The Journal of Fourier Analysis and Applications Volume 13, Issue 6, 2007 Image Projection and Representation on S n Emil Saucan, Eli Appleboim, and Yehoshua Y. Zeevi Communicated by Stephan Dahlke ABSTRACT. In signal processing, communications, and other branches of information technolo- gies, it is often desirable to map the higher-dimensional signals on S n . In this article we introduce a novel method of representing signals on S n . This approach is based on geometric function theory, in particular on the theory of quasiregular mappings. The importance of sampling is underlined, and new geometric sampling theorems for general manifolds are presented. 1. Introduction One of the major applications of harmonic analysis and in particular Fourier analysis, is signal processing. In recent years it became common amongst the signal processing community, to consider signals as Riemannian manifolds embedded in higher-dimensional spaces. Usually, the embedding manifold is taken to be R n yet, other possibilities are also considered. For instance, in [26] images are considered as surfaces embedded in higher-dimen- sional manifolds, where a gray scale image is a surface in R 3 , and a color image is a surface embedded in R 5 , each color channel representing a coordinate of the ambient space. In both cases the intensity, either gray scale or color, is considered as a function of the two spatial coordinates (x,y) and thus the surface may be equipped with a metric induced by this function. The question of smoothness of the function is in general omitted, if numerical schemes are used for the approximations of derivatives, whenever this is necessary. A major advantage of such a viewpoint of signals is the ability to apply mathematical tools traditionally used in the study of Riemannian manifolds, for signal processing as well. For example, in medical imaging it is often convenient to treat CT/MRI scans, as Riemannian surfaces in R 3 . One can then borrow techniques from differential geometry and geometric analysis, such as quasiconformal/quasiisometric maps between Riemannian manifolds, in Math Subject Classifications. Primary: 68U10, 30C65, 57R05; secondary: 57R10, 57R12, 53C21. Keywords and Phrases. Image representation, geometric sampling, quasiregular mapping, Alexander’s method, spherical metric. © 2007 Birkhäuser Boston. All rights reserved ISSN 1069-5869 DOI: 10.1007/s00041-006-6906-z