Scale Space - Gaussian Derivatives

Gaussian Derivatives

At any scale in scale space, we can apply local derivative operators to the scale-space representation:

Due to the commutative property between the derivative operator and the Gaussian smoothing operator, such scale-space derivatives can equivalently be computed by convolving the original image with Gaussian derivative operators. For this reason they are often also referred to as Gaussian derivatives:

Interestingly, the uniqueness of the Gaussian derivative operators as local operations derived from a scale-space representation can be obtained by similar axiomatic derivations as are used for deriving the uniqueness of the Gaussian kernel for scale-space smoothing.