Gradients and edges, recalled

The gradient is an arrow. At any point, it points toward the direction a function grows fastest. Its length says how fast. Calculus computes that arrow directly, for a smooth surface. But an image has no "in between" two pixels — there's nothing there for a derivative to measure. So the fix is to compare a pixel to its actual neighbors instead. That comparison stands in for the derivative calculus would have computed. A plain, single check — one pixel minus its left neighbor — works. But a single noisy pixel throws it off completely.

Sobel and Prewitt fix that noise problem the same way. Instead of checking just two pixels, they check several and average, using one 3\times3 grid — three rows times three columns of numbers, the same "times" as ordinary multiplication. That grid blends a horizontal derivative (the same left-versus-right check) with a vertical smoothing pass. One bad pixel gets watered down by its neighbors, instead of skewing the answer on its own. Sobel weights the middle row of that smoothing pass a little more heavily than Prewitt does. That's a small tweak, not a different idea. Either kernel's output is large exactly where a human eye would call something an edge. It's near zero on a flat, quiet patch. The arrow's length really is measuring "how edge-like is this spot."

For a smooth function f(x,y), the gradient \nabla f = \left(\partial f/\partial x,\ \partial f/\partial y\right) is a vector. It points in the direction f increases fastest, and its magnitude |\nabla f| equals that fastest rate of increase. An image I(x,y) is defined only at integer pixel coordinates, so its partial derivatives can't be taken in the calculus sense. digital-image-processing's fix is a finite-difference approximation: compare a pixel to its neighbors, computed by convolution with a small kernel instead of by calculus.

Two named 3\times3 kernels compute this approximation while resisting noise. Each one differentiates in one direction and smooths in the other, at the same time. The Prewitt kernel for the horizontal component is H_x^P = \begin{bmatrix}-1&0&1\\-1&0&1\\-1&0&1\end{bmatrix}, a horizontal difference averaged uniformly down each column. The Sobel kernel uses the identical structure, but weights the center row more heavily: H_x^S = \begin{bmatrix}-1&0&1\\-2&0&2\\-1&0&1\end{bmatrix}. This is a small refinement, giving slightly better noise resistance at the same cost. Sobel is the one nearly every real system reaches for. A vertical kernel H_y — rotate either grid 90° — gives the second gradient component. Combine both, |\nabla I| = \sqrt{G_x^2+G_y^2}, and you get exactly this section's opening claim, made computable: large where the image changes fast in either direction, near zero on a flat patch.