Scale Space Implementation - Separability

Separability

Using the separability property of the Gaussian kernel

the N-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel along each dimension

where

and the standard deviation of the Gaussian is related to the scale parameter according to .

Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case.