This page hosts ImageJ  plug-ins developed by Dimiter Prodanov and fellows of the Belgian Node.

Laplacian of Gaussian (LoG)

[screenshot]

Theory

Laplacian of Gaussian convolves an image with the trace of the Hessian operator, that is the Laplacian.

Gaussian convolution solves the diffusion problem:

$$ \frac{\partial L \left(x,y \right) }{\partial s} –
\frac{1}{2} \nabla^2 L\left(x,y \right) = 0  $$

With an initial condition the original image
$$
L \left(x, y, 0 \right) = I(x,y)
$$

This is an example of the so-called Scale space approach. The special properties of the Gaussian kernel rely on the fact that it is a  generic solution  (i.e. a Green function) of the diffusion equation

In the context of image processing, these properties have been used to construct the mathematical apparatus of scale space theory independently by  Ijima and his disciples in Japan and  Witkin  in Europe. The approach has been used to robustly detect edges (Hildreth and Marr).

$$
M_s(x,y)= \frac{\partial L}{\partial s} =  \frac{x^2+y^2 – 2 s}{2 \pi s^3} e^{- \frac{x^2+y^2 }{2 s}}
$$

By duality

$$ 
M \star I = – \nabla^2  I \star L
$$

where the RHS has to be interpreted as a formal operation. Nevertheless,  in distributional sense the equation is well defined.

 

 

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