# Geometric approach to sampling and communication

@article{Saucan2010GeometricAT, title={Geometric approach to sampling and communication}, author={Emil Saucan and Eli Appleboim and Yehoshua Y. Zeevi}, journal={ArXiv}, year={2010}, volume={abs/1002.2959} }

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.

## 13 Citations

Geometric Sampling of Images, Vector Quantization and Zador's Theorem

- Mathematics, Computer Science
- 2010

The relevance of the geometric method to the vector quantization of images is single out and it is given a concrete and candidate for the optimal embedding dimension in Zador’s Thorem.

Isometric Embeddings in Imaging and Vision: Facts and Fiction

- MathematicsJournal of Mathematical Imaging and Vision
- 2011

This work investigates the relevance of a result of Burago and Zalgaller regarding the existence of PL isometric embeddings of polyhedral surfaces in ℝ3 and shows that their proof does not extended directly to higher dimensions.

Metric curvatures and their applications I

- Mathematics
- 2015

We present, in a natural, developmental manner, the main types of metric curvatures and investigate their relationship with the notions of Hausdorff and Gromov-Hausdorff distances, which by now have…

A Simple Sampling Method for Metric Measure Spaces

- MathematicsArXiv
- 2011

We introduce a new, simple metric method of sampling metric measure spaces, based on a well-known "snowflakeing operator" and we show that, as a consequence of a classical result of Assouad, the…

Geometric Wavelets for Image Processing: Metric Curvature of Wavelets

- Mathematics, Computer Science
- 2010

There is an inverse relationship between local scale and local curvature in images, which allows curvature as a geometrically motivated automatic scale selection in signal and image processing, this being an incipient bridging of the gap between the methods employed in Computer Graphics and Image Processing.

Generalized Ricci curvature based sampling and reconstruction of images

- Mathematics, Computer Science2015 23rd European Signal Processing Conference (EUSIPCO)
- 2015

We introduce a novel method of image sampling based on viewing grayscale images as manifolds with density, and sampling them according to the generalized Ricci curvature introduced by Bakry, Emery…

Metric Ricci curvature for $PL$ manifolds

- MathematicsArXiv
- 2012

A fitting version of the Bonnet-Myers theorem is proved, for surfaces as well as for a large class of higher dimensional manifolds, and a metric notion of Ricci curvature is introduced.

Network Topology vs. Geometry: From persistent Homology to Curvature

- Mathematics
- 2017

Among the methods for data analysis, and mainly for the understanding of the shape of the data, as obtained by sampling of some underlying structure (the intended final object of such a study) the so…

Metric Curvatures Revisited: A Brief Overview

- Mathematics
- 2014

We survey metric curvatures, special accent being placed upon the Wald curvature, its relationship with Alexandrov curvature, as well as its application in defining a metric Ricci curvature for PL…

Homotopic Object Reconstruction Using Natural Neighbor Barycentric Coordinates

- Computer Science, Mathematics2010 International Symposium on Voronoi Diagrams in Science and Engineering
- 2010

This paper addresses the problem of 2D object reconstruction from arbitrary linear cross sections through continuous deformations of line intersections in the plane and defines Voronoi diagram based barycentric coordinates on the edges of n-sided convex polygons as the area stolen by any point inside a polygon from the Vor onoi regions of each open oriented line segment bounding the polygon.

## References

SHOWING 1-10 OF 102 REFERENCES

Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

- MathematicsJournal of Mathematical Imaging and Vision
- 2008

This work presents new sampling theorems for surfaces and higher dimensional manifolds and shows how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis.

Communication in the presence of noise

- Computer ScienceProceedings of the IEEE
- 1984

A method is developed for representing any communication system geometrically and a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect.

Geometric Sampling For Signals With Applications to Images 1 SampTA 2007 – June 1-5 , 2007 , Thessaloniki , Greece

- Mathematics
- 2007

We present a non uniform version of the sampling theorems introduced in [15], [16]. In addition, we show that the geometric sampling scheme produces sparse sampling for gray scale images.

Sampling, data transmission, and the Nyquist rate

- Computer Science
- 1967

It is argued that only stable sampling is meaningful in practice, and it is proved that stable sampling cannot be performed at a rate lower than the Nyquist, and data cannot be transmitted as samples at a Rate of 2W per second, regardless of the location of sampling instants, the nature of the set of frequencies which the signals occupy, or the method of construction.

A mathematical theory of communication

- MathematicsBell Syst. Tech. J.
- 1948

In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a…

Nonuniform sampling and antialiasing in image representation

- Computer ScienceIEEE Trans. Signal Process.
- 1993

A unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented. A nonuniform sampling…

Sampling-50 years after Shannon

- Computer ScienceProceedings of the IEEE
- 2000

The standard sampling paradigm is extended for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets, and variations of sampling that can be understood from the same unifying perspective are reviewed.

Uncertainty principles and signal recovery

- Computer Science
- 1989

The uncertainty principle can easily be generalized to cases where the “sets of concentration” are not intervals, and for several measures of “concentration” (e.g., $L_2 $ and $L-1 $ measures).

Sphere Packings, Lattices and Groups

- MathematicsGrundlehren der mathematischen Wissenschaften
- 1988

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to…