Laplacian Based Methods (continued)

Depending on how 2 nd order derivatives are approximated, it is possible to derive many other impulse responses such as.

When edges are detected by looking for zero crossing points of the zero crossing contours which represents boundaries between regions tend to be continuous lines. As a result edge thinning necessary in gradient based methods is not needed in Laplacian based methods. However choosing all zero crossing points as edges tends generate a large numbers of edge points

The disadvantage of Laplacian based methods is that it generates many false edge contours which typically appear in regions where local variance of image is small.

As a special case, consider a uniform background region so that is constant.

Since is zero and we detect edges from zero crossing points of , any small perturbation in is likely to cause false edge contours.

One method to remove many of these false edge contours is to require that the local variance is sufficiently large at an edge point .

The local variable is estimated by

closely related to where

Since is compared with a threshold scaling factors are neglected. In addition , needs to be computed for which are zero crossing points of .

The Laplacian based edge detection system that does not produce many false edge contours can be implemented as.

Requiring crosses zero at edge point can be interpreted as edge thinning.

With this interpretation we can implement the system shown above by computing the first and then by detecting the zero crossing point of only at these points where is above the chosen threshold.