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* 2nd Order: [[Laplacian]], [[Marr-Hildreth algorithm|Marr-Hildreth]], zero-crossings of the second-order derivative in the gradient direction.
Currently, the Canny operator (or variations of this operator) is most commonly used
In his original work, Canny studied the problem of designing an optimal pre-smoothing filter for edge detection, and then showed that this filter could be well approximated by a first-order Gaussian derivative kernel.
Canny also introduced the notion of non-maximum suppression, which means that edges are defined as points where the gradient magnitude assumes a maximum in the gradient direction.
On a discrete grid, the non-maximum suppression stage can be implemented by estimating the gradient direction using first-order derivatives, then rounding off the gradient direction to multiples of 45 degrees, and finally comparing the values of the gradient magnitude in the estimated gradient direction.
A more refined approach to obtain edges with sub-pixel accuracy is by detecting
zero-crossings of the second-order directional derivative in the gradient direction
:<math>L_x^2 \, L_{xx} + 2 \, L_x \, L_y \, L_{xy} + L_y^2 \, L_{yy} = 0,</math>
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where <math>L_x</math>, <math>L_y</math> ... <math>L_{yyy}</math> denote partial derivatives computed from
a [[scale-space]] representation <math>L</math> obtained by smoothing the original image with a [[Gaussian]] kernel .
[[Category:Image processing]]
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