6.1- Introduction
6.2 -Gradient Based Edge Detection
        6.2.1 Roberts
        6.2.2 Sobel
        6.2.3 Prewitt

6.3 - Laplacian Based Edge Detection
6.4 - Canny’s Edge Detection

6.3  LAPLACIAN BASED EDGE DETECTION      

The edge points of an image can be detected by finding the zero crossings of the second derivative of the image intensity. The idea is illustrated for a 1D signal in Figure 5.3.1. However, calculating 2^nd derivative is very sensitive to noise. This noise should be filtered out before edge detection. To achieve this, “Laplacian of Gaussian” is used. This method combines Gaussian filtering with the Laplacian for edge detection.

Figure 5.3.1 1st and 2nd derivative of an edge illustrated in one dimension.
The first graph represents an edge in 1D.

    In Laplacian of Gaussian edge detection there are mainly three steps:

·         Filtering,

·         Enhancement,

·         and detection.

Gaussian filter is used for smoothing and the second derivative of which is used for the enhancement step. The detection criterion is the presence of a zero crossing in the second derivative with the corresponding large peak in the first derivative.

    In this approach, firstly noise is reduced by convoluting the image with a Gaussian filter. Isolated noise points and small structures are filtered out. With smoothing; however; edges are spread. Those pixels, that have locally maximum gradient, are considered as edges by the edge detector in which zero crossings of the second derivative are used. To avoid detection of insignificant edges, only the zero crossings whose corresponding first derivative is above some threshold, are selected as edge point. The edge direction is obtained using the direction in which zero crossing occurs.

    The output of the Laplacian of Gaussian (LoG) operator; h(x,y); is obtained by the convolution operation:

where
 

is commonly called the mexican hat operator.
 

    In the LoG there are two methods which are mathematically equivalent:

·         Convolve the image with a gaussian smoothing filter and compute the Laplacian of the result,

·         Convolve the image with the linear filter that is the Laplacian of the Gaussian filter.

    This is also the case in the LoG. Smoothing (filtering) is performed with a Gaussian filter, enhancement is done by transforming edges into zero crossings and detection is done by detecting the zero crossings.
 

Click here for the Laplacian Based Edge Detector Applet