Chapter 3: One-dimensional discrete scale-space theory

Chapter 3 in Scale-Space Theory in Computer Vision answers the question: Which convolution kernels share the property of never introducing new local extrema in a signal? Qualitative properties of such kernels are pointed out, and a complete classification is given.

These results are then used for showing that there is only one reasonable way to define a scale-space for one-dimensional discrete signals, namely by discrete convolution with a family of kernels called the discrete analogue of the Gaussian kernel. This scale-space can equivalently be described as the solution to a semi-discretized version of the diffusion equation. The conditions that single out this scale-space are essentially non-creation of local extrema combined with a semi-group assumption and the existence of a continuous scale parameter. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

The commonly adapted technique with a sampled Gaussian may lead to undesirable effects (scale-space violations). This result exemplifies the fact that properties derived in the continuous case might be violated after discretization.