Speckle-based X-ray microtomography via preconditioned Wirtinger flow

TL;DR

This work addresses the challenge of obtaining quantitative, three-dimensional X-ray phase images without multiple measurements or strong assumptions. It introduces preconditioned Wirtinger flow (PWF), a gradient-based, physics-informed reconstruction that accounts for partial coherence and uses a speckle diffuser to achieve single-shot phase retrieval of the complex sample field. By incorporating a preconditioning filter and an oversampling-based regularization window, PWF delivers high-fidelity phase and attenuation maps and enables accurate 3D refractive-index tomography, outperforming conventional speckle-tracking methods in both fidelity and spatial resolution. The method shows robustness across diverse samples and holds promise for bench-top and potentially polychromatic X-ray imaging, offering a practical path to single-shot, quantitative X-ray phase tomography with reduced measurement burden and radiation dose.

Abstract

Quantitative phase imaging has been extensively studied in X-ray microtomography to improve the sensitivity and specificity of measurements, especially for low atomic number materials. However, obtaining quantitative phase images typically requires additional measurements or assumptions, which significantly limits the practical applicability. Here, we present preconditioned Wirtinger flow (PWF): an assumption-free, single-shot quantitative X-ray phase imaging method. Accurate phase retrieval is demonstrated using a specialized gradient-based algorithm with an accurate physical model. Partial coherence of the source is taken into account, extending the potential applications to bench-top sources. Improved accuracy and spatial resolution over conventional speckle tracking methods are experimentally demonstrated. The various samples are explored to demonstrate the robustness and versatility of PWF.

Figures (16)

• Figure 1: Speckle–based X-ray microtomography with a partially coherent source.a, Schematic of the experimental setup. Partially coherent source provides an additional blurring effect on the intensity image. The $*$ symbol indicates image convolution. $L_1=\qty{3}{\mm}$ and $L_2=\qty{20}{\mm}$ are used throughout the paper. An experimentally measured speckle image of the toothpick with glass beads sample is presented on the detector plane. b, Schematic of preconditioned Wirtinger Flow (PWF). The vectors are defined as $\bm{\psi}:=\left[\psi_r \right]_{1\leq r \leq m}$, where $m$ is the number of pixels in the image. Although the variables and operators are represented in vector-matrix form here for visualization purposes, the actual algorithm is performed based on element-wise calculations and Fourier transforms (see Algorithm \ref{['algorithm:PWF']}). $z=z'+iz"$ is a complex number. c, Reconstructed phase and attenuation from the measured speckle pattern (see Fig. \ref{['fig2']} for details)
• Figure 2: Field reconstruction results of a toothpick with glass beads.a--d, Reconstructed phase (a), horizontal phase gradient (b), and attenuation (c), and the remaining root-mean-square error (RMSE, d) from PWF with a single measurement ($K$ = 1). e--h, The reconstructed counterparts from LCS with 12 measurements at different diffuser positions ($K$ = 12) quenot2021implicit.
• Figure 3: Tomographic reconstruction results of a toothpick with glass beads.a--b, The real (a) and imaginary (b) parts of the reconstructed 3D refractive index distribution, $n(\mathbf{r}) = 1-\delta(\mathbf{r})+i\beta(\mathbf{r})$, from PWF with a single measurement ($K$ = 1). Please refer to Video S1 for the results from different cross sections. The yellow arrows indicate the interface between the toothpick and the adhesive. The white arrow symbols indicate the rotation axis orientations. The intersection of two orthogonal slices is shown as a solid line in the background. c--d, The reconstructed counterparts from a LCS with 12 measurements in different diffuser positions ($K$ = 12) quenot2021implicit. Note that the color scale is different from the PWF results (a and b). The red arrows indicate the reconstruction artifacts. e, Line profiles along the intersection lines shown in the backgrounds. The expected $\delta$ and $\beta$ values of the beads are shown as dotted lines. f, The Fourier shell correlation (FSC) of the reconstructed tomograms. The intersections with the 1/4 criterion (dotted line) are pointed by the arrows with the corresponding spatial resolutions.
• Figure 4: Reconstructed quantitative phase tomogram $\delta(\mathbf{r})$ of various samples.a, A cumin seed; b, a dried shrimp; c, a dried anchovy; and d, a piece of cork. The red and yellow arrows highlights the structural contrast of $\delta$, and fine structures, respectively. All colorbar units are $10^{-6}$. Please refer to Videos S2--S5 for the results from different cross sections. The reconstructed $\beta(\mathbf{r})$ are shown in Fig. \ref{['figS:fig4atten']}.
• Figure S1: Coherence speckle generation in partially coherent X-ray microtomographya, Setup diagram. The source, sample, diffuser and detector are placed from the front. The definitions of the variables used are as follows: $D_0$, the size of the source; $\theta_0$, the emission angle of the source; $\theta_1$, the diffraction angle of the sample; $\theta_2$, the diffraction angle of the diffuser; $\theta_3$, the detection angle at the detector; $\Delta_0$, $\Delta_1$, and $\Delta_2$ are the lateral propagation distances due to $\theta_0$, $\theta_1$, and $\theta_2$, respectively; and $L_0$, $L_1$, and $L_2$ are the distances between the elements. Note that the angles can be negative. The blue and red arrows represent positive and negative angles, respectively. b, Valid $(\theta_1,\theta_2)$ pairs to construct a speckle grain. The blue and red lines represent the bounds of the equations \ref{['eqS:thetaConditionPixel']} and \ref{['eqS:thetaConditionSource']}, respectively, and the shaded area represents the $(\theta_1,\theta_2)$ pairs that satisfy the equations. The constants $A$ and $B$ are the intercepts of $\theta_2$, which are $\lambda/(4p) + \lambda/(2D_0)$ and $\lambda(L_0+L_1)/(2D_0L_2) + \lambda/(2D_0)$, respectively. c, Same plot as b, but when $B<A$ and $L_1=0$, the overlapped area is constrained only by Eq. \ref{['eqS:thetaConditionSource']}.