Formulation
Mathematically, the operator uses two 3×3 kernels which are convolved with the original image to calculate approximations of the derivatives - one for horizontal changes, and one for vertical. If we define as the source image, and and are two images which at each point contain the horizontal and vertical derivative approximations, the latter are computed as:
where here denotes the 2-dimensional convolution operation.
Since the Prewitt kernels can be decomposed as the products of an averaging and a differentiation kernel, they compute the gradient with smoothing. For example, can be written as
The x-coordinate is defined here as increasing in the "right"-direction, and the y-coordinate is defined as increasing in the "down"-direction. At each point in the image, the resulting gradient approximations can be combined to give the gradient magnitude, using:
Using this information, we can also calculate the gradient's direction:
where, for example, Θ is 0 for a vertical edge which is darker on the right side.
Read more about this topic: Prewitt Operator
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