The 2D Analytic Signal

This technical report covers a fundamental problem of 2D local phase based signal processing: the isotropic generalization of the analytic signal (D. Gabor) for two dimensional signals. The analytic signal extends a real valued 1D signal to a complex valued signal by means of the classical 1D Hilbert transform. This enables the complete analysis of local phase and amplitude information. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals. In case of 2D image signals the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n + 1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform (Riesz transform). In this technical report we present the 2D analytic signal as a novel generalization of the 2D monogenic signal which now extends the original 2D signal to a multivector valued signal in conformal space by means of higher order Hilbert transforms and by means of a hybrid matrix geometric algebra representation. The 2D analytic signal can be interpreted in conformal space which delivers a descriptive geometric interpretation of 2D signals. One of the main results of this work is, that all 2D signals exist per se in a 3D projective subspace of the conformal space and can be analyzed by means of geometric algebra. In case of 2D image signals the 2D analytic signal enables now the rotational invariant analysis of lines, edges, corners and junctions.

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