Gradient operators are commonly used in edge detection. Usually, proper smoothing processing is performed on the
original image when a gradient operator is applied. Generally, the smoothing processing is embedded in the gradient
operator, such that each component of the gradient operator can be decomposed into some smoothing processing and a
discrete derivative operator, which is defined as the difference of two adjacent values or the difference between the two
values on the two sides of the position under check. When the image is smoothed, the edges of the main objects are also
smoothed such that the differences of the adjacent pixels across edges are lowered down. In this paper, we define the
derivative of f at a point x as f'(x)=g(x+Δx)-g(x-Δx), where g is the result of smoothing f with a smoothing filter, and Δx is an increment of x and it is properly selected to work with the filter. When Δx=2, sixteen gradient directions can be obtained and they provide a finer measurement than usual for gradient operators.