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Diffraction Tomography for a Generalized Incident Field

Clemens Kirisits, Noemi Naujoks, Otmar Scherzer

TL;DR

This work extends diffraction tomography (DT) beyond the standard plane-wave illumination by introducing a generalized incident field described via an angular spectrum, enabling focused-beam illumination. By deriving an adapted Fourier diffraction relation and formulating a two-step inversion that first recovers Fourier-domain data through a regularized inversion of a compact operator and then performs Fourier inversion, the authors provide a practical reconstruction pipeline for $2$D DT with arbitrary illumination. Numerical experiments show that focusing can degrade stability, but accurately modeling the illumination improves reconstruction when plane-wave assumptions fail and demonstrates the method's value for ultrasound/optical tomography with tailored beams. Overall, the paper broadens the applicability of DT to customized illumination schemes and offers a regularized, two-step framework for robust reconstruction under nonplane-wave conditions.

Abstract

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

Diffraction Tomography for a Generalized Incident Field

TL;DR

This work extends diffraction tomography (DT) beyond the standard plane-wave illumination by introducing a generalized incident field described via an angular spectrum, enabling focused-beam illumination. By deriving an adapted Fourier diffraction relation and formulating a two-step inversion that first recovers Fourier-domain data through a regularized inversion of a compact operator and then performs Fourier inversion, the authors provide a practical reconstruction pipeline for D DT with arbitrary illumination. Numerical experiments show that focusing can degrade stability, but accurately modeling the illumination improves reconstruction when plane-wave assumptions fail and demonstrates the method's value for ultrasound/optical tomography with tailored beams. Overall, the paper broadens the applicability of DT to customized illumination schemes and offers a regularized, two-step framework for robust reconstruction under nonplane-wave conditions.

Abstract

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.
Paper Structure (19 sections, 5 theorems, 85 equations, 11 figures, 1 algorithm)

This paper contains 19 sections, 5 theorems, 85 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Forward Model
  3. Conceptual setup
  4. A generalized incident field
  5. The Born approximation
  6. Comparison of the forward models
  7. A Fourier diffraction relation
  8. Rotating the incident field
  9. Inverse problem
  10. Identification of the data in $k$-space
  11. Frequency coverage and Fourier inversion
  12. Numerical implementation
  13. Simulation of measurements
  14. Discrete backpropagation
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f\in L^2(\mathds{R}^2)$ with $\operatorname{supp}(f)\subset\mathcal{B}_{r_s}$ and $a\in L^2(\mathbb S^1)$. Then, for $k\in(-k_0,k_0)$ and with the definitions introduced above, we have

Figures (11)

  • Figure 1: Comparison of experimental setups. In both cases the scattering potential $f$ is supported in $\mathcal{B}_{r_s}$ while measurements are taken at $r_2= r_M>r_s$. (a) Conventional DT setup: Incident field $u^{\text{inc}}_{\mathbf{s}}$ is a plane wave propagating in direction $\mathbf{s}$. Additional data are typically generated by varying $\mathbf{s}$ or rotating the object. (b) DT setup considered here: Generalized incident field $u^{\text{inc}}$ of the form \ref{['eqn: Inc']}, e.g. a focused beam, making a full $360^\circ$ rotation during data acquisition.
  • Figure 2: Incident illumination scenarios. (a) and (b) visualize waves $\left|\operatorname{Re}(u^{\text{inc}})\right|$ of the form described in \ref{['eqn: Inc']} with a Gaussian profile given via \ref{['eq:gaussian_a']}, for different amplitudes $A$. (c) shows a plane wave $u^{\text{inc}}_{\mathbf{s_0}}$ propagating in direction $\mathbf{s}_0 = (0,-1)^\intercal$. The wavelength in all illumination settings is $\lambda = \frac{2\pi}{k_0} = 1$.
  • Figure 3: Comparison of the forward models. An object is specified as the characteristic function $f_3\coloneqq\mathbf{1}_{\mathcal{B}_3}$. The scattered wave under Born approximation from \ref{['eqn: u']} is calculated using plane wave illumination on the one hand and focused imaging on the other hand. For illumination we used a downward propagating plane wave $u^{\text{inc}}_{\mathbf{s_0}}$, $\mathbf{s}_0 = (0,-1)^\intercal$ and a focused beam from \ref{['eq:gaussian_a']} with parameter $A = 10$. In both cases the wave number is chosen as $\lambda = \frac{2\pi}{k_0}=1$ and the measurements are taken at $\{\mathbf{r}\in\mathds{R}^2:r_2=-6\}$.
  • Figure 4: Construction of the 2D frequency coverage. The coverage $T((-k_0,k_0)\times \mathcal{I})$ for different sub-intervals $\mathcal{I}\subseteq[-\pi,\pi]$ is illustrated. The resulting shadowed coverages are built from a union of infinitely many semicircles of radius $k_0$ whose centres lie on the dashed line $-k_0\mathbf{s}(\varphi)$, $\varphi\in\mathcal{I}$. Some of these semicircles are depicted in blue. (c) shows the full coverage $T(\mathcal{U})$ as the union of (a) and (b).
  • Figure 5: Level sets of Banach indicatrix. (a) and (b) illustrate the the sets $A_1,A_2\subset T(\mathcal{U})$ as defined in \ref{['eq:A1']} and \ref{['eq:A2']}. (c) shows the union of the sets and the level sets of the Banach indicatrix $\operatorname{Card}(T^{-1} (\mathbf y) )$.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Example 1
  • Remark 1
  • Theorem 1
  • Proof 1
  • Definition 1
  • Lemma 3.1
  • Proof 2
  • Theorem 2
  • Theorem 3
  • Proof 3
  • ...and 2 more