Laplacian
This group provides second order derivative operators.
- DobFilter2d: Appproximates a Laplacian as the difference of two box filters applied on the same two-dimensional image.
- DobFilter3d: Appproximates a Laplacian as the difference of two box filters applied on the same three-dimensional image.
- MorphologicalLaplacian: Approximates an image Laplacian by using morphological operations.
- RecursiveLaplacian2d: Computes the Laplacian of a two-dimensional image with a recursive algorithm.
- RecursiveLaplacian3d: Computes the Laplacian of a three-dimensional image with a recursive algorithm.
For a two-dimensional function F(x,y), one might define the Laplacian operator as :
$$ \Delta^2(x,y) = \frac{\partial^2I(x,y)}{\partial^2x} + \frac{\partial^2I(x,y)}{\partial^2y} $$
The Laplacian is an isotropic second-order derivative operator. It is approximated
in the discrete case, on a 4-connected grid, as:
$$ L(i,j) = I(i-1,j) + I(i+1,j) + I(i,j-1) + I(i,j+1) - 4I(i,j) $$ Using the Laplacian operator to measure the edge strength and then using a threshold strategy to mark edge points has severe drawbacks:
Because of this sensitivity to noise, a lot of isolated points are enhanced along with the edges.
To remove the noise, the ZeroCrossing2d algorithm can be applied afterwards, or the input image should be filtered first. Zero-crossings are points where the Laplacian switches from positive to negative values, and vice versa. Compared to thresholding of a gradient, zero-crossings present the advantage of producing closed curves of 1 pixel thickness. This is most interesting when a precise object boundary detection is the goal.
$$ L(i,j) = I(i-1,j) + I(i+1,j) + I(i,j-1) + I(i,j+1) - 4I(i,j) $$ Using the Laplacian operator to measure the edge strength and then using 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 .
Because of this sensitivity to noise, a lot of isolated points are enhanced along with the edges.
To remove the noise, the ZeroCrossing2d algorithm can be applied afterwards, or the input image should be filtered first. Zero-crossings are points where the Laplacian switches from positive to negative values, and vice versa. Compared to thresholding of a gradient, zero-crossings present the advantage of producing closed curves of 1 pixel thickness. This is most interesting when a precise object boundary detection is the goal.