What is the complexity in Marr-Hildreth (Laplacian of aGaussian) filter?

Question

what is the drawback in Laplacian of Gaussian filter? why are we going for Difference of gaussian?

Answer 1

There is no drawback in Laplace of Gaussian. I use it all the time. Difference of Gaussians is an approximation, but both need the same amount of computation:

• LoG: convolution with the second derivative along x of a Gaussian + convolution with the second derivative along y of a Gaussian

• DoG: convolution with a Gaussian - convolution with another Gaussian

Each of those convolutions is a separable operation, so both require 4 1D convolutions, and 1 intermediate image to store one of the two results.

Many people implement these operations differently, for example the LoG as a convolution with a Gaussian and then with a discrete Laplace operator. This is, again, an approximation, and could be slightly faster.

There are also separable approximations to the DoG (which require thus only 2 1D convolutions), but these are much less isotropic (which means not invariant to rotations of the image).

Little known fact: as the two sigmas in the Difference of Gaussians approach each other, the approximation becomes more similar to the Laplace of Gaussian.

EDIT: I have just posted a more elaborate answer over at Signal Processing.