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Abstract
Simultaneous measurement of X-ray ptychography and fluorescence microscopy allows high-resolution and high-sensitivity observations of the microstructure and trace-element distribution of a sample. In this paper, we propose a method for improving scanning fluorescence X-ray microscopy (SFXM) images, in which the SFXM image is deconvolved via virtual single-pixel imaging using different probe images for each scanning point obtained by X-ray ptychographic reconstruction. Numerical simulations confirmed that this method can increase the spatial resolution while suppressing artifacts caused by probe imprecision, e.g., probe position errors and wavefront changes. The method also worked well in synchrotron radiation experiments to increase the spatial resolution and was applied to the observation of S element maps of ZnS particles.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Visualizing the distributions of trace elements and microscopic structures in materials and cells provides essential clues for understanding their functions. Analytical methods utilizing focused electron beams and X-rays are useful for visualizing samples at the microscale. In particular, scanning fluorescence X-ray microscopy (SFXM) [1] using synchrotron radiation is effective for visualizing the microstructures and determining the elemental distributions of thick micrometer-scale samples. In SFXM, X-rays focused by optical elements, such as Fresnel zone plates (FZPs), total reflection focusing mirrors, and refractive lenses, are used to scan a sample plane, and the elemental distribution is obtained by measuring the X-ray fluorescence intensity at each scanning point. X-ray fluorescence microscopy has the advantage of higher sensitivity to trace elements than transmission X-ray microscopy and is used in various fields—particularly in the life sciences [2–4]. However, the spatial resolution is limited to several tens of nanometers due the optical elements, which is inferior to the resolution of electron microscopy. In addition, SFXM is unsuitable for observing structures composed of light elements (C, H, O, N, etc.) with low fluorescence quantum yields.
X-ray ptychographic and fluorescence microscopy is based on the simultaneous measurement of SFXM and X-ray ptychography [5], which is a scanning coherent X-ray diffraction imaging technique [6–11]. In X-ray ptychography, a coherent X-ray beam is used to scan the sample plane so that the illumination areas overlap; at each scanning point, a diffraction intensity pattern is collected by a two-dimensional detector placed behind the sample. By performing a phase-retrieval calculation on the obtained diffraction intensity pattern dataset, a complex transmission function of the sample is reconstructed, providing a distribution of the amount of X-ray absorption and the amount of X-ray phase shift, which is sensitive to light elements. Furthermore, the resolution of the reconstructed image depends on the detectable scattering angle and is not limited by the fabrication accuracy of the focusing optical element. Therefore, X-ray ptychographic and fluorescence microscopy allows the observation of the distribution of trace elements in a sample and microstructures composed of light elements with high sensitivity and resolution. Additionally, this approach is expected to shorten the measurement time and reduce the damage to the sample compared with separate measurements [11].
Another significant advantage of X-ray ptychographic and fluorescence microscopy is that the probe information provided by the ptychographic reconstruction can be used for the deconvolution of SFXM images. Deconvolution of scanning microscope images requires knowledge of the precise intensity distribution of the probe, which is not straightforward. X-ray ptychography allows us to reconstruct a complex wavefield illuminating a sample by using extended reconstruction algorithms [12]. The use of the reconstructed probe intensity distribution for the deconvolution of the SFXM image can increase both the contrast and the resolution [6,7]. However, when deconvolution is applied to data with errors, such as probe position error or probe wavefront fluctuations during measurement, distortions and/or artifacts, which are difficult to distinguish from the real structure, appear in the image. Various measures have been taken to reduce these errors, such as stage position control using laser interferometry [13] and temperature control of the optical system [14]. However, experimentally suppressing these errors remains a challenge.
In recent years, the development of ptychographic reconstruction algorithms has allowed the accurate reconstruction of probe information, such as the probe position error and wavefront variations. Several algorithms have been developed for probe position correction, including annealing [15], cross-correlation [16], and intensity gradient (IG) methods [17]. These algorithms can estimate the position error on the subpixel order using the image-shift method proposed by Maiden et al. [18]. Although the assumption of probe invariance is a strong constraint in ptychographic reconstruction, the orthogonal probe relaxation (OPR) method [19] relaxes this constraint by linking probes at different scanning points using singular value decomposition, which can be used to reconstruct different probes at each scanning point. Applying this information to the deconvolution process can alleviate the problems caused by inaccurate probe information described above, but this has not been demonstrated. Virtual single-pixel imaging (v-SPI) [20], which was recently developed by Zhang et al., is suitable for this purpose. Many deconvolution methods, such as the Wiener filter [21] and Richardson–Lucy method [22,23], require a Fourier transform of the target image in the process of filtering in the inverse space and convolving with a blur kernel. This implicitly imposes the condition that the positional relationship between the scanning points on the sample plane and the pixels of the sample image is consistent and that the probe is constant at all scanning points. In contrast, v-SPI has the advantage that a different blur kernel (or probe image) can be set for each pixel because deconvolution is performed by solving linear equations.
In this paper, we propose a v-SPI-based deconvolution method to improve X-ray fluorescence microscopy images by utilizing different probe images for each scanning point obtained via X-ray ptychographic reconstruction. We first investigated the effectiveness of the proposed method for SFXM images with probe position errors through a numerical simulation. Then, the method was applied to a scanning transmission X-ray microscopy (STXM) image to evaluate its effects on the probe intensity and wavefront fluctuations. Finally, the method was applied to an SFXM image of ZnS particles acquired using X-ray ptychographic and fluorescence microscopy.
2. Principle
v-SPI is a deconvolution method based on the principle of single-pixel imaging (SPI) [24,25], in which a sample image is reconstructed from the measurement data of a single-pixel detector. In SPI, the incident light on the sample is modulated using a spatial light modulator, and a single-pixel detector records the response of the sample to each illumination pattern. The scheme is expressed as follows:
where $A\in \mathbb {R}^{M\times N}$ is a matrix of $M$ modulated illumination patterns consisting of $N$ pixels, $\mathbf {x}\in \mathbb {R}^{N\times 1}$ is a vector containing $N$ pixels representing the reconstructed image, and $\mathbf {b}\in \mathbb {R}^{M\times 1}$ is a vector consisting of $M$ measurement values. The image of the sample is reconstructed by solving Eq. (1).In v-SPI, we consider the pixel value of the measured image, i.e., $b_{i}(i=1, \ldots, M)$, to be equal to the integral of the blurred image value modulated by the set of discrete probe positions $\pi _{i}$. This value is mathematically equivalent to the integral of the value of the ideal sample image $\mu$ multiplied by the convolution of the set of discrete probe positions $\pi _{i}$ with a blur kernel $h_{i}$ [20]. These relationships are expressed as follows:
A schematic of the proposed method is shown in Fig. 1. In X-ray ptychography, the image of a sample is reconstructed by performing a phase-retrieval calculation on the diffraction intensity pattern $I_{i}\in \mathbb {R}^{p\times q}\ (i=1, \ldots, M)$ collected using a two-dimensional detector. Using the aforementioned algorithms, the positions and complex wavefield of the probe, which differ among scanning points, are reconstructed simultaneously. The set of discrete probe positions $\pi _{i}(\mathbf {r})$ is obtained as the reconstructed probe position, and a blur kernel $h_{i}(\mathbf {r})\in \mathbb {R}^{p\times q\times M}$ is obtained by taking the square of the absolute value of the complex wavefield (See Supplement 1, 1. Process for obtaining blur kernel from reconstructed probe function). The convolution of these gives the illumination pattern $\pi _{i}(\mathbf {r})\ast h_i(\mathbf {r})\in \mathbb {R}^{P\times Q\times M}$, which is arranged to form the matrix $A\in \mathbb {R}^{M\times N}(N=P\times Q)$. Using this matrix and vector $\mathbf {b}\in \mathbb {R}^{M\times 1}$ obtained from the scanning microscope image, we can formulate Eq. (1). The vector $\mathbf {x}\in \mathbb {R}^{N\times 1}$ obtained as the solution of the equation gives $\mu ^{\prime }(\mathbf {r})\in \mathbb {R}^{P\times Q}$, which is the result of deconvolution incorporating the correction for errors estimated via X-ray ptychography.
Fig. 1. Schematic of the deconvolution of SFXM images using v-SPI with correction for the probe position and probe variation reconstructed by X-ray ptychography.
For solving the linear equation, any algorithm can be selected in terms of noise level, amount of data to be handled, and computational speed [25]. In this study, we used the least-squares method with QR-factorization (LSQR) [26] owing to its fast convergence to the solution.
3. Numerical simulation
To evaluate the effectiveness of the proposed method for scanning microscopy images measured in the presence of probe position errors, we performed a numerical simulation of X-ray ptychographic and fluorescence microscopy. As a probe to irradiate the sample, we used 5-keV X-rays focused using an FZP with an outermost zone width of 50 nm. Typical protein (H$_{48.6}$C$_{32.9}$N$_{8.9}$O$_{8.9}$S$_{0.6}$, 1.35 g cm$^{-3}$) [27] with a thickness of 2 $\mu$m was employed as the model of the sample. Structures consisting of H, C, N, and O with low fluorescence quantum yields and trace amounts of S were observed using X-ray ptychography and SFXM, respectively. Figure 2(a) shows the sample structures of sulfur and other elements (pixel size: 12.5 $\times$ 12.5 nm$^2$). The diffraction intensity patterns were collected with a two-dimensional detector (pixel size: 75 $\times$ 75 $\mu$m$^2$) located 3.2 m downstream of the sample. As the sample function, we used a multiplication of the two sample functions shown in Fig. 2(a). The intensity of the fluorescent X-rays $I_{f}$ at each scanning point was calculated using the following equation [28]:
Fig. 2. (a) Sample image of the protein (H$_{48.6}$C$_{32.9}$N$_{8.9}$O$_{8.9}$S$_{0.6}$) with a thickness of 2 $\mu$m used in the simulation (left: structures of S; right: structures of elements other than S). (b) Reconstructed images by X-ray ptychography obtained using the ePIE and IG method (left: phase image; middle: transmission image; right: probe-intensity distribution). (c) Ideal SFXM image for S K$\alpha$ emission. (d) Simulated SFXM image without probe position error. (e) Simulated SFXM image with probe position error. (f) Deconvolution results of the SFXM image with probe position error (left: without position correction; middle: with correction using the probe position reconstructed via the IG method; right: with correction using the actual probe position). (g) Comparison of normalized residual errors plotted with respect to the LSQR iteration number. (h) Comparison of FRCs between the simulated SFXM images (dashed line) and the deconvolution results (solid line).
Figure 2(b) shows the phase-shift distribution in the sample plane (phase image), transmittance distribution in the sample plane (transmission image), and probe-intensity distribution obtained via ptychographic reconstruction. In both the phase and transmission images, structures composed of elements other than sulfur were reconstructed, whereas those composed of sulfur were difficult to observe. Figures 2(c–e) show an ideal SFXM image (without noise, blurring, and probe position error), simulated SFXM image without probe position error, and simulated SFXM image with probe position error (<10 nm), respectively. These images show structures composed of sulfur that were difficult to observe using X-ray ptychography. However, compared with the ideal image, the simulated images exhibited reduced contrast and blurred structures. We can also see that Fig. 2(e), which was subjected to probe position error, has slightly rougher contours than Fig. 2(d). Figure 2(f) shows the deconvolution results of the simulated SFXM image with probe position error (Fig. 2(e)) obtained using v-SPI under three different conditions: without probe position correction (left), with correction using the probe position estimated by the IG method (middle), and with correction using the actual probe position (right). All the results exhibited a similar contrast to the ideal image (Fig. 2(c)), indicating that v-SPI worked adequately. However, significant artifacts were observed in the images without position correction. In comparison, the two images acquired with position correction had fewer artifacts and finer structures.
We present graphs of the normalized residual error $\frac {\Arrowvert A\mathbf {x}-\mathbf {b}\Arrowvert }{\Arrowvert \mathbf {b}\Arrowvert }$ ($\Arrowvert \cdot \Arrowvert$ denotes the L$_2$ norm) with respect to the LSQR iteration number in Fig. 2(g). The normalized residual errors at the last iteration were 1.00 $\times$ 10$^{-2}$, 5.93 $\times$ 10$^{-3}$, and 5.48 $\times$ 10$^{-3}$ for the three conditions: without correction for the probe position, with correction using the position reconstructed via the IG method, and with correction using the actual position, respectively. This indicates that the position correction reduced the residual error by >40%. Figure 2(h) shows Fourier ring correlations (FRCs) [30] of the simulated SFXM images (Figs. 2(d) and (e)) and deconvolution results (Fig. 2(f)). Although the difference between the SFXM images with and without position error is slight, the deconvolution results show improved FRCs in the spatial frequency range of 0.005 to 0.015 nm$^{-1}$. Comparing the deconvolution results with and without position correction, those with position correction show further improvement in the spatial frequency range higher than 0.008 nm$^{-1}$. These results indicate that simultaneous implementation of deconvolution and probe position correction is effective in improving the resolution of scanning microscope images. In addition, the results obtained using the reconstructed probe positions were comparable to those obtained using accurate positions, indicating that the proposed method has practical utility.
Note that ringing artifacts near the edges of the deconvolution results (Fig. 2(f)) are attributed to boundary effects [31]. The boundary effects are caused when pixels near the edges have sample information outside the field of view due to the spatial extent of a blur kernel. Such information becomes the source of error in deconvolution processes, resulting in ringing artifacts [32]. Zhang et al. addressed boundary effects in v-SPI by cropping image edges with a width equal to the FWHM of the blurring kernel [20]. Our results show ringing artifacts in regions wider than the probe FWHM ($\sim$50 nm), which seems to be due to the presence of probe sidelobes. Although these ringing artifacts may degrade the FRCs in the higher spatial frequency range, we can compare the FRCs of the deconvolution results because the noises are observed in the same way. Therefore, the ringing artifacts do not affect the above discussion.
4. Experiments
4.1 Ta test chart
Next, we investigated the effectiveness of the proposed method for measurement data with probe variations. The proposed method can be applied to scanning X-ray microscope images based on any principal measurement (e.g., transmission method or photoelectron yield method) as long as the measurement can be performed simultaneously with X-ray ptychography. Thus, we applied the proposed method to an STXM image of a 200-nm-thick Ta test chart prepared from diffraction intensity patterns obtained via X-ray ptychography measurements.
Measurements were performed on BL29XU at SPring-8 using a previously reported optical system [33]. X-rays monochromatized to 5 keV by a Si(111) double-crystal monochromator via harmonic cut mirrors irradiated a triangular aperture having side lengths of approximately 10 $\mu$m, which was fabricated by applying a focused ion beam to 20-$\mu$m-thick Pt foil polished on both sides (TDC Co., Ltd.). The X-ray beam illuminated an FZP (Applied Nanotools Inc.) with an outermost zone width of 50 nm and a diameter of 180 $\mu$m, which was located 108 mm downstream of the aperture, and its size was reduced by a factor of 2 at the sample position 54 mm downstream of the FZP. The FZP was positioned approximately 50 $\mu$m off-axis, and the first-order diffraction light was extracted using an order-sorting aperture (OSA) with a 10-$\mu$m-diameter circular aperture of 25-$\mu$m-thick Ta. The diffraction intensity patterns of the sample were collected using a two-dimensional detector (EIGER 1M, Dectris Ltd.) with a pixel size of 75 $\times$ 75 $\mu$m$^2$, which was placed approximately 3.2 m downstream of the sample. The sample was raster-scanned using a piezo stage (P-621.ZLC and P-621.1CL, PI GmbH Co.) at 99 $\times$ 99 points with a spacing of 500 nm. The flux of the X-rays incident on the sample was approximately 1.0 $\times$ 10$^7$ photons s$^{-1}$ and the exposure time at each scanning point was 1 s. The total measurement time was approximately 10.5 h. Diffraction intensity patterns of 21 $\times$ 21 pixels were used for phase-retrieval calculations, resulting in a pixel size of 500 $\times$ 500 nm$^2$ for the reconstructed image. In the reconstruction, OPR, which can correct for probe variation, was used in combination with ePIE, and the number of eigenprobes in OPR was set to 5. An STXM image was prepared from the bright-field intensities of the diffraction intensity patterns. Deconvolution of the STXM image via v-SPI was performed under three conditions: using an average probe at all scanning points without correction for probe intensity variations, using the average probe with correction for OPR-reconstructed intensity variations, and using different probes at each scanning point with correction for OPR-reconstructed intensity variations. For each condition, 150 LSQR iterations were performed.
Figures 3(a) and (b) show the STXM image of the sample and the result of the X-ray ptychographic reconstruction using OPR (left: transmission image; middle: phase image; right: probe intensity distribution averaged over all scanning points), respectively. The intensity of the STXM image was normalized by the average intensity of the reconstructed probes to facilitate comparison with the transmission image of X-ray ptychography. In the STXM image, the structure of the sample is significantly blurred owing to the large probe size of approximately 5 $\mu$m. In addition, some of the stripes, which were due to fluctuations in the incident X-ray intensity, extend laterally, which is the direction of the raster scan. Figure S1 shows the results of the ptychographic reconstruction using only ePIE without consideration of probe variation. Artifacts that appear to be caused by intensity fluctuations are present throughout the transmission image, and the phase image exhibits a non-uniform distribution. This suggests that both the intensity and wavefront of the probe fluctuated during the measurement. In contrast, the image reconstructed using OPR (Fig. 3(b)) exhibited fewer artifacts in the transmission image and a relatively uniform distribution in the phase image. These results indicate that the correction of probe variation via OPR was effective for the measurement data.
Fig. 3. (a) STXM image of the Ta test chart. (b) Results of ptychographic reconstruction using the OPR (left: transmission image of the sample; middle: phase image of the sample; right: average intensity distribution of probes individually reconstructed at all scanning points). (c)–(e) Deconvolution results of STXM images obtained using v-SPI ((c) using the average probe without correction for probe intensity variation, (d) using the average probe with correction for OPR-reconstructed intensity variation, and (e) using different probes for each scanning point with correction for OPR-reconstructed intensity variation). (f) Comparison of the normalized residual errors for the different conditions.
Figures 3(c–e) show the deconvolution results of the STXM image obtained via v-SPI under the following conditions: using the probe averaged over all scanning points without correction for probe intensity variations (Fig. 3(c)), using the average probe with correction for OPR-reconstructed intensity variations (Fig. 3(d)), and using a different probe for each scanning point with correction for OPR-reconstructed intensity variations (Fig. 3(e)). In Fig. 3(c), where no correction for intensity fluctuations is performed, the artifacts caused by intensity fluctuations are emphasized and spread over the entire image, whereas in Figs. 3(d) and (e), where correction for intensity variation is performed, the artifacts are significantly reduced. Furthermore, the regions surrounded by the solid lines in Figs. 3(d) and (e) show that the artifacts were reduced by correcting for fluctuations in the probe intensity distributions. Even though a 5-$\mu$m probe was used, a 1-$\mu$m-wide structure is observed in the region surrounded by the dashed line in Fig. 3(e), indicating that the resolution was significantly improved by the deconvolution. Additionally, the normalized residual errors shown in Fig. 3(f) confirm that correcting for intensity fluctuations significantly reduced the residual error and that correcting for the probe intensity distribution further reduced the residual error. The normalized residual errors at the last iteration were 2.79 $\times$ 10$^{-3}$, 4.13 $\times$ 10$^{-4}$, and 3.08 $\times$ 10$^{-4}$ for the three conditions: using the average probe without intensity correction, using the average probe with intensity correction, and using different probes for each scanning point with intensity correction, respectively. This implies that the residual error was reduced by approximately 89% with the correction for probe variation. These results indicate that incorporating the probe variation correction into deconvolution improves the image quality and that the proposed method works well for experimental data.
In this experiment, we succeeded in correcting for slow probe changes that occurred over a long measurement time of 10.5 h. However, our method is not expected to be effective for data in which rapid probe changes are present. This limitation is imposed by the principle of the OPR algorithm, which reconstructs probe changes by exploiting the information redundancy introduced by the overlap of adjacent illumination areas. Thus, if the probe changes drastically between adjacent scanning points, it is difficult to reconstruct the change.
4.2 ZnS particles
Finally, the proposed method was applied to an SFXM image that was obtained via X-ray ptychographic and fluorescence microscopy. ZnS particles (Kishida Chemical Co., Ltd.) dispersed on an SiN membrane were used as a sample. The experiment was performed on BL27SU B branch at SPring-8 using the optical system shown in Fig. 4(a). X-rays monochromatized to 2.5 keV by a Si(111) channel-cut monochromator via a harmonic cut mirror irradiated a 155-$\mu$m-diameter pinhole with a 50-$\mu$m-diameter beam stop made of 10-$\mu$m-thick Ta. The X-rays illuminated an FZP (diameter of 106.7 $\mu$m, outermost zone width of 412 nm, NTT Advanced Technology Corporation). Using a 200-$\mu$m-thick Si OSA with a square aperture having side lengths of 10 $\mu$m, only first-order diffraction light was extracted to illuminate the sample located at the focal spot, which was approximately 88 mm downstream of the FZP. A two-dimensional detector (SOPHIAS-L) [34] with a pixel size of 30 $\times$ 30 $\mu$m$^2$ located approximately 1.13 m downstream of the sample was used to collect the diffraction intensity patterns. Simultaneously, the intensity of S K$\alpha$ emission from the sample was measured using a silicon drift detector (SDD, Amptek Inc.) with an active area of 25 mm$^2$. The SDD was placed approximately 1 mm behind the sample and approximately 1 mm off the optical axis, with the detector plane parallel to the direction of the optical axis. Its solid angle was estimated to be approximately 0.86 sr using the approximate formula [35]. The sample was raster-scanned using a piezo stage (P-621.ZLC and P-621.1CL, PI GmbH Co.) at the 7(H) $\times$ 21(V) points, with a spacing of 250 nm. The incident X-ray flux of the sample was approximately 4.0 $\times$ 10$^7$ photons s$^{-1}$, and the exposure time was 5 s scan$^{-1}$. The total measurement time was approximately 1.5 h. For the phase-retrieval calculation, mixed-state ePIE [36], which allows reconstruction using data measured with partially coherent X-rays, and the IG method were used. The number of mixed-state modes was set to 8. Using diffraction intensity patterns of 598 $\times$ 598 pixels, a reconstructed image with a pixel size of approximately 31.25 $\times$ 31.25 nm$^2$ was acquired. In this analysis, we used a probe image with a pixel size of 31.25 $\times$ 31.25 nm$^2$ for deconvolution and obtained an image with a pixel size of 250 $\times$ 250 nm$^2$ by performing 8 $\times$ 8 binning of the resulting image. Deconvolution was performed under two conditions: with and without correction of the probe position reconstructed using the IG method. For each condition, eight iterations of LSQR were performed. During the iteration, a constraint was imposed such that pixels with values less than zero were set to zero to prevent the negative divergence of pixel values.
Fig. 4. (a) Schematic of the optical system used for the X-ray ptychographic and fluorescence measurements. (b) Probe intensity distributions and (c) phase and absorption images of ZnS particles reconstructed using the mixed-state ePIE and IG method. (d) SFXM image obtained by measuring the emission of S K$\alpha$ from the sample. (e) Deconvolution results of the SFXM image obtained using v-SPI (left: without position correction; right: with position correction based on the IG method).
Figures 4(b) and (c) show the probe intensity distributions (sum of all mixed-state modes) and sample images (left: phase image; right: absorption image), respectively, reconstructed using the mixed-state ePIE and IG method. The probe exhibited a stretched shape in the vertical direction owing to defocusing caused by optical alignment errors and partial coherence of the X-rays. In the phase and absorption images, a larger area was reconstructed than the field of view of the SFXM image, as indicated by the dashed lines in the figures, because of the contribution of the probe sidelobes to the reconstruction. Figures S2a–c show a STXM image of the ZnS particles prepared from the diffraction intensity pattern dataset, the deconvolution results of the STXM image via v-SPI, and a graph of the normalized residual errors plotted with respect to the LSQR iteration number, respectively. The results indicated that correction using the probe position reconstructed via the IG method made the particle contours clearer, and the normalized residuals were significantly reduced. These findings suggest that correction using the estimated probe position was effective for the measurement data. Figures 4(d) and (e) show the SFXM image and deconvolution results obtained using v-SPI (left: without probe position correction; right: with correction using the probe position reconstructed via the IG method), respectively. In addition to contrast enhancement, deconvolution allowed the observation of particle contours that were hardly observed in the original SFXM image, suggesting an increase in resolution. Although there was no apparent improvement in the image quality after the correction for the probe position, the areas indicated by arrows in the figures indicate different intensity distributions. The normalized residual errors without and with probe position correction were 5.53 $\times$ 10$^{-2}$ and 5.34 $\times$ 10$^{-2}$, respectively, indicating a reduction of approximately 3.4%. This suggests that the deconvolution result with probe position correction yielded a more accurate elemental distribution.
In this experiment, no clear improvement in image quality was observed in the deconvolution results of the SFXM images with probe position correction, which can be attributed to the lack of signal-to-noise ratio (SNR) in the X-ray ptychography and/or SFXM measurement. Insufficient SNR of X-ray ptychography measurements leads to inaccurate reconstruction of probe images and probe positions, resulting in inaccurate deconvolution results. However, the deconvolution results of a STXM image (Fig. S2) showed a clear improvement in image quality and a decrease in normalized residual error by performing probe position correction, even though the same probe image and probe positions were used as in the deconvolution of the SFXM image (Fig. 4(e)). This suggests that the SNR of the SFXM measurement, rather than the X-ray ptychography measurement, was insufficient. To increase the SNR of SFXM measurements, it is necessary to increase the incident X-ray intensity, solid angle of the detector, and exposure time per scanning point. Among these, increasing the exposure time is relatively easy because it does not require modification of optical systems. However, increasing the measurement time carries the risk of fluctuations in the probe and increases the amount of beam drift. The proposed method alleviates these problems and is considered adequate for observing trace elements with high resolution and accuracy.
5. Conclusion
We have proposed a deconvolution method to improve X-ray fluorescence microscopy images by utilizing different probe images for each scanning point obtained via X-ray ptychographic reconstruction. The proposed method is based on the v-SPI algorithm, which allows the use of a different blur kernel for deconvolution of each pixel of the image. By performing deconvolution with probe position correction on the SFXM image obtained via numerical simulation, the normalized residual error was reduced by >40%, and an improvement in the FRC in the high spatial frequency range was confirmed. Deconvolution of the experimentally obtained STXM image with correction for probe variation resulted in significant reductions of artifacts and normalized residual error (by approximately 89%), and 1-$\mu$m structures were observed using a 5-$\mu$m probe. When our method was applied to an SFXM image of ZnS particles obtained via X-ray ptychographic and fluorescence microscopy, the contours of the particles became more apparent, and the normalized residual error was reduced by 3.4%.
Our method is expected to contribute to the investigation of biological functions, as it effectively improves the resolution and quantification of trace element distributions that are difficult to observe via X-ray ptychography. Furthermore, the interactive use of data derived from multimodal measurements can provide information on samples that cannot be obtained from individual measurements [37,38]. We expect the proposed method to become increasingly useful, as improvements in the quality of individual measurement data are paramount for maximizing the efficiency of multimodal measurement analyses.
However, it is still difficult to achieve a resolution that exceeds the performance of focusing optical elements in scanning X-ray microscopy as the resolution of deconvolution results is limited by the maximum spatial frequency of incident X-rays. We plan to overcome this challenge by adding high spatial frequency information to the probe using an object with fine structures (e.g., a pinhole with sharp edges) placed just upstream of the sample. To achieve this scheme using conventional deconvolution methods, it was necessary to maintain a constant positional relationship between the incident X-rays and the object so that the probe does not change during the measurement. Our method alleviates this requirement and thus has potential to contribute to SFXM image acquisition at single-nanometer resolution.
Funding
Japan Society for the Promotion of Science (JP18H05253, JP22K05296, JP23H05403, JP23KJ0137).
Acknowledgments
We thank Dr. Jungmin Kang for his help in synchrotron experiments. The X-ray ptychographic and fluorescence microscopy measurements of ZnS particles were performed on BL27SU at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (Proposal Nos. 2022A1465, 2022A1377, and 2022B1269).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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