Reconstruction of Convex Bodies from Moments

Astrid Kousholt, Julia Schulte*

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Abstract

We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.

Original languageEnglish
JournalDiscrete & Computational Geometry
Volume65
Issue1
Pages (from-to)1-42
Number of pages42
ISSN0179-5376
DOIs
Publication statusPublished - Jan 2021

Keywords

  • Convex body
  • Geometric moment
  • Legendre moment
  • Reconstruction
  • Uniqueness
  • Stability
  • SHAPE
  • SETS

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