Q17Digital Image Processing

An edge in an image is a physical location where the pixel intensity violently and suddenly changes (e.g., a black car against a white sky). To mathematically detect this, we must execute First-Order spatial calculus. The Gradient of an image $f(x,y)$ at any point is a vector mathematically pointing in the direction of the greatest rate of change:

$$\nabla f=\left[\begin{array}{c}{G}_x\\ G_y\end{array}\right]=\left[\begin{array}{c}\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\end{array}\right]$$

The absolute Magnitude of the edge is calculated as $M(x,y)=\sqrt{G_x^2+G_y^2}$. In digital pixels, calculus derivatives are violently approximated using discrete convolution masks.

The Prewitt Operator

It utilizes a basic $3\times 3$ mask to approximate the derivative.

$G_x=\left[\begin{array}{ccc}-1& 0& 1\\ -1& 0& 1\\ -1& 0& 1\end{array}\right]$.

It treats all pixels in the row equally. It is mathematically simple but highly susceptible to random high-frequency noise.

The Sobel Operator (The Standard)

The Sobel operator executes a brilliant mathematical modification. It forcefully injects a weight of 2 into the absolute center of the mask:

$G_x=\left[\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right]$.

By doubling the weight of the center pixel, the Sobel operator mathematically executes a slight Gaussian smoothing effect simultaneously with the derivative. This violently suppresses random salt-and-pepper noise, resulting in significantly thicker, cleaner, and more robust edge detection compared to the primitive Prewitt.