Introducing Sobel 

The gradient-based method based on the first-order derivatives is called the Sobel edge detector. The Sobel edge detector calculates the first-order derivatives of the image separately for the x axis and y axis. Sobel uses two 3 x 3 kernels that convolve over the original image to calculate the derivatives. For image A, Gx and Gy are two images that represent the horizontal and vertical derivative approximations:

The * character indicates the 2D signal processing convolution operation.

The Sobel kernels compute the gradient with smoothing, as it can be decomposed a product of the averaging and differentiation kernels.

Sobel computes the gradient using smoothing. For example, * can be written as follows:

Here, the x-coordinate shows an increase in a right direction, and the y-coordinate shows an increase in a downward direction.

The resulting gradient approximations at each point in the image can be merged...