To the Graduate Council: I am submitting herewith a dissertation ...

Chapter 3: A Feature Saliency Descriptor for Registration and Pose Recovery 24λ 1e3e2λ3λ2e1Fig. 3.1. The ellipsoid describing orientation uncertainty and the associated eigendecomposition.In 3D the matrix T can be decomposed into its eigenvalues λ , λ , )(1 2λ3andeigenvectors e , e , ) as follows:(1 2e3T=[ e e e ]123⎡λ1⎢⎢0⎢⎣00λ200 ⎤⎡e⎢0⎥⎥⎢eλ ⎥⎢3 ⎦⎣eT1T2T3⎤⎥⎥⎥⎦(3.1)So thatTTTT λ1e1e1+ λ2e2e2+ λ3e3e3= , where λ1 ≥ λ2≥ λ3are the principal axes ofthe orientation ellipsoid. The saliency of a feature is determined by the size and shapeof the uncertainty ellipsoid and depends directly on these eigenvalues. As expressed in(3.1) T has the characteristic representation of a tensor.Point features are encoded using the so-called ball tensor, geometrically described bya circle in 2D and a sphere in 3D. Curve elements (Curvels) in 2D, consisting of thepair point + tangent vector, are encoded using the covariance matrix of the tangent