# 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).