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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 Image


to suppress the noise before using Laplace for edge detection:
Image


The first equal sign is due to the fact that
Image


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


and Image


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


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


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

Image
Image


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


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.
Image

http://www.codeproject.com/KB/GDI-plus/Laplace_Gaussion_edge.aspx
http://www.codeproject.com/KB/GDI-plus/Image_Processing_Lab.aspx
http://fourier.eng.hmc.edu/e161/lectures/gradient/node10.html

MUST Creative Engineering Laboratory

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