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 a convolution with a Gaussian kernel of width $\sigma$
In $N = 2$ to suppress the noise before using Laplace for edge detection
The first equal sign is due to the fact that
So we can obtain the Laplacian of Gaussian $\bigtriangleup G_{\sigma}(x,y)$ first and then convolve it with the input image. To do so, first consider
and
Note that for simplicity we omitted the normalizing coefficient $1/\sqrt{2\pi \sigma^2}$. Similarly we can get
Now we have LoG as an operator or convolution kernel defined as
There is an example of LoG operator of size $5\times5$ below:
