Generalized Fourier Diffraction Theorem and Filtered Backpropagation for Tomographic Reconstruction

Abstract

This paper concerns diffraction-tomographic reconstruction of an object characterized by its scattering potential. We establish a rigorous generalization of the Fourier diffraction theorem in arbitrary dimension, giving a precise relation in the Fourier domain between measurements of the scattered wave and reconstructions of the scattering potential. With this theorem at hand, Fourier coverages for different experimental setups are investigated taking into account parameters such as object orientation, direction of incidence and frequency of illumination. Allowing for simultaneous and discontinuous variation of these parameters, a general filtered backpropagation formula is derived resulting in an explicit approximation of the scattering potential for a large class of experimental setups.

Figures (14)

• Figure 1: Measurement setup for $d=3$ with the open interval $I^+(f)$ marked in red along the $r_3$ axis. The incident field has direction ${\bf s}$ and wavelength $2\pi/k_0$.
• Figure 2: 2D Fourier coverage for incidence direction varying according to ${\bf s}(t) = (\cos t, \sin t)$ where $t \in [\pi/4, 3\pi/4]$ (left), $t \in [0,\pi]$ (center) and $t \in [0,2\pi]$ (right). Measurements are taken at $r_2=r_{\mathrm{M}}$ with $r_{\mathrm{M}} \in I^+$, recall \ref{['eq:I(f)']}. The Fourier coverage (light red) is a union of infinitely many semicircles, some of which are depicted in red. Their centers lie on the dashed blue curve.
• Figure 3: 2D Fourier coverage for a rotating object, incidence direction ${\bf s}=(0,1)$ and measurements taken at $r_2=r_{\mathrm{M}} \in I^+$. The Fourier coverage (light red) is a union of infinitely many semicircles, some of which are depicted in red.
• Figure 4: 2D Fourier coverage for a rotating object, incidence direction ${\bf s}=(1,0)$ and measurements taken at $r_2=r_{\mathrm{M}} \in I^+$. The Fourier coverage (light red) is a union of infinitely many semicircles, some of which are depicted in red.
• Figure 5: 3D Fourier coverage for a full rotation of the object about the $r_1$-axis with incidence direction ${\bf s}=(0, 1, 0)$. Left and center: 3D visualization. Right: 2D cross section through $y_1y_2$-plane. In this case there is no difference in the Fourier coverage between $r_{\mathrm{M}} \in I^+$ or $r_{\mathrm{M}} \in I^-$.

Theorems & Definitions (33)

• Theorem 1
• Proof 1
• Theorem 2
• Proof 2
• Remark 1
• Theorem 3
• Proof 3
• Lemma 3.1
• Proof 4
• Lemma 3.2