Laplacian of Gaussian (LoG)

 

As Laplace operator may detect edges as well as noise (isolated, out-of-range), it may be desirable to smooth the image first by convolution with a Gaussian kernel of width 

 

 

 

to suppress the noise before using Laplace for edge detection:

 

 

 

The first equal sign is due to the fact that

 

其中第一步变换是因为高斯函数的对称性保证的，具有可交换性也是高斯模板的高度规整性使然吧.. 同理最后一步也是如此

 

 

So we can obtain the Laplacian of Gaussian  first and then convolve it with the input image. To do so, first consider

 

 

 

and

 

 

 

Note that for simplicity we omitted the normalizing coefficient . Similarly we can get

 

 

 

Now we have LoG as an operator or convolution kernel defined as

 

 

 

The Gaussian  and its first and second derivatives  and  are shown here:

This 2D LoG can be approximated by a 5 by 5 convolution kernel such as

 

 

 

The kernel of any other sizes can be obtained by approximating the continuous expression of LoG given above. However, make sure that the sum (or average) of all elements of the kernel has to be zero (similar to the Laplace kernel) so that the convolution result of a homogeneous regions is always zero.

The edges in the image can be obtained by these steps:

• Applying LoG to the image
• Detection of zero-crossings in the image
• Threshold the zero-crossings to keep only those strong ones (large difference between the positive maximum and the negative minimum)

The last step is needed to suppress the weak zero-crossings most likely caused by noise.