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Reconstruction of polytopes from the modulus of the Fourier transform with small wave length

  • Konrad Engel EMAIL logo and Bastian Laasch ORCID logo
Published/Copyright: August 30, 2022

Abstract

Let 𝒫 be an n-dimensional convex polytope and let 𝒮 be a hypersurface in n . This paper investigates potentials to reconstruct 𝒫 , or at least to compute significant properties of 𝒫 , if the modulus of the Fourier transform of 𝒫 on 𝒮 with wave length λ, i.e.,

| 𝒫 e - i 1 λ 𝐬 𝐱 𝐝𝐱 | for  𝐬 𝒮 ,

is given, λ is sufficiently small and 𝒫 and 𝒮 have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of 𝒫 with wave length λ when λ 0 . The theory of X-ray scattering of nanoparticles motivates this study, since the modulus of the Fourier transform of the reflected beam wave vectors is approximately measurable on a half sphere in experiments.

MSC 2010: 42B10; 52B11; 81U40

Funding source: European Social Fund

Award Identifier / Grant number: ESF/14-BM-A55-0006/19

Award Identifier / Grant number: ESF/14-BM-A55-0006/19

Funding statement: This work was partly supported by the European Social Fund (ESF) and the Ministry of Education, Science and Culture of Mecklenburg-Western Pomerania (Germany) within the project NEISS – Neural Extraction of Information, Structure and Symmetry in Images under grant no. ESF/14-BM-A55-0006/19.

Acknowledgements

We would like to thank the referee for helpful comments.

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Received: 2020-11-07
Revised: 2022-05-02
Accepted: 2022-06-12
Published Online: 2022-08-30
Published in Print: 2022-10-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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