Image Analysis by Conformal Embedding
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This work presents new ideas in isotropic multi-dimensional phase based signal theory. The novel approach, called the conformal monogenic signal, is a rotational invariant quadrature filter for extracting local features of any curved signal without the use of any heuristics or steering techniques. The conformal monogenic signal contains the recently introduced monogenic signal as a special case and combines Poisson scale space, local amplitude, direction, phase and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation between the Radon transform and the generalized Hilbert transform. The main idea of the conformal monogenic signal is to lift up n-dimensional signals by inverse stereographic projections to a n-dimensional sphere in ℝn+1 where the local signal features can be analyzed with more degrees of freedom compared to the flat n-dimensional space of the original signal domain. As result, it delivers a novel way of computing the isophote curvature of signals without partial derivatives. The philosophy of the conformal monogenic signal is based on the idea to use the direct relation between the original signal and geometric entities such as lines, circles, hyperplanes and hyperspheres. Furthermore, the 2D conformal monogenic signal can be extended to signals of any dimension. The main advantages of the conformal monogenic signal in practical applications are its compatibility with intrinsically one dimensional and special intrinsically two dimensional signals, the rotational invariance, the low computational time complexity, the easy implementation into existing software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.
KeywordsUnit sphere Signal processing Generalized Hilbert transform Riesz transform Radon transform Isotropic Local phase based signal analysis Clifford analysis Monogenic signal Analytic signal Isophote curvature Poisson scale space Stereographic projection Conformal space
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