# Edge Detection: Zero X Laplacian

Edge detection is possibly the most common method for segmenting objects in medical images. Typically, these algorithms find edges that form a closed contour, which completely bound an object. Currently, MIPAV offers two edge detection algorithms: Zero X Laplacian. and Edge Detection: Zero X Non-Maximum Suppression.

## Contents

### Background

The Laplacian is a 2-D isotropic measure of the 2-nd spatial derivative of an image and can be defined as:

Equation 1

 $\triangledown^2 f=\frac {\partial^2 f}{\partial x^2} + \frac {\partial^2f}{\partial y^2}$

in 2D images and

Equation 2

 $\triangledown^2 f=\frac {\partial^2 f}{\partial x^2} + \frac {\partial^2f}{\partial y^2} + \frac {\partial^2f}{\partial z^2}$

in 3D images.

The Laplacian of an image highlights regions of rapid intensity change and therefore can be used for edge detection. Zero X Laplacian algorithm finds edges using the zero-crossing property of the Laplacian. The zero crossing detector looks for places in the Laplacian of an image where the value of the Laplacian passes through zero - i.e. points where the Laplacian changes its sign. Such points often occur at the edges in images - i.e. points where the intensity of the image changes rapidly.

Figure 1 shows that in the approach of a change in intensity, the Laplacian response is positive on the darker side, and negative on the lighter side. This means that at a reasonably sharp edge between two regions of uniform but different intensities, the Laplacian response is:

• zero at a long distance from the edge,
• positive just to one side of the edge,
• negative just to the other side of the edge,
• zero at some point in between, on the edge itself. However, as a second order derivative, the Laplacian is very sensitive to noise, and thus, to achieve the best result, it should be applied to an image that has been smoothed first. This pre-processing step reduces the high frequency noise components prior to the differentiation step.

The concept of the Zero X Laplacian algorithm is based on convolving the image with 2D Gaussian blur function, first, and then applying the Laplacian. The 2D Gaussian blur function can be defined as

Equation 3

 $h(x,y)= -exp \left \{ \frac {x^2 + y^2} {2 \sigma^2} \right \}$

where -sigma is a standard deviation. This function blurs the image with the degree of blurring proportional to sigma.

In that case the 2-D Laplacian of the Gaussian (that is the second derivative of h with respect to r) is centered on zero and with Gaussian standard deviation sigma has the form of

Equation 4

 $\triangledown^2 f= - \left \{ \frac {r^2 - \sigma^2} {\sigma^4} \right \}exp \left \{ - \frac {r^2} {2\sigma^2} \right \}$ 