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A typical edge might be (for instance) the border between a block of red color and a block of yellow; in contrast a '''[[line (mathematics)|line]]''' can be a small number of pixels of a different color on an otherwise unchanging background. There will be one edge on each side of the line. Edges play quite an important role in all applications of image processing.
== Simple edge model ==
Edges in natural images are normally caused by one of the following effects:
* focal blur caused by a finite [[depth-of-field]].
* [[penumbra|penumbral blur]] caused by shadows created by light sources of non-zero radius.
* [[shading]] at a smooth object edge.
Although this is not really exact, we can use the [[error function]] <math>\operatorname{erf}</math> to model each of these effects in pracitse.
Thus, an 1-dimensional image <math>f</math> which has exactly one edge placed at <math>0</math> can be modelled as follows:
<math>f(x) = \frac{I_r - I_l}{2} \left( \operatorname{erf}\left(\frac{x}{\sqrt{2}\sigma}\right) + 1\right) + I_l</math>
Then, left of the edge the intensity is <math>I_l = \lim_{x \rightarrow -\infty} f(x)</math>, and right of the edge it is
<math>I_r = \lim_{x \rightarrow \infty} f(x)</math>; <math>\sigma</math> is called the blur scale of the edge.
Please note that <math>f</math> can be written as a [[convolution]] <math>f = u * g_\sigma</math> where <math>u</math> is a [[step function]] and <math>g_\sigma</math> is the [[gaussian kernel]] with [[standard deviation]] <math>\sigma</math>:
<math>u(x) := \left\{
\begin{matrix}
I_l, & \mathrm{if} \; x \leq 0\\
I_r, & \mathrm{otherwise}
\end{matrix}
\right.</math>
== Detecting an edge ==
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