Discrete Laplace operator

The Laplacian filter, or discrete Laplacian is a filter for edge detection, which approximates the Laplacian (sum of the pure second derivatives ):

Under an edge of one now refers to a curve along which the gradient of the image is always pointed in the normal direction. The vector field is thus free of sources in the region of the edge. An edge can thus be adjusted only if the following equation is satisfied:

Thus one seeks the zero crossings of a Laplacian filtered image. Here, however, it should be noted that homogeneous areas are null. That is, the Laplacian filter only provides a superset of the possible connections.

Operation

In the figure alongside a noisy signal is shown, from which the second derivative was calculated. Here the edge immersed in a zero crossing of the signal. A discrete signal gn and GNM the Laplacian operator is applied by a convolution. Use can use the following simple convolution masks:

For the 2D filter, there is a second variant, which additionally responsive in contrast to the upper variant at 45 ° edges:

This convolution mask is obtained by discretization of the differential quotients. At the end of the article is seen examples of the application of the Laplacian filter.

Transfer function and isotropy of the filter

The transfer function ( Fourier transform ) of the ideal Laplacian Δ is:

A discretized Laplace operator must be approximated as well as possible this parabolic transfer function.

The figure below shows the transfer function of the first 2D Laplacian filter. You can clearly see the anisotropy and the high-pass nature of the transfer function. As a formula it is:

Shows similarity to an ideal transfer function of the Laplace operator.

You come to a more isotropic approximation of the Laplace operator, if one chooses a different representation of the Laplacian filter:

Here is the 3 × 3 binomial filter (smoothing filter) and a " unit filter " / δ - pulse, which reflects the image onto itself ( the point answer is zero everywhere, except for the central pixel. , Where she is 1). The transfer function of this filter is:

This transfer function is also included in the figure to the right. It turns out that it is much more isotropic than the first version.

Sample Images

Image filtered with the simple filter mask (click to enlarge)

Image filtered by the second filter mask (click to enlarge)