Edge Detection

A photo is a function: brightness at every pixel. Edges are where brightness changes fast, which is exactly where its partial derivatives are large.

I(x,y)\text{ brightness}

\lVert\nabla I\rVert

How it works

Partial derivatives of an image

In one sentence

Edges are where image brightness $I(x, y)$ changes fastest; the gradient magnitude $\lVert\nabla I\rVert = \sqrt{I_x^2 + I_y^2}$ , estimated with finite differences such as the Sobel filter, is large there.

General formula

{\htmlData{sym=0}{\lVert\nabla I\rVert}} = \sqrt{\left({\htmlData{sym=1}{\frac{\partial I}{\partial x}}}\right)^2 + \left({\htmlData{sym=2}{\frac{\partial I}{\partial y}}}\right)^2}

$I_x$ is how fast brightness changes left to right and $I_y$ top to bottom; together they give the steepness in the steepest direction.

What the symbols mean

$\lVert\nabla I\rVert$
Edge strength
I is the image's brightness at each pixel. This is how fast it changes there; edges are where it is large.
$\frac{\partial I}{\partial x}$
Horizontal change
How fast brightness changes moving right.
$\frac{\partial I}{\partial y}$
Vertical change
How fast brightness changes moving down.

Treat brightness $I(x, y)$ as a surface. Edges are its cliffs.
A Sobel kernel estimates $\partial I/\partial x$ by subtracting the pixels on the left from those on the right.
The gradient magnitude combines both directions. Threshold it and you get an edge map.
Blurring first removes pixel noise, because derivatives amplify small jitters.

Learn the math

Iterated IntegralsCalc III · Lesson 31, Multiple Integrals Diffusion in Image ProcessingPDE · Lesson 19, Beyond the Basics