# Filters (Spatial) Laplacian

Edge detection is the identification of meaningful discontinuities in gray level or color images. Edges are formed between two regions that have differing intensity values. This algorithm calculates the laplacian of an image (or VOI of the image) using the second derivatives (Gxx, Gyy, and Gzz [3D]) of the Gaussian function at a user-defined scale sigma [standard deviation (SD)] and convolving it with image. The convolution of the second derivatives of the Gaussian with an image is a robust method of extracting edge information. By varying the SD, a scale-space of edges can easily be constructed.

## Contents

### Background

Figure 1 shows the 1D signal of an object. The first derivative of the object is shown next. Last, the Laplacian (defined later) is shown. Note that the zero-crossing of the second derivative corresponds to the edges of the object.

The laplacian function is defined as:'

$\triangledown^2G(x,y) = \frac {\partial^2G}{\partial x^2} + \frac {\partial^2G}{\partial y^2}$

An efficient method of calculating the laplacian is to analytically calculate the second derivatives of Gaussian and convolve the sum with the image. The equation for this calculation follows:

#### Image types

You can apply this algorithm to all image data types except complex and to 2D, 2.5D, and 3D images.

#### Special notes

The resulting image is, by default, a float type image.

To achieve 2.5D blurring (each slice of the volume is processed independently) of a 3D dataset, select Process each slice independently (2.5D) in the Laplacian dialog box (Figure 3).

#### References

Tony Lindeberg, "Linear Scale-Space I: Basic Theory," Geometry-Driven Diffusion in Computer Vision, Bart M. Ter Har Romeney, ed. (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1994), pp. 1-38.

J. J. Koenderink, "The Structure of Images," Biol Cybern 50:363-370, 1984.

Raphael C. Gonzalez and Richard E. Woods, Digital Image Processing (Boston: Addison-Wesley, 1992).

### Applying the Laplacian algorithm

To run this algorithm, complete the following steps:

1. Select Algorithms > Filter > Laplacian. The Laplacian dialog box opens (Figure 3).
2. Complete the fields in the dialog box.
3. When complete, click OK. The algorithm begins to run.
A pop-up window appears with the status. The following message appears
"Calculating the Laplacian."
When the algorithm finishes, the pop-up window closes. Depending on whether you selected New Image or Replace Image, the results either appear in a new window or replace the image to which the algorithm was applied.
.

 X Dimension Specifies the standard deviation (SD) of Gaussian in the X direction. Y Dimension Specifies the SD of Gaussian in the Y direction. Z Dimension Specifies the SD of Gaussian in the Z direction. Amplification factor Applies, by default, the classical laplacian factor of 1.0; values greater than 1.0 and less than 1.2 enable the laplacian to act as a high-pass, or high-boost, filter. Use image resolutions to normalize Z scale Normalizes the Gaussian to compensate for the difference if the voxel resolution is less than the voxel resolution inplane. Process each edge independently (2.5D) Blurs each slice of the dataset independently. Output edge image Produces a binary image that represents the edges defined by the 0 crossings of the laplacian Threshold edge noise between <_> and <_> Limits the threshold edge noise to the range that you specify between the 0 crossings of the laplacian; the default range is -10 to 10. New image Shows the results of the algorithm in a new image window. Replace image Replaces the current active image with the results of the algorithm. Whole image Applies the algorithm to the whole image. VOI region(s) Applies the algorithm to the volumes (regions) delineated by the VOIs. OK Applies the laplacian algorithm according to the specifications in this dialog box. Cancel Disregards any changes that you made in this dialog box and closes the dialog box. Help Displays online help for this dialog box.