Parent Topic: ALGORITHM

LAPLACIAN FILTER

The Laplacian filters are high-pass filters which act as a local edge detector. A characteristic of the Laplacian is that it is zero at points where the gradient is a maximum or a minimum. Therefore points detected as gradient edges would generally not be detected as edge points with the Laplacian filter. Another characteristic of Laplacian operators is that a single grey level transition may produce two distinct peaks, one positive and one negative, in the Laplacian which may be offset from the gradient location.

The Laplacian filter detects edges, regardless of direction. It produces sharper edges than most other edge detection filters.

The Laplacian of a function f(x,y) is

  L(f(x,y)) = (d**2)f / d(x**2) + (d**2)f / d(y**2)

   where, 

    (d**2)f / d(x**2) is the second partial derivative of f with respect to x
 
   and,

    (d**2)f / d(y**2) is the second partial derivative of f with respect to y

The second partial derivatives can be approximated by,

  (d**2)f / d(x**2) = f(x+1) - 2f(x) + f(x-1)

  and

  (d**2)f / d(y**2) = f(y+1) - 2f(y) + f (y-1)

The Laplacian can therefore be approximated by,

  L(f(x,y)) = f(x+1,y) + f(x-1,y) + f(x,y+1) + f(x,y-1) - 4f(x,y)

Parent Topic: ALGORITHM
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