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234 Chapter 6 ■ Thinning

From the results so far, these assumptions appear to be at least approxi-
mately true.
Further work on this idea has developed over the past decade. Most
recently, researchers have developed a high-speed algorithm for approximat-
ing the force fields and have extended the force-based thinning idea into three
dimensions [Brunner, 2008].

6.6.3 Subpixel Skeletons

The force-based thinning method has been implemented and tested on a
number of images, both artificial and scanned. The results in all cases are either
promising or excellent. The subpixel accurate skeletons provide substantially
more information about the geometry of the object. There will often be areas
in the object where the forces are not actually zero, but are small or known to
be changing. By splitting each pixel into four pixels, a more accurate force can
be calculated for each such pixel, and the region where the force is zero can be
estimated. If this fails, each subpixel can be further split into four, and so on
(Figure 6.23).

(a) (b) (c)

(d) (e) (f)
Figure 6.23: Subpixel skeletons. (a) The original image. (b) The level-1 skeleton, showing
zero crossings. (c) A subpixel section of a gap in the level-0 skeleton. (d) The level-1
skeleton. (e) Subpixel force magnitudes in a gap in the level-1 skeleton. (f) The final
skeleton with all gaps filled in.

This is expensive, since for each subpixel the visible pixels must be deter-
mined, and then the force accumulated. One way to speed this up is to compute
the forces based on the lines formed by the boundary pixel instead of using