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Super-Resolution Structured-Illumination X-Ray Microscopy based on Fourier Decomposition

Stefan Schwaiger, Lennart Forster, Martin Dierolf, Franz Pfeiffer, Benedikt Günther

TL;DR

The paper addresses the limit imposed by detector pixel size in X-ray microscopy by introducing a full-field structured-illumination method grounded in Fourier decomposition. By scanning a 2D grating across one period, the authors create a Dirac-comb–modulated illumination whose Fourier transform yields a superposition of shifted replicas of the sample spectrum, enabling retrieval of information beyond the native pass-band. They demonstrate this approach on a resolution test pattern, achieving a 2.1× resolution enhancement and an SR projection with an effective pixel size of $0.64\,\mu$m, and extend the method to X-ray tomography with sustained Fourier-space correlation beyond the native Nyquist. The technique is multimodal, supporting phase-contrast and dark-field reconstructions from the same data via methods such as UMPA, while delivering an additional super-resolved transmission channel. This work offers a non-destructive, large-field SR imaging route for applications in materials science and biomedicine, with potential improvements through optimized gratings, absorption-based illumination, and GPU-accelerated computation to tackle larger datasets.

Abstract

We present a structured-illumination technique for full-field super-resolution transmission X-ray microscopy, which employs Fourier spectral decomposition inspired by established methods in visible-light microscopy. A 2D grating creating this illumination is stepped across one period to acquire a set of images at unique illumination positions. The Fourier domain of each image is described as a linear combination of replicated sample information at each frequency harmonic. As this superposition is created independently of detection, it contains spatial information exceeding native detector resolution. Recovering the encoded high-frequency components enables the population of an expanded frequency space. We demonstrate the presence of additional sample information in the Fourier spectrum and introduce a method to recover it. We achieve a resolution improvement by a factor of 2.1 for the projection image of a resolution test pattern. We further demonstrate seamless integration into standard X-ray tomography acquisition schemes. The acquisition is inherently multimodal, as phase-contrast and dark-field images can be computed from the same data using methods such as unified modulated pattern analysis, while providing an additional super-resolved transmission channel. These results indicate broad potential for non-destructive testing and biomedical imaging, as they alleviate pixel-size limitations in photon-counting detectors and sample-size restrictions imposed by optical magnification.

Super-Resolution Structured-Illumination X-Ray Microscopy based on Fourier Decomposition

TL;DR

The paper addresses the limit imposed by detector pixel size in X-ray microscopy by introducing a full-field structured-illumination method grounded in Fourier decomposition. By scanning a 2D grating across one period, the authors create a Dirac-comb–modulated illumination whose Fourier transform yields a superposition of shifted replicas of the sample spectrum, enabling retrieval of information beyond the native pass-band. They demonstrate this approach on a resolution test pattern, achieving a 2.1× resolution enhancement and an SR projection with an effective pixel size of m, and extend the method to X-ray tomography with sustained Fourier-space correlation beyond the native Nyquist. The technique is multimodal, supporting phase-contrast and dark-field reconstructions from the same data via methods such as UMPA, while delivering an additional super-resolved transmission channel. This work offers a non-destructive, large-field SR imaging route for applications in materials science and biomedicine, with potential improvements through optimized gratings, absorption-based illumination, and GPU-accelerated computation to tackle larger datasets.

Abstract

We present a structured-illumination technique for full-field super-resolution transmission X-ray microscopy, which employs Fourier spectral decomposition inspired by established methods in visible-light microscopy. A 2D grating creating this illumination is stepped across one period to acquire a set of images at unique illumination positions. The Fourier domain of each image is described as a linear combination of replicated sample information at each frequency harmonic. As this superposition is created independently of detection, it contains spatial information exceeding native detector resolution. Recovering the encoded high-frequency components enables the population of an expanded frequency space. We demonstrate the presence of additional sample information in the Fourier spectrum and introduce a method to recover it. We achieve a resolution improvement by a factor of 2.1 for the projection image of a resolution test pattern. We further demonstrate seamless integration into standard X-ray tomography acquisition schemes. The acquisition is inherently multimodal, as phase-contrast and dark-field images can be computed from the same data using methods such as unified modulated pattern analysis, while providing an additional super-resolved transmission channel. These results indicate broad potential for non-destructive testing and biomedical imaging, as they alleviate pixel-size limitations in photon-counting detectors and sample-size restrictions imposed by optical magnification.
Paper Structure (13 sections, 22 equations, 8 figures)

This paper contains 13 sections, 22 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Demonstrating the Presence of Additional Information
  4. Demonstration of Super-Resolution
  5. Super-Resolution Tomography
  6. Discussion
  7. Methods
  8. Presence of Additional Image Information
  9. Recovering additional information
  10. Experimental Setup for Test Pattern Acquisition
  11. Experimental Setup for Tomographic Acquisition
  12. Reconstruction
  13. Resolution Assessment

Figures (8)

  • Figure 1: Descriptive overview of the frequency landscape and its implications. (a) Exemplary region of the periodically modulated acquisition. (b) Modulation transfer function. The white circle indicates the region of support of $>\qty{10}{\percent}$ modulation. (c) The logarithmic absolute value of the discrete Fourier transform of a single acquired image under structured illumination. Selected peaks in the Dirac comb are labeled. The second-order harmonic is marked at its aliased position. Each column (d--f) discusses an associated separated component marked in (c). The components are separated and zero-padded to twice their size to satisfy the Nyquist criterion for the super-resolved pass-band. The rows (i-iii) depict the same state for each component. (i) Separated components. Their information contribution is assumed to be zero outside of the region of support inferred from (b) and marked by the solid white circle. The additional constraint of the Nyquist frequency is shown by the white dashed square. The origin of each sample replica is marked by a white cross. (ii) The components of (i) shifted to their true position. The marked origin of the replica coincides with the origin of Fourier space. The shifted information exceeds both the previous region of support and the Nyquist limit. (iii) The absolute value of the inverse Fourier transform of (ii). Evidently, the feature modulation increases along the dimension of the shift vector.
  • Figure 1: Native projection image of the resolution test pattern. The colorbar indicates the normalized intensity.
  • Figure 2: Reconstruction and analysis for a resolution test pattern projection. (a) Reconstructed super-resolved projection of the pattern. The height map is normalized by a reference reconstruction. (b) Native image of the center region of the Siemens star in the middle of the test pattern marked in green in (a). (c) Second-order super-resolved image of the same region. (d) Modulation of the line pattern marked in light blue in (a) as a function of spatial frequency. The error margins indicate the propagated standard deviation of the noise. A modulation of $\qty{10}{\percent}$ is chosen as the resolution criterion. The modulation in the reconstructed super-resolution (SR) images is superior across the entire frequency range and extends beyond the native Nyquist limit. (e) Native image of the L-shaped resolution target, marked in orange in (a). The 0.5, 0.75 and 1 features are not resolved. The 2 features are well resolved. (f) Super-resolved image of the L-shaped resolution target marked in orange in (a). The 2, 1 and 0.75 features are resolved. The 0.5 features remain unclear.
  • Figure 2: Super-resolved projection image of the resolution test pattern. The colorbar indicates the normalized intensity.
  • Figure 3: Native and super-resolved tomographic reconstruction. (a) Native downsampled projection. (b) Second-order super-resolution reconstruction of the downsampled projection. The colorbars in (a) and (b) are marked at $0.7$ and $1.1$ in arbitrary units of normalized intensity. (c) Region-of-interest in a tomographic slice of the native downsampled reconstruction. (d) Region-of-interest in a tomographic slice of the second-order super-resolution reconstruction of the downsampled projections. The red arrows in the tomographic reconstructions highlight features that are indistinguishable in the native image in (c) but become discernible in the super-resolved image in (d). The two ticks in the colorbars in (c) and (d) mark the linear attenuation at $0$ and $10$ in units of . (e) Fourier ring correlation computed on (a) and (b) and respective neighboring projections. Features are considered resolved as long as their correlation exceeds the half-bit criterion van_heel_fourier_2005. In the super-resolution (SR) reconstruction, the correlation is sustained beyond the Nyquist limit of the native image.
  • ...and 3 more figures