Segmentation (Image Processing) - Important Concepts in Signal Processing, Image Processing and Data Compression - page 35

One-dimensional hierarchical signal segmentation
Witkin's seminal work 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.
Image segmentation and primal sketch
There have been numerous research works in this area, out of which a few have now
reached a state where they can be applied either with interactive manual intervention
(usually with application to medical imaging) or fully automatically. The following is a
brief overview of some of the main research ideas that current approaches are based
upon.
The nesting structure that Witkin described is, however, specific for one-dimensional
signals and does not trivially transfer to higher-dimensional images. Nevertheless, this
general idea has inspired several other authors to investigate coarse-to-fine schemes for
image segmentation. Koenderink proposed to study how iso-intensity contours evolve
over scales and this approach was investigated in more detail by Lifshitz and Pizer.
Unfortunately, however, the intensity of image features changes over scales, which
implies that it is hard to trace coarse-scale image features to finer scales using iso-
intensity information.
Lindeberg studied the problem of linking local extrema and saddle points over scales,
and proposed an image representation called the scale-space primal sketch which makes
explicit the relations between structures at different scales, and also makes explicit which
image features are stable over large ranges of scale including locally appropriate scales
for those. Bergholm proposed to detect edges at coarse scales in scale-space and then
trace them back to finer scales with manual choice of both the coarse detection scale and
the fine localization scale.
Gauch and Pizer studied the complementary problem of ridges and valleys at multiple
scales and developed a tool for interactive image segmentation based on multi-scale
watersheds. The use of multi-scale watershed with application to the gradient map has
also been investigated by Olsen and Nielsen and been carried over to clinical use by Dam
Vincken et al. proposed a hyperstack for defining probabilistic relations between image
structures at different scales. The use of stable image structures over scales has been
furthered by Ahuja and his co-workers into a fully automated system.