Laplacian

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:

• 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 privilege 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.

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.