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In Figure 18,
a Laplace operator transforms the amplitude edges
in the image cube on the left
to zero crossings of a bipolar wavelet
in the image cube on the right.
Given an image volume ,
the Laplace operator
zeroes the amplitude where f is constant.
Analogous to the gradient magnitude operator case,
a Laplace operator is conveniently approximated
by its finite difference approximations, such as those shown in
Table 3.
In general,
the masks vary in their moments and consequently in their
isotropy, robustness and resolving power.
In this chapter,
I use a threedimensional version of
the onedimensional and twodimensional Laplace filters
of Figure 19.
consFoltLap
Figure 18
Laplace operator applied to constant amplitude image.
The output of the Laplace operator on the right delineates
the edges successfully.
Edges are indicated by the zerocrossing (grey) of a bipolar wavelet.
Constant input regions result in zero output values (grey).
lapGeneral
Figure 19
Generalized Laplace operator.
The onedimensional filter on the left is constructed by
subtracting a wide and a narrow triangle of equal area.
Similarly,
the twodimensional filter on the right is constructed by
subtracting a wide and a narrow pyramid of equal volume.
Table 3:
Finitedifference approximations of the 2D Laplace operator.

1 


1 
4 
1 


1 
0
$ 

1 
8 
1 

1 
1 
1
$ 

2 
4 
2 

1 
2 
1
$ 

1 
1 
1 
1 
1 
1 
24 
1 
1 
1 
1 
1 
1 
1 
1 
1 
The finitedifference representations of Table 3
approximate
a generalized Laplace operator
that combines the nabla operator
with a presmoothing by a Gaussian:
where controls the width of the Gaussian kernel.
In the Fourier domain,
the generalized Laplace operator (5)
is
The factor (u^{2} + v^{2}) is the Fourier transform of the nabla operator;
the exponential is the transform of the Gaussian lowpass filter.
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Stanford Exploration Project
3/8/1999