Geometric Sampling of Infinite Dimensional Signals

Saucan, Emil

doi:10.1007/bf03549534

Cited by 3 publications

(2 citation statements)

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“…More precisely, to construct a thick triangulation of a manifold, one chooses its vertices according to a (metric) sampling density given by its curvature. This is, by now, a subject explored (and exploited) in detail -see [71], [81], [85], [104], [88], [86], [101], [102].…”

Section: Computer Graphicsmentioning

confidence: 99%

Metric curvatures and their applications I

Saucan

2015

Geometry, Imaging and Computing

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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 been widely adopted in the fields of Imaging, Vision and Graphics. In addition we present a number of applications to the fields above, as well as to Communication Networks and Regge Calculus. Further connections with such established notions as excess and fatness are also investigated. While the present paper represents essentially a survey, a number of possible applications are presented here for the first time, for instance to a numerically feasible quantification of a notion of quasi-flatness of manifolds, with applications in Imaging; and to the introduction of curvatures, in particular the Lipschitz-Killing curvature measures, for almost Riemannian manifolds, with a view to their usage in Regge Calculus, as well as in Graphics. Further directions of study are also suggested.

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Section: Computer Graphicsmentioning

confidence: 99%

Metric curvatures and their applications I

Saucan

2015

Geometry, Imaging and Computing

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“…Recently, driven mainly by what has become by now a common practice amongst the image processing community, namely to regard images as Riemannian manifolds embedded in higher dimensional spaces (usually R n or S n ) -see e.g. [22], [70], [64], [27], [55] -a geometric approach to sampling and its implications has emerged [26], [56], [57], [52].…”

Section: Introductionmentioning

confidence: 99%

A Simple Sampling Method for Metric Measure Spaces

Saucan¹

2011

Preprint

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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 sampling of doubling metric spaces is bilipschitz equivalent to that of subsets of some R N . Moreover, we compare this new method with two other approaches, in particular to one that represents a direct application of our triangulation method of metric measure spaces satisfying a generalized Ricci curvature condition.

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