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Second Order Methods

All of the previous edge detectors have approximated the first order derivatives of pixel values in an image.

It is also possible to use second order derivatives to detect edges.

A very popular second order operator is the Laplacian operator.

The Laplacian of a function f(x,y), denoted by tex2html_wrap_inline3450, is defined by:

Once more we can use discrete difference approximations to estimate the derivatives and represent the Laplacian operator with the tex2html_wrap_inline3352 convolution mask shown in Fig 25.  
Fig. 25 Laplacian operator convolution mask

However there are disadvantages to the use of second order derivatives.

The problems that the presence of noise causes when using edge detectors means we should try to reduce the noise in an image prior to or in conjunction with the edge detection process.

We have already discussed some methods of reducing or smoothing noise in the Image Processing Section.

Some of these methods may be of use here.

Another smoothing method is Gaussian smoothing

Lets look at the Gaussian smoothing process first.

The Gaussian distribution function in two variables, g(x,y), is illustrated in Fig. 26 and is defined by

where tex2html_wrap_inline3456 is the standard deviation representing the width of the Gaussian distribution.


Fig. 26 The Gaussian distribution in two variables

Having smoothed the image with a Gaussian operator we can now take the Laplacian of the smoothed image:


Fig. 27 The LOG operator

Thus the edge pixels in an image are determined by a single convolution operation.

This method of edge detection was first proposed by Marr and Hildreth at MIT who introduced the principle of the zero-crossing method.

The basic principle of this method is to find the position in an image where the second derivatives become zero. These positions correspond to edge positions as shown in Fig. 28.


Fig. 28 Steps of the LOG operator

NOTE also:

A related method of edge detection is that of applying the Difference of Gaussian (DOG) operator to an image.

Another important recent edge detection method is the Canny edge detector.

Canny's approach is based on optimising the trade-off between two performance criteria:

The optimisation can be formulated by maximising a function that is expressed in terms of

next up previous
Next: Edge Linking Up: Detecting Edge Points Previous: Gradient based methods

tex2html_wrap_inline2984 David Marshall 1994-1997