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

Abstract

Abstract Let 𝒫 {\mathcal{P}} be an n-dimensional convex polytope and let 𝒮 {\mathcal{S}} be a hypersurface in ℝ n {\mathbb{R}^{n}} . This paper investigates potentials to reconstruct 𝒫 {\mathcal{P}} , or at least to compute significant properties of 𝒫 {\mathcal{P}} , if the modulus of the Fourier transform of 𝒫 {\mathcal{P}} on 𝒮 {\mathcal{S}} with wave length λ, i.e., | ∫ 𝒫 e - i ⁢ 1 λ ⁢ 𝐬 ⋅ 𝐱 ⁢ 𝐝𝐱 | for ⁢ 𝐬 ∈ 𝒮 , \biggl{\lvert}\int_{\mathcal{P}}e^{-i\frac{1}{\lambda}\mathbf{s}\cdot\mathbf{x% }}\,\mathbf{dx}\biggr{\rvert}\quad\text{for }\mathbf{s}\in\mathcal{S}, is given, λ is sufficiently small and 𝒫 {\mathcal{P}} and 𝒮 {\mathcal{S}} have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of 𝒫 {\mathcal{P}} with wave length λ when λ → 0 {\lambda\rightarrow 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.