Chapter 4: Discrete scale-space theory in higher dimensions

In chapter 4 in Scale-Space Theory in Computer Vision, the one-dimensional scale-space theory from chapter 3 is generalized to discrete signals of arbitrary dimension. The treatment is based upon the assumptions that

• the scale-space representation should be defined by convolving the original signal with a one-parameter family of symmetric smoothing kernels possessing a semi-group property, and
• local extrema must not be enhanced when the scale parameter is increased continuously.

Given these requirements, the scale-space representation must satisfy a semi-discretized version of the diffusion equation. In a special case the representation is given by convolution with the one-dimensional discrete analogue of the Gaussian kernel along each dimension.