Geometric approach to sampling and communication

Relationships that exist between the classical, Shannon-type, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.Comment: 19 pages, submitted for publicatio

A Minimal Set of Koopman Eigenfunctions -- Analysis and Numerics

This work provides the analytic answer to the question of how many Koopman eigenfunctions are necessary to generate the whole spectrum of the Koopman operator, this set is termed as a \emph{minimal set}. For an $N$ dimensional dynamical system, the cardinality of a minimal set is $N$. In addition, a numeric method is presented to find such a minimal set. The concept of time mappings, functions from the state space to the time axis, is the cornerstone of this work. It yields a convenient representation that splits the dynamic into $N$ independent systems. From them, a minimal set emerges which reveals governing and conservation laws. Thus, equivalency between a minimal set, flowbox representation, and conservation laws is made precise. In the numeric part, the curse of dimensionality in samples is discussed in the context of system recovery. The suggested method yields the most reduced representation from samples justifying the term \emph{minimal set}

From Normal Surfaces to Normal Curves to Geodesics on Surfaces

This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well

J Math Imaging Vis DOI 10.1007/s10851-007-0048-z Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

Abstract We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis