Editing Image segmentation (section)

=== One-dimensional hierarchical signal segmentation ===

Witkin's seminal work<ref>Witkin, A. P. "Scale-space filtering", Proc. 8th Int. Joint Conf. Art. Intell., Karlsruhe, Germany,1019–1022, 1983.</ref><ref>A. Witkin, "[https://ieeexplore.ieee.org/abstract/document/1172729/ Scale-space filtering: A new approach to multi-scale description]," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing ([[ICASSP]]), vol. 9, San Diego, CA, March 1984, pp. 150–153.</ref> in scale space included the notion that a one-dimensional signal could be unambiguously segmented into regions, with one scale parameter controlling the scale of segmentation.

A key observation is that the zero-crossings of the second derivatives (minima and maxima of the first derivative or slope) of multi-scale-smoothed versions of a signal form a nesting tree, which defines hierarchical relations between segments at different scales. Specifically, slope extrema at coarse scales can be traced back to corresponding features at fine scales. When a slope maximum and slope minimum annihilate each other at a larger scale, the three segments that they separated merge into one segment, thus defining the hierarchy of segments.