Laplacian
[Edge Detection]
For a 2-D function F(x,y), one might define the Laplacian operator as :
![\[ \nabla^2(x,y) = \frac{\partial^2I(x,y)}{\partial^2x} + \frac{\partial^2I(x,y)} {\partial^2y} \]](form_111.png)
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Classes | |
| class | SoDoBFilterProcessing |
 SoDoBFilterProcessing engine More... | |
| class | SoMorphologicalLaplacianProcessing |
| SoMorphologicalLaplacianProcessing engine More... | |
| class | SoRecursiveLaplacianProcessing |
| SoRecursiveLaplacianProcessing engine applies a recursive algorithm for the determination of the laplacian. More... | |
Detailed Description
For a 2-D function F(x,y), one might define the Laplacian operator as :
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The Laplacian is an isotropic, second-order derivative operator. It is approximated in the discrete case, with the 4-connected grid, as .
![\[ L(i,j) = I(i-1,j) + I(i+1,j) + I(i,j-1) + I(i,j+1) - 4T(i,j) \]](form_112.png)
The normalization parameter allows one to divide the value of the Laplacian by the sum of the absolute values of the masks. It reduces the noise sensitivity, avoids overflow, but lowers the contrast. Only main edges will appear on the Laplacian image. When no normalization is performed, there may be overflow, but edges will be much more evident.
Use the magnitude of the Laplacian to measure the edge strength, and then use a threshold strategy to mark edge points has severe drawbacks:
- A second-order derivative is more sensitive to noise than a first-order derivative, thus making the distinction between edge and noise is more difficult on the Laplacian.
- The Laplacian carries no directional information.
As it is not directional, the Laplacian does not privileged one direction more than another. However, it is even more sensitive to noise than the gradient. Because of this sensitivity to noise, a lot of isolated points are enhanced along with the edges.