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AN INTUITIVE EXPLANATION OF MEDIAL AXIS TRANSFORMATION

 


Images A-D show some ideas of Harry Blum, inventor of the medial axis transform. Whereas ripples on water traverse through
each other (A), one also find 'waves' that can block each other (B). As example, Blum imagined fires starting at two points in a field of grass. They spread and eventually meet along a line (the medial axis) exactly equidistant between the two starting points (B). They stop because no grass is left behind where either of the fires already swept by. A fire, started along a triangular outline (C), stops along the 'Y'-shaped medial axis inside the figure. Blum showed that grass fires are equivalent to tracing the centres of the largest (maximal) discs that can just fit into the outline figure (D). (E) shows the medial axis transform (MAT) of a human figure.


Our HST model uses Blum's concept, but has a shunting mechanism to let computational 'grass fires' burn only a fraction of the 'grass' they spread through in the 'field' of image pixels. Fires that burn very small fractions of grass (weak shunting), resemble ripples on water (F). When fires burn all the grass (strong shunting), they behave as blocking wave fronts (G). We call this tunable method of medial axis transformation the Hybrid Symmetry Transform, or HST.


HST is elegant for several reasons. It is based on a silicon shunting inhibition network, proposed by Carver Mead (
Analog VLSI and Neural Systems,1989). It further deals with noise in a particular way. Consider  the MAT or HST for a trilobed amoeba shape (H). With noise dots added, the MAT is highly scattered (I) and resembles the medial axes between the closest data points (similar to HST with strong shunting). But with HST shunting set between weak and strong values, the original medial axis transform of the amoeba shape reappears (J).


Why would a study of human visual perception be concerned with medial axis transformation? I. Kovacs et al.
(Vis. Res., 38(15/16), 1998) have shown that human visual sensitivity changes drastically depending on where they look within an outline figure (the white line figure in K is known as a cardioid shape, and its HST is shown in black). The HST output in both vertical and horizontal cross sections (L) closely matches Kovac's computational results. It also closely matches human perceptual sensitivity changes (M) measured for that particular outline figure. With these specific parameters values, the HST predicts human perceptual sensitivity maps for other figures. We applied the HST to reveal some of the implicit visual structure in Ryoanji (100k PDF. Van Tonder, Lyons & Ejima, Nature, 419:359-360. 2002, and 60k PDF supplementary material), a well known Japanese Zen garden. T.S. Lee et al. (Vis.Res., 38(15/16), 1998) found evidence that neurons in the primate visual system respond to medial axes. In a collaboration with Dr. S. Oka, we used EEG analysis to show that symmetry axes play a role in human shape detection (342K PDF. Oka, Van Tonder & Ejima, Vision Research, 41:3791-3803, 2001).


Extracting the darkest 'ridges' from the medial axis transform gives the medial axes of that figure (N). Medial axes are used as compact descriptions of more complex shapes in engineering applications involving pattern recognition. Links to many other industrial uses for medial axis transformation can be found HERE.

Read more about the HST model developed in this research 293K PDF. Van Tonder & Ejima, IEEE SMC B, 33(3):535-541, 2003, .

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