Torsion Scale Space Images
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Space curves are useful to study since they can effectively and efficiently represent free-form 3-D surfaces and objects. For example, contours of zero Gaussian or mean curvature on a 3-D surface are space curves invariant under similarity transforms of the surface.
A Torsion Scale Space image is a multi-scale organization of the torsion zero-crossing points of a space curve (or 3-D contour) as it evolves. Intuitively, torsion is a local measure of how non-planar a space curve is. Contour evolution (or smoothing) is achieved by first parametrizing using arc-length. This involves sampling the contour at equal intervals and recording the 3-D coordinates of each sampled point. The result is a set of 3 coordinate functions (of arc-length) which are then convolved with a Gaussian filter of increasing width or standard deviation. Next step is to compute torsion on each smoothed space curve. As a result, torsion zero-crossing points can be recovered and mapped to the TSS image in which the horizontal axis represents the arc-length parameter on the original contour, and the vertical axis represents the standard deviation of the Gaussian filter.
The features recovered from a TSS image for matching are the maxima and minima of its zero-crossing contours. The matching of two TSS images consists of finding the optimal horizontal shift of the extrema in one of the TSS images that would yield the best possible overlap with the extrema of the other TSS image. The matching cost is then defined as the sum of pairwise distances (in TSS) between corresponding pairs of extrema.
The left figure above shows a space curve depicting a chair with red points showing the locations of its torsion zero-crossing points during evolution, and the right figure shows the construction of its TSS image. Details of this technique can be found in published papers (such as SCIA'97 and CVIU, Oct 97).
F.Mokhtarian@ee.surrey.ac.uk Jan 2004