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- A Theory of Multi-Scale, Curvature and Torsion Based Shape Representation for Planar and Space Curves, October 1990 Farzin Mokhtarian
This thesis presents a theory of multi-scale, curvature and torsion based shape representation for planar and space curves. The theory presented has been developed to satisfy various criteria considered useful for evaluating shape representation methods in computer vision. The criteria are: invariance, uniqueness, stability, efficiency, ease of implementation and computation of shape properties. The regular representation for planar curves is referred to as the curvature scale space image and the regular representation for space curves is referred to as the torsion scale space image. Two variants of the regular representations, referred to as the renormalized and resampled curvature and torsion scale space images, have also been proposed. A number of experiments have been carried out on the representations which show that they are very stable under severe noise conditions and very useful for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation. \n Planar or space curves are described at varying levels of detail by convolving their parametric representations with Gaussian functions of varying standard deviations. The curvature or torsion of each such curve is then computed using mathematical equations which express curvature and torsion in terms of the convolutions of derivatives of Gaussian functions and parametric representations of the input curves. Curvature or torsion zero-crossing points of those curves are then located and combined to form one of the representations mentioned above. \n The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. This thesis contains a number of theorems about evolution and arc length evolution of planar and space curves along with their proofs. Some of these theorems demonstrate that evolution and arc length evolution do not change the physical interpretation of curves as object boundaries and others are in fact statements on the global properties of planar and space curves during evolution and arc length evolution and their representations. Other theoretical results shed light on the local behavior of planar and space curves just before and just after the formation of a cusp point during evolution and arc length evolution. Together these results provide a sound theoretical foundation for the representation methods proposed in this thesis.

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