Laplace operator

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 $f({\bf x})$, 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 three-dimensional version of the one-dimensional and two-dimensional 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 zero-crossing (grey) of a bipolar wavelet. Constant input regions result in zero output values (grey).

 

lapGeneral
Figure 19 Generalized Laplace operator. The one-dimensional filter on the left is constructed by subtracting a wide and a narrow triangle of equal area. Similarly, the two-dimensional filter on the right is constructed by subtracting a wide and a narrow pyramid of equal volume.

   

-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 finite-difference representations of Table 3 approximate a generalized Laplace operator that combines the nabla operator with a pre-smoothing by a Gaussian:   where controls the width of the Gaussian kernel. In the Fourier domain, the generalized Laplace operator (5) is

The factor (u² + v²) is the Fourier transform of the nabla operator; the exponential is the transform of the Gaussian low-pass filter.