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Representation and Visualisation of Local Orientation Estimates

Before discussing implementation details for orientation estimation, we will take a moment to familiarise ourselves with the unique nature of the problem. You will notice that the needles depicted in Figure 1 do not have arrowheads like normal vectors. This is because orientation, like tangency, is a tex2html_wrap_inline2490 -periodic measurement. That is to say, the two complex numbers tex2html_wrap_inline2492 and tex2html_wrap_inline2494 represent the same orientation. Averaging of these two vectors, however, will result in total cancellation. At first, it may seem that simply restricting the allowable orientation estimation values to a particular interval would eliminate this problem. A little thought will reveal that such a method cannot work. The correct way to represent measures of orientation strength and angle, as discovered by G. H. Granlund in [Gra78], is to simply double the angle of each orientation estimate. While doubling the angle is unattractive for visualisation purposes, mathematically it provides us with a meaningful representation for averaging, differentiation, and other related operations. For example, the two orientation estimates tex2html_wrap_inline2492 and tex2html_wrap_inline2494 would get mapped to tex2html_wrap_inline2500 and tex2html_wrap_inline2502 , respectively, the latter of which becomes tex2html_wrap_inline2504 . Since angle doubling correctly maps these two estimates to the same vector, the averaging operation is meaningful. After averaging, the angle is of course halved for purposes of visualisation.

Markus Weber
Tue Jan 7 15:44:13 PST 1997