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Raster Scan Diffraction Tomography

Peter Elbau, Noemi Naujoks, Otmar Scherzer

Abstract

Diffraction tomography is a widely used inverse scattering technique for quantitative imaging of weakly scattering media. In its conventional formulation, diffraction tomography assumes monochromatic plane wave illumination. This assumption, however, represents a simplification that often fails to reflect practical imaging systems such as medical ultrasound, where focused beams are used to scan a region of interest of the human body. Such measurement setups, combining focused illumination with scanning, have not yet been incorporated into the diffraction tomography framework. To bridge this gap, we extend diffraction tomography by modeling incident fields as Herglotz waves, thereby incorporating focused beams into the theory. Within this setting, we derive a new Fourier diffraction relation, which forms the basis for quantitative tomographic reconstruction from scanning data. Using this result, we systematically analyze how different scan geometries influence the reconstruction.

Raster Scan Diffraction Tomography

Abstract

Diffraction tomography is a widely used inverse scattering technique for quantitative imaging of weakly scattering media. In its conventional formulation, diffraction tomography assumes monochromatic plane wave illumination. This assumption, however, represents a simplification that often fails to reflect practical imaging systems such as medical ultrasound, where focused beams are used to scan a region of interest of the human body. Such measurement setups, combining focused illumination with scanning, have not yet been incorporated into the diffraction tomography framework. To bridge this gap, we extend diffraction tomography by modeling incident fields as Herglotz waves, thereby incorporating focused beams into the theory. Within this setting, we derive a new Fourier diffraction relation, which forms the basis for quantitative tomographic reconstruction from scanning data. Using this result, we systematically analyze how different scan geometries influence the reconstruction.
Paper Structure (14 sections, 4 theorems, 72 equations, 8 figures, 1 table)

This paper contains 14 sections, 4 theorems, 72 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The forward model
  3. First-order scattering model
  4. Herglotz wave illumination
  5. Scanning measurements
  6. Fourier diffraction theorem for scanning measurements
  7. Fourier reconstruction
  8. Naive backpropagation
  9. Advanced backpropagation
  10. Fourier coverages for 2D scan geometries
  11. Transmission imaging
  12. Reflection imaging
  13. Oblique imaging
  14. Conclusions

Key Result

Theorem 1

Let $d\in\mathds{N}\setminus\{1\}$ and $f\in L^1(\mathds{R}^d)$ be a function with $\mathop{\mathrm{supp}}\nolimits(f)\subseteq \mathcal{B}^d_{r}$. Further, assume the incident wave is represented by eq:ui_rigidm with $a\in L^2(\mathbb{S}^{d-1}_{k_0})$. Choose an orientation $\nu \in \mathbb{S}^{d-1 for $k\in\mathcal{B}^{d}_{k_0}\cap e_d^\perp$ and $\xi\in\mathcal{B}^d_{k_0}\cap \nu^\perp$, where

Figures (8)

  • Figure 1: Schematic overview of the general scan geometry in $d=2$. (a) A focused beam propagates in the direction $\omega\in\mathbb{S}^{1}$ and scans an object by shifting the focal point along the line $\nu^\perp$ orthogonal to the direction $\nu\in\mathbb{S}^1$. The resulting scattered waves are measured on a receiver line $x_2 = L$ outside the object. (b) Reflection imaging with the beam pointing downward, $\omega=\nu=-e_2$. (c) Transmission imaging with the beam pointing upward, $\omega=\nu=-e_2$.
  • Figure 2: Illustration of the classical Fourier diffraction theorem stated in \ref{['eq:FDT']}. For plane-wave illumination with direction $s\in\mathbb{S}_{k_0}^{d-1}$, the Fourier transform of the scattering potential is accessible on a downward-oriented semicircle for $d=2$ or hemisphere for $d=3$, both of radius $k_0$ and centered at $-s\in\mathbb{S}_{k_0}^{d-1}$.
  • Figure 3: Illustration of the symmetry with respect to the scan plane $\nu^\perp$. (a) All directions $\sigma\in\mathbb{S}^{d-1}_{k_0}$ contribute to the Fourier diffraction relation when the beam profile $a\in L^{2}(\mathbb{S}^{d-1}_{k_0})$ is fully supported. (b) If $a(\sigma)\neq 0$ only for $\sigma\in S_{\omega}$, then only directions in $S_{\omega}$ contribute. Here, the subset $\Sigma_1$ contains directions whose reflections lie outside $S_{\omega}$, whereas for $\Sigma_2$ the reflections remain inside.
  • Figure 4: Illustration of the decomposition of $S_\omega\subseteq \mathds{R}^2$ corresponding to \ref{['ex:Ytilde']}. The two half-planes defined by the directions $\omega = (1/\sqrt{2}, -1/\sqrt{2})$ and $H_\nu\omega = (1/\sqrt{2}, 1/\sqrt{2})$ divide the semicircle $S_\omega$ into two arcs, $\Sigma_1$ and $\Sigma_2$. Because the reflection at the line $\nu^\perp$ (here, the $y_1$-axis) maps every point in $\Sigma_2$ to the opposite vertical half-plane, the part of $\Sigma_2$ lying above the $y_1$-axis constitutes $\tilde{\Sigma}$.
  • Figure 5: Perpendicular vs. tilted scan in transmission imaging: the figure illustrates how the total coverage $\mathcal{Y}^\mathrm{adv}=\mathcal{Y}_1\subseteq \mathds{R}^2$ changes with the scan configuration in transmission imaging. The beam propagates along $\omega = e_2$, while its focal point moves along $\nu^\perp$ with normal $\nu = (\cos\theta,\ \sin\theta) \in \mathbb{S}^1$ for different angles $\theta\in[0,\pi]$. The corresponding set $-\Sigma_1\subseteq S_{- e_2}$ is shown as a dashed orange arc. The perpendicular scan, $\omega=\nu=e_2$, achieves maximum coverage.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Example 1: Gaussian beam
  • Theorem 1: Fourier transform of the measurements
  • Proof 1
  • Remark 1: Connection to Synthetic Aperture Diffraction Tomography
  • Definition 1: Reduced measurements
  • Example 2: The sets $\Sigma_1$ and $\Sigma_2$ for perpendicular and parallel scans
  • Corollary 1: Fourier diffraction relation for scanning measurements
  • Definition 2: Naive backpropagation
  • Example 3: Naive backpropagation for perpendicular scans
  • Theorem 2: Fourier coverage in two dimensions
  • ...and 4 more