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X-ray imaging from nonlinear waves: numerical reconstruction of a cubic nonlinearity

Markus Harju, Suvi Anttila, Teemu Tyni

TL;DR

The paper tackles recovering a time-dependent potential $q(x,t)$ in a nonlinear wave equation from boundary measurements in a $2$-D setting by leveraging higher-order linearization to link DN-map derivatives to the Radon transform $\mathscr{R}(q)$. It introduces a direct, non-iterative Radon-based reconstruction pipeline aided by spectral regularization of numerical derivatives, and provides stability analyses for the regularization. A comparative pointwise reconstruction is implemented to benchmark the Radon approach. Numerical experiments show that the regularized Radon method accurately recovers the support and structure of $q$ and remains robust under noise and limited-angle data, enabling practical X-ray–style imaging from nonlinear wave data.

Abstract

We study an inverse boundary value problem for the nonlinear wave equation in $2 + 1$ dimensions. The objective is to recover an unknown potential $q(x, t)$ from the associated Dirichlet-to-Neumann map using real-valued waves. We propose a direct numerical reconstruction method for the Radon transform of $q$, which can then be inverted using standard X-ray tomography techniques to determine $q$. Our implementation introduces a spectral regularization procedure to stabilize the numerical differentiation step required in the reconstruction, improving robustness with respect to noise in the boundary data. We also give rigorous justification and stability estimates for the regularized spectral differentiation of noisy measurements. A direct pointwise reconstruction method for $q$ is also implemented for comparison. Numerical experiments demonstrate the feasibility of recovering potentials from boundary measurements of nonlinear waves and illustrate the advantages of the Radon-based reconstruction.

X-ray imaging from nonlinear waves: numerical reconstruction of a cubic nonlinearity

TL;DR

The paper tackles recovering a time-dependent potential in a nonlinear wave equation from boundary measurements in a -D setting by leveraging higher-order linearization to link DN-map derivatives to the Radon transform . It introduces a direct, non-iterative Radon-based reconstruction pipeline aided by spectral regularization of numerical derivatives, and provides stability analyses for the regularization. A comparative pointwise reconstruction is implemented to benchmark the Radon approach. Numerical experiments show that the regularized Radon method accurately recovers the support and structure of and remains robust under noise and limited-angle data, enabling practical X-ray–style imaging from nonlinear wave data.

Abstract

We study an inverse boundary value problem for the nonlinear wave equation in dimensions. The objective is to recover an unknown potential from the associated Dirichlet-to-Neumann map using real-valued waves. We propose a direct numerical reconstruction method for the Radon transform of , which can then be inverted using standard X-ray tomography techniques to determine . Our implementation introduces a spectral regularization procedure to stabilize the numerical differentiation step required in the reconstruction, improving robustness with respect to noise in the boundary data. We also give rigorous justification and stability estimates for the regularized spectral differentiation of noisy measurements. A direct pointwise reconstruction method for is also implemented for comparison. Numerical experiments demonstrate the feasibility of recovering potentials from boundary measurements of nonlinear waves and illustrate the advantages of the Radon-based reconstruction.

Paper Structure

This paper contains 11 sections, 5 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Reconstruction methods
  3. Higher order linearization method
  4. Selecting the boundary values
  5. Reconstruction of the Radon transform of $q$
  6. Pointwise reconstruction of $q$
  7. Numerical implementation
  8. Discretization of the forward model and evaluating the DN-map
  9. Numerical approximation of the mixed derivatives
  10. Numerical implementation of the inverse problem
  11. Numerical examples

Key Result

Lemma 1

Let $G\in \mathrm{Lip}_c(\mathbb{R}^{n+1})$. Let $t_0\in \mathbb{R}$ and $\tau>0$. There exists $C>0$ (depending only on $\mathrm{supp} (G)$) such that Here, $C$ is independent of $\theta\in \mathbb{S}^{n-1}$ and $\eta\in \mathbb{R}$. In particular, the integral on the left converges uniformly to $\mathscr{R}(G)(t_0,\theta, \eta)$ when $\tau \to \infty$.

Figures (8)

  • Figure 1: Comparison of regularized differentiation via high-frequency truncation (red dashed) and Gaussian filtering (blue dot-dashed). The exact derivatives are shown in solid black. Left: Noise level $\sigma=10^{-3}$. Truncation cutoffs: $\xi_{\max}=64$ (1st derivative) and $40$ (2nd derivative). Gaussian parameters: $\alpha=5\times10^{-5}$ (1st derivative) and $8\times10^{-5}$ (2nd derivative). RMS-errors: $e_\mathrm{trunc}=0.016$, $e_\mathrm{Gauss}=0.013$ (1st derivative), and $1.54$ vs. $1.16$ (2nd derivative). Right: Noise level $\sigma=10^{-4}$. Truncation cutoffs: $\xi_{\max}=200$ (1st derivative) and $87$ (2nd derivative). Gaussian parameter: $\alpha=5\times10^{-5}$ (both derivatives). RMS-errors: $0.0089$ vs. $0.0035$ (1st derivative) and $0.61$ vs. $0.49$ (2nd derivative). For the 1st derivative the methods are visually indistinguishable, while for the 2nd derivative the Gaussian filter better suppresses overshoots at jump discontinuities, smoothing the jump in the process.
  • Figure 2: Left: The causal set where waves can be sent and measured lies between the two blue cones within the cylinder $\Omega\times[0,T]$ (gray). This is the optimal set, which can be reached from the lateral boundary $\Sigma$ by light rays and from where light can emanate to the boundary. Right: Illustration of how the Radon transform of $q$ is obtained. The space–time domain $\Omega\times[0,T]$ is shown as the gray cylinder. A (linear) wave $v_1$ (red plane, bottom-right to top-left) propagates through the domain. Auxiliary waves $v_0$ (gray planes, top-right to bottom-left) intersect $v_1$ along line segments with constant time inside $\Omega$ (solid black lines). These intersection lines are the ones over which $\mathscr{R}(q)$ can be computed. Their projections to the initial time plane $t=0$ are indicated by black dashed lines. If $q$ is time-independent, then by varying $v_0$ while fixing a single wave $v_1$ one can recover $\mathscr{R}(q)(\theta_0,\eta_0)$ for all $\eta_0$ at a fixed angle $\theta_0\in\mathbb{S}^1$.
  • Figure 3: Sinograms of Example \ref{['eq: example lungs']}. Left: True $\mathscr{R}(q)$ computed with Matlab's radon. Middle: reconstructed $\mathscr{R}(q)^{\text{rec}}$ by finite differences. Right: reconstructed $\mathscr{R}(q)^{\text{rec}}$ via spectral differentiation. The finite difference approximation of $D^3_{\varepsilon_1}\widetilde{g}(0)$ amplifies measurement error. A better approximation to the true sinogram is obtained via regularized differentiation as $R_\alpha^{3}\widetilde{g}(0)$. Here $\alpha=0.01$.
  • Figure 4: Example \ref{['eq: example bump']}. 1st: Precise unknown potential. 2nd: Pointwise reconstruction using finite differences. 3rd: Radon reconstruction using finite differences. 4th: Radon reconstruction using regularized spectral differentiation. In the unregularized cases, in the pointwise approach noise dominates the reconstruction compared to the back-projection, where the location and rough shape of the potential can be detected. By regularizing the differentiation step of the reconstruction, the location, shape, and size of the potential function are captured accurately.
  • Figure 5: Example \ref{['eq: example lungs']}, smooth bump functions supported on a disc and two ellipses (tilted by $\pm 22.5^\circ$ from the vertical axis). 1st: Precise unknown potential. 2nd: Pointwise reconstruction using finite differences. 3rd: Radon reconstruction using finite differences. 4th: Radon reconstruction using regularized spectral differentiation. Location, shape, and size of the potential functions are captured accurately, but finer overlapping features appear blurred.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • ...and 2 more