Local analysis of multi-dimensional signals
Prof. Dr. Gerald Sommer
Universität Kiel (Germany)


Abstract: Local spectral representations, that is local amplitude and local phase, are well-known in 1D signal analysis and are also widely used in image analysis. They are derived in the complex domain from a real valued signal by applying bandpass operators as the Gabor filter. Regrettably, their use in the case of multi-dimensional signals is seriously limited because of algebraic reasons. In a long-term research programme our group is searching for ways to overcome these limitations. By choosing the right embedding of a multi-dimensional signal into an adequate algebraic framework, given by the geometric algebra of Euclidean spaces, we learned that the monogenic signal can generalize the analytic signal from 1D to intrinsically 1D structures in the nD case. One side effect is that not only the local spectral representations can be derived but also the geometry, that is the orientation of the 1D structure. Besides, a linear scale-space emerges from the Poisson kernel, which in the case of a monogenic extension, constitutes a monogenic scale-space. Regrettably, the monogenic signal is limited to i1D structures. These are of parabolic type in the framework of differential geometry. By starting with a local curvature tensor representation of the signal instead of the scalar grey value, we can extend the local monogenic signal model to all types of differential geometry and i2D (elliptic and hyperbolic). Besides, a monogenic curvature signal represents local spectral and geometric features of i2D signals. This not only generalizes the Hilbert transform by a Riesz transform of second order. Additionally, a monogenic curvature scale-space emerges as linear scale-space of i2D signals. The monogenic signal turns out to be a special case of the presented more general one.