## Abstract

The theory of the previous chapters is applicable to multidimensional signals, but to arrive at their full formulation, some topics, mainly multidimensional groups, must be further developed. In this chapter this will be done for gratings and especially for lattices, first with an efficient representation of gratings and then that of lattices, and then we will consider the problem of generating all subgroups of a given group. Other topics that are further investigated are those of *cells*, with their multiplicity of representations, and the evaluation of *sum* and *intersection*, which is formulated in the framework of integer matrices.

The final advanced topic is *change of signal dimensionality*, which is encountered, e.g., in television scanning, where a time-varying image (a 3D signal) is converted to a 1D signal. This topic has not been developed in the literature in operator form and represents an original contribution of the Unified Signal Theory.

## Keywords

Dimensionality Reduction Voronoi Cell Coordinate Change Elementary Operation Signal Extension## References

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