Scale-Space for Discrete Images

Tony Lindeberg

Technical report ISRN KTH/NA/P--88/08--SE.

Shortened version in Proc. 6th Scandinavian Conference on Image Analysis, (Oulo, Finland), pages 1098--1107, Jun. 1989.


This article addresses the formulation of a scale-space theory for one-dimensional discrete images. Two main subjects are treated:
  1. Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output image does not exceed the number of local extrema (or zero-crossings) in the original image?
  2. How should one create a multi-resolution family of representations with the property that an image at a coarser level of scale never contains more structure than an image at a finer level of scale?
We propose that there is only one reasonable way to define a scale-space for discrete images comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T(n; t) = e^{-t} I_n(t),, where $I_n$ are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

Some obvious discretizations of the continuous scale-space theory are discussed in view of the results presented. An important result is that scale-space violations might occur in the family of representations generated by discrete convolution with the sampled Gaussian kernel.

PDF: (PDF 1.5 Mb)

Earlier technical report: (PDF 8.3 Mb)

Extended journal version: (Abstract)

Tony Lindeberg