Low level image processing is mainly based on 2D signal theory. That means that images are embedded as real functions over the reals. Accordingly, the 2D Fourier transform yields Hermitian C-valued functions. In this framework, several approaches for the analysis of the local image structure exist, e.g. the analytic signal resp. quadrature filters which are based on the partial Hilbert transform, the structure tensor resp. the tensor of inertia, and steerable filters.

All these approaches suffer from the inadequate algebraic framework. Especially the local symmetries in a 2D signal cannot be represented completely in the complex algebra. Consider e.g. the analytic signal: in 1D, the fundamental idea is to have spectral properties like amplitude and phase in a local context, i.e. in the spatial domain. The 2D extension of this idea is insufficient, since it is based on the Hilbert transform which is depending on the orientation (i.e. it is not isotropic). Accordingly, the structure tensor (which is either based on quadrature filters or on products of derivatives) has to be a non-linear approach. The idea of coding the local structure in the phase of the filter response (as in the case of the 1D analytic signal) is lost. Also the steerable filters combine filter responses with different symmetries in one component. The extension of the algebra in which the 2D signal processing is embedded makes it possible to design capable approaches which do not have these drawbacks. Lately, we proposed a (partial) solution to this problem in the form of the monogenic signal which is derived from Clifford analysis and is based on the Riesz transform. From the monogenic signal, we get local amplitude, local orientation, and local phase, where the phase is adopted from the 1D analytic signal. Therefore, the monogenic signal is an adequate method for the analysis of intrinsically 1D structures (i.e. signals which are constant in one direction). For the algebraic embedding, we chose the algebra of quaternions resp. the Clifford algebra which is mainly used in Clifford analysis for 3D monogenic functions. The main drawback of this choice is that geometric ideas suffer from this embedding. Therefore, we embedded the 2D signal processing (including the monogenic signal) into the 3D geometric
algebra and extended the approach by an appropriate concept for intrinsically 2D signals.

mfe@ks.informatik.uni-kiel.de