Scale Space Implementation - The Discrete Gaussian Kernel

The Discrete Gaussian Kernel

A more refined approach is to convolve the original signal by the discrete Gaussian kernel

where

and denotes the modified Bessel functions of integer order. This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.

This filter can be truncated in the spatial domain as for the sampled Gaussian

or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform:

With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation.

As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of, while for larger values of it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window.