
• s: 0,1,1,0,0,0
• h: 0,1,1,-1,-1,0

The same reasoning applies to columns. The Sobel operators are composed of a second filter in the opposite direction: 1,2,1 that smoothes the data. So the global effect: it differentiates in one direction, smoothes in the other. THe resulting value will be high or low, positive or negative, depending on the type of contour in the image.

The (close-to-)zero values are features not detected by the derivative (hopefully non-edges). High amplitude values and their signs are hopefully the strength of the edge and the grading (positive: black-to-white, negative: white-to-black).