The Gradient and Sobel derivatives

A key building block in computer vision is finding edges and this is closely related to finding an approximation to derivatives in an image. From basic calculus, it is known that a derivative shows the variation of a given function or an input signal with some dimension. When we find the local maximum of the derivative, this will yield regions where the signal varies the most, which for an image might mean an edge. Hopefully, there's an easy way to approximate a derivative for discrete signals through a kernel convolution. A convolution basically means applying some transforms to every part of the image. The most used transform for differentiation is the Sobel filter [1], which works for horizontal, vertical, and even mixed partial derivatives of any order.

In order to approximate the value for the horizontal derivative, the following sobel kernel matrix is convoluted with an input image:

This means that, for each input pixel, the calculated value of its...