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 userdefined 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 scalespace 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 zerocrossing of the second derivative corresponds to the edges of the object.
The laplacian function is defined as:'
<math> \triangledown^2G(x,y) = \frac {\partial^2G}{\partial x^2} + \frac {\partial^2G}{\partial y^2} </math>
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
Refer to the following references for more information about this algorithm:
Tony Lindeberg, "Linear ScaleSpace I: Basic Theory," GeometryDriven Diffusion in Computer Vision, Bart M. Ter Har Romeney, ed. (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1994), pp. 138.
J. J. Koenderink, "The Structure of Images," Biol Cybern 50:363370, 1984.
Raphael C. Gonzalez and Richard E. Woods, Digital Image Processing (Boston: AddisonWesley, 1992).
Applying the Laplacian algorithm
To run this algorithm, complete the following steps:
 Select Algorithms > Filter > Laplacian. The Laplacian dialog box opens (Figure 3).
 Complete the fields in the dialog box.
 When complete, click OK. The algorithm begins to run.
 A popup window appears with the status. The following message appears
 "Calculating the Laplacian."
 When the algorithm finishes, the popup 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 highpass, or highboost, 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.

See also:
 Fast Fourier Transformation (FFT)
 Filters (Spatial): Adaptive Noise Reduction
 Filters (Frequency)
 Filters (Spatial): Adaptive Path Smooth
 Filters (Spatial) Anisotropic Diffusion
 Filters (Spatial): CoherenceEnhancing Diffusion
 Filters (Spatial): Gaussian Blur
 Filters (Spatial): Gradient Magnitude
 Filters (Spatial): Haralick Texture
 Filters (Spatial) Laplacian
 Filters (Spatial): Local Normalization
 Filters (Spatial): Mean
 Filters (Spatial): Median
 Filters (Spatial): Mode
 Filters (Spatial): Nonlinear Noise Reduction
 Filters (Spatial): Nonmaximum Suppression
 Filters (Spatial): Regularized Isotropic (Nonlinear) Diffusion
 Filters (Spatial): Slice Averaging
 Filters (Spatial): Unsharp Mask