Laplacian Based Methods

The Objective of an edge detection algorithm into locate the regions where the intensity is changing rapidly.

In 1-D case this corresponds to searching for regions where is large.

For grad based methods, is considered large when is > threshold.

Another possible way is to assume that is large wherever it reaches a local extremum ie whenever has a zero crossing.

Declaring zero crossing as edges results in a large no. of points being declared to be edge points. Since there is no check on the magnitude of , any small ripple in is enough to generate an edge point.Due to this sensitivity to noise, the application of a noise reduction system, prior to edge detection is very desirable in processing images with a background noise.

A generalization of to a 2-D function for the purpose of edge detection is the Laplacian given by

Zero crossing of occur at edge pts of f(x,y)

of second derivative action No edge thinning is required as Zero crossing themselves are the edge location.

For a 2-D sequence f(n₁ ,n₂ ) the partial second derivatives and can be replaced by 2^nd order differences. This is discussed in the following section.