Invertibility of the Fourier Diffraction Relation in Raster Scan Diffraction Tomography

TL;DR

This work analyzes the invertibility of the Fourier diffraction relation in raster-scan diffraction tomography with focused beams. By formulating a detailed forward model under the Born approximation and deriving a Fourier diffraction relation, the authors introduce coupling sets $F_y$ and a graph-theoretic framework to study unique recoverability of Fourier coefficients across dimensions. They prove that in dimensions $d\ge3$, all Fourier coefficients involved are generically uniquely determined from the data, while in $d=2$ recoverability is restricted and non-uniqueness arises on a substantial region; in 3D a determinant-based local solvability argument supports robust reconstructibility under mild nondegeneracy conditions. The results provide a rigorous understanding of the theoretical limits of raster-scan diffraction tomography and guide the development of reconstruction algorithms in practical imaging systems.

Abstract

Diffraction tomography aims to recover an object's scattering potential from measured wave fields. In the classical setting, the object is illuminated by plane waves from many directions, and the Fourier diffraction theorem gives a direct relation between the Fourier transform of the scattering potential of the object and the Fourier transformed measurements. In many practical imaging systems, however, focused beams are used instead of plane waves. These beams are then translated across the object to bring different regions of interest into focus. The Fourier diffraction relation adapted to this setting differs in one crucial point from the plane-wave case: while certain Fourier coefficients of the measurements still directly correspond to individual Fourier coefficients of the scattering potential, others are given by linear combinations of two Fourier coefficients of the scattering potential. This article investigates which Fourier coefficients of the scattering potential can be uniquely recovered from these relations. We show that in dimensions higher than two, all coefficients appearing in the equations are typically uniquely determined. In two dimensions, however, only part of the Fourier coverage is uniquely recoverable, while on the remaining subset, distinct coefficients can produce identical data.

Figures (5)

• Figure 1: Schematic overview of the general scan geometry in two dimensions. A focused incident beam propagates in a direction $\omega\in\mathbb{S}^{1}$ and scans an object by moving the focal point along a line $\nu^\perp$ orthogonal to the direction $\nu\in\mathbb{S}^1$. The resulting scattered waves are measured at every point on a receiver line $\{x\in\mathds{R}^2\mid x_2 = L\}$ outside the object.
• Figure 2: (a) Illustration of the symmetry with respect to the scan plane $\nu^\perp$. The subset $\Sigma_1$ (blue) contains directions whose reflections lie outside $S_{\omega}$, whereas for $\Sigma_2$ (red) the reflections remain inside. (b) Region in Fourier space that can be accessed by the measurements. It is obtained as the union of semicircles centered at $-S_\omega = -(\Sigma_1 \sqcup \Sigma_2)$. Representative semicircles are colored according to their centers: blue for $\Sigma_1$ and red for $\Sigma_2$. The dark gray region corresponds to $\mathcal{Y}_1$ (centers in $-\Sigma_1$), and the light gray region to $\mathcal{Y}_2$ (centers in $-\Sigma_2$). In this configuration, the two regions do not intersect. In general, however, $\mathcal{Y}_1$ and $\mathcal{Y}_2$ do not have to be disjoint, see \ref{['lem:intY']}.
• Figure 3: Illustration of the coupling set $F_y$ defined in \ref{['eq:Fy']}. The full spheres $\mathbb{S}^{d-1}_{k_0}$ and $\mathbb{S}^{d-1}_{k_0}-y$ are shown in light orange and light blue, respectively, while the subsets $\Sigma_2$ and $S_{e_d}-y$ are highlighted in saturated colors. (a) For $d=2$, the intersection $\Sigma_2\cap(S_{e_d}-y)$ consists of at most two points, whose projection onto the line $y+\mathbb{R}\nu$ yields the coupling set $F_y$. (b) For $d>2$, the dashed gray curve represents $\mathbb{S}^{d-1}_{k_0}\cap(\mathbb{S}^{d-1}_{k_0}-y)$, while the dashed red curve highlights the restricted intersection $\Sigma_2\cap(S_{e_d}-y)$. Their projections onto $y+\mathbb{R}\nu$ give the segment $L_y$ and the coupling set $F_y$, respectively.
• Figure 4: Illustration of the construction from \ref{['lem:pointsonline']}. Starting with a point $y+w$ in a neighborhood $U$ of $y$, we perturb it slightly in the direction $\nu$ to obtain $y+w+\delta\nu$ for a sufficiently small $\delta>0$ such that the associated coupling sets $F_{y+w}$ and $F_{y+w+\delta\nu}$ intersect. From this intersection we select two points, $z+w$ and $z+w+\varepsilon\nu$. Together, the four points form a closed $4\times4$ system.
• Figure 5: Illustration of the intersection $(\sigma_1+S_{e_2})\cap (\sigma_2+S_{e_2})$ for different positions of the centers. If both centers lie on the upper semicircle, the intersection is given by their sum; otherwise, the intersection lies in the origin.

Theorems & Definitions (58)

• Lemma 2.1
• Definition 1
• Lemma 3.1
• Proof 1
• Definition 2
• Remark 1
• Definition 3
• Lemma 3.2
• Proof 2
• Definition 4