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 , is defined by:

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

Fig. 25 Laplacian operator convolution mask

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

• (We should note that first derivative operators exaggerate the effects of noise.) Second derivatives will exaggerated noise twice as much.
  
• No directional information about the edge is given.

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

• Gaussian smoothing is performed by convolving an image with a Gaussian operator which is defined below.
  
• By using Gaussian smoothing in conjunction with the Laplacian operator, or another Gaussian operator, it is possible to detect edges.

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

• The shape of the distribution and hence the amount of smoothing can be controlled by varying .
  
• In order to smooth an image f(x,y), we convolve it with g(x,y) to produce a smoothed image s(x,y) i.e. s(x,y) = f(x,y)*g(x,y).

 

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:

• Therefore the total operation of edge detection after smoothing on the original image is .
  
• It is simple to show that this operation can be reduced to convolving the original image f(x,y) with a ``Laplacian of a Gaussian'' (LOG) operator , which is shown in Fig. 27.

 

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

• The Gaussian function firstly smooths or blurs any step edges.
  
• The second derivative of the blurred image is taken; it has a zero-crossing at the edge.
  
• NOTE: Blurring is advantageous here:
  
  
  • Laplacian would be infinity at (unsmoothed) step edge.
    
  • Edge position still preserved.

NOTE also:

• LOG operator is still susceptible to noise, but the effects of noise can be reduced by ignoring zero-crossings produced by small changes in image intensity.
  
• LOG operator gives edge direction information as well as edge points - determined from the direction of the zero-crossing.

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

• computed by applying two Gaussian operators with different values of to an image and forming the difference of the resulting two smoothed images.
  
• It can be shown that the DOG operator approximates the LOG operator
  
• Evidence exists that the human visual system uses a similar method.

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:

• Good edge detection -- there should be low probabilities of failing to mark real edge points and marking false edge points.
  
• Good edge localisation -- the positions of edge points marked by the edge detector should be as close as possible to the real edge.

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

• The signal-to-noise ratio of the image,
  
• The localisation of the edges
  
• A probability that the edge detector only produces a single response to each actual edge in an image.