X-ray deconvolution microscopy

Recent advances in single-photon-counting detectors are enabling the development of novel approaches to reach micrometer-scale resolution in x-ray imaging. One example of such a technology are the MEDIPIX3RX-based detectors, such as the LAMBDA which can be operated with a small pixel size in combination with real-time on-chip charge-sharing correction. This characteristic results in a close to ideal, box-like point spread function which we made use of in this study. The proposed method is based on raster-scanning the sample with sub-pixel sized steps in front of the detector. Subsequently, a deconvolution algorithm is employed to compensate for blurring introduced by the overlap of pixels with a well defined point spread function during the raster-scanning. The presented approach utilizes standard laboratory x-ray equipment while we report resolutions close to 10 μm. The achieved resolution is shown to follow the relationship p n with the pixel-size p of the detector and the number of raster-scanning steps n. © 2016 Optical Society of America OCIS codes: (040.0040) Detectors; (100.0100) Image processing; (100.6640) Superresolution; (110.6960) Tomography; (170.7440) X-ray imaging; (180.7460) X-ray microscopy. References and links 1. P. Delpierre, “A history of hybrid pixel detectors, from high energy physics to medical imaging,” J. Instrum. 9(5), C05059 (2014). 2. T. Hatsui, and H. Graafsma, “X-ray imaging detectors for synchrotron and XFEL sources,” IUCrJ 2, 371–383 (2015). 3. P. Horowitz, and J. Howell, “A scanning x-ray microscope using synchrotron radiation,” Science 178(61), 608– 611 (1972). 4. G. Schmahl, D. Rudolph, B. Niemann, and O. Christ, “Zone-plate X-ray microscopy,” Q. Rev. Biophys. 13(3), 297–315 (1980). 5. A. Sakdinawat, and D. Attwood, “Nanoscale X-ray imaging,” Nature Photon. 4(12), 840–848 (2010). 6. H. Kang, H. Yan, R. 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Introduction
In recent years, advances in semiconductor detector technology have lead to the development of various photon-counting hybrid-pixel detectors [1,2].Common to these systems is the fact that each incoming photon is detected, processed and counted individually by the readout ASIC.Hybrid-pixel detectors typically use directly converting semiconductor radiation sensors which produce signals that are proportional to the energy of the incident photons.Therefore, photoncounting hybrid-pixel detectors are equipped with at least one energy threshold which allows to cut off electronic noise.This feature leads to an advantage of photon-counting systems, namely their signal-to-noise ratio (SNR) being only limited by the Poisson statistic of the radiation source.
X-ray microscopy methods are becoming increasingly important in various fields of applied research.Both materials science and life science benefit from the exceptionally high spatial resolution in the order of a few micrometers down to the nanometer range.Imaging at a nanometer scale typically utilizes monochromatic radiation and additional x-ray optical elements [3][4][5] to focus the x-ray beam in the nanometer range.Because of the required high beam intensity, such experiments are usually carried out at modern synchrotron x-ray sources.Traditionally, soft x-rays have been used for transmission x-ray microscopy (TXM) but ongoing research lead to the development of focussing devices that can produce spot sizes in the nanometer regime even for hard x-rays in the order of some 10 keV [6,7] making TXM principally feasible even at higher energies.One limiting factor to consider is that transmission contrast in TXM decreases substantially at energies higher than 15 -20 keV due to the very small sample size and the chemical composition, often consisting mostly of light elements.In addition to TXM, other methods have been developed that increase both contrast and resolution, the most prominent of which being coherent diffraction imaging (CDI) or ptychographic imaging.These methods put strong additional requirements on beam quality and coherence and involve challenging image reconstruction algorithms [8,9].
However, typical technical and biomedical applications require the use of hard x-rays in the range of 20 − 100 keV due to the size and chemical composition of the samples.Intermediate resolution in the range of a few micrometers is considered most important regarding the size of typical structures within such samples.This range of feature resolution is achieved either using an optical lens coupled to a CCD with very small pixel size to magnify the x-ray image projected onto a scintillator crystal [10] or directly using the magnification of a cone-beam geometry [11].Other investigators have illustrated this approach to reach resolution down to the sub-micrometer level [10].The above methods are regularly combined with computed tomography (CT) techniques in order to obtain 3-dimensional (3D) information.Nevertheless, this approach suffers from the limited photon flux of the employed microfocus sources.This limitation leads to rather long exposure times, especially in tomography applications which makes rapid inspection of large numbers of samples difficult to achieve.Additionally the geometric magnification limits the maximum sample size and the equipment for such microscopes tends to be rather cost intensive.
In clinical CT systems on the other side, an enhancement of spatial resolution is obtained using techniques such as the flying focal spot, where a periodic movement of the focal spot in the x-ray source leads to an increased resolution both along the scan axis and in-plane axis of a CT scan [12].The resolution within the image plane is typically increased using a so-called 'highresolution comb', a periodic grid that is placed in front of the detector reducing the geometric aperture of the detector pixels.Recent CT systems may offer a combination of both of these methods and also have 2D high-resolution comb extended in the image plane as well as in the z-plane [13] to yield an almost isotropic resolution of approximately 250 µm.However, using high-resolution combs strongly reduces the dose efficiency of these systems which goes along with an increased noise level in the reconstructed images.Hence, this technique is typically only used in cases where high contrast objects with small feature size are of interest, such as fine bone structures found in the ear [13].
In the presented work, we demonstrate a novel approach to obtaining resolution in the range of 10 µm.The presented technique does not require any optical elements, nor microfocussources to obtain the high-resolution images.Instead we make use of the box-like point spread function (PSF) that is achieved by the single-photon-counting systems based on the MEDIPIX3RX (Developed within the Medipix3 collaboration based at CERN, Switzerland) operated in charge summing mode (CSM) which is a real-time on-chip correction for chargesharing effects.This feature can be used to obtain images with a deep sub-pixel resolution as explained in the following section.

Materials and methods
The proposed new method is called X-ray Deconvolution Microscopy (XDM) in the following, where we refer to a similar approach in optical fluorescence microscopy [14].The fundamental principle is based on raster-scanning the sample in front of the detector with sub-pixel sized steps.A schematic overview of our approach can be seen in Fig. 1(a) -(d).The total scanning range equals exactly one pixel-pitch of the detector.Generally, translating the detector instead of the sample is equivalent.However, scanning the sample has the advantage that field inhomogeneities of the x-ray source are easily cancelled out after a flat-field correction of the detector.Thereby, after proper re-ordering of the rastered images, information about the sample structure on a sub-pixel scale can be obtained.Formally, this procedure can be described in the following way: Let I D x,y be the intensity recorded by a detector at its intrinsic pixel coordinate (x, y) given in units of the pixel size p.Suppose an object O(ξ , υ) is to be measured at super-resolution using M × N raster steps in the x and y -direction.Then the intensity I SR Mx+m,Ny+n in the rastered superresolution image is constructed via where T(...) denotes the translation of the object by a given distance in the x and y -direction.However, the raster-scanning introduces a strong correlation between neighbouring pixels depending both on the total number of steps and on the width of the initial detector PSF.If the PSF of an original pixel is known, one can estimate the PSF of the raster-scanned image and apply a deconvolution method, such as the Richardson -Lucy algorithm [15,16], to restore the latent image.
Considering the sampling and the following deconvolution steps in XDM one needs to exactly know the underlying PSF of the imaging system.Despite the small physical pixel size p = 55 µm the PSF of the used MEDIPIX3RX-based detector system can be shown to be sufficiently close to a box-function when operated in CSM.Since the pixel and sensor geometry are symmetric in the x-and y-direction, the measured PSF can be characterized by a single 1D box-shaped line-spread function (LSF), see Fig. 2, that extends the same way along the detector rows and columns: where x is the pixel coordinate on the detector.The resulting PSF in the detector plane is then described by a two-dimensional box function with symmetry in the x-and y-direction.Figure 2 shows the measured LSF along the detector columns.The LSF was determined following the slanted-edge method as explained in [17].After n-fold raster-scanning, the pixel size in the XDM image becomes p XDM = p/n and the LSF imposed on the latent XDM image is estimated to be again a box-function but with a width of n pixels in the XDM image space: The assotiated PSF is used to deconvolve the raster-scanned images in order to obtain the XDM images.
In order for these results to be valid the influence of the finite source size must be negligible.This influence can be described by the geometrical unsharpness ∆ given by where s is the source size, o the source-to-object distance, d the object-to-detector distance and M is the geometric magnification achieved by the set-up.For equations ( 2) and (3) to be valid, the blurring that is introduced by the geometrical unsharpness should vanish within the noise floor of the detector images.In practice, the condition of sufficiently small ∆ is realized by maximizing o and minimizing d.After sorting of the recorded raster-stepping images an image is obtained with pixel size of p/n that can be described by spatially oversampling the sample structures with respect to the detector pixels.A Richardson-Lucy deconvolution step is performed to restore the latent XDM image (e).
In this context it is important to ensure a sufficient positioning accuracy and repeatability of the linear stages used to perform the raster-scanning.The estimation of the PSF used to restore the XDM image given in equation ( 3) relies on a equidistant and reproducible rasterscanning of the sample.Any uncertainty in the positioning of the sample will result in decrease in achievable resolution.Such inaccuracies lead to a differently shaped PSF, which may cause deconvolution to fail.
Recent photon-counting detectors, namely the systems based on the MEDIPIX3RX [18] readout chips, offer additional features like a configurable pixel size, energy discrimination capability with up to 8 individual bins and the possibility to correct charge-sharing effects on-chip in real-time.Especially the latter is a unique feature of the MEDIPIX3RX, called charge-summing Fig. 2. Experimentally determined LSF along the pixel columns of the MEDIPIX3RXbased LAMBDA detector in charge summing mode, assembled with a 300 µm thick Si sensor at 100 V bias (solid line).The LSF was obtained using the slanted-edge method.As a reference, a theoretical box-like LSF with a width of one pixel is illustrated (dashed line).mode.In this mode, when the signal from a photon hit is shared between neighbouring pixels, the chip sums these signals together and assigns the hit to whichever pixel measured the largest signal.Generally, the pixel with the largest signal will be the pixel that was initially hit by the photon.After the correct hit allocation, CSM results in an improved spectral response of the detector and yields a box-shaped PSF matching the detector pixel size.
In particular, the LAMBDA detector system (Developed by DESY, Hamburg and Technische Universität München, Munich, Germany) [19], which is based on the MEDIPIX3RX, was used throughout this work.The particular assembly is equipped with a 300 µm thick Si sensor at 100 V bias.The LAMBDA was operated with a pixel size of 55 µm and an energy threshold of 8 keV was used to cut off the electronic noise in CSM.
An XRD source (Manufactured by Seifert, GE Inspection Technologies) was used with a geometrical source size of 0.4 × 0.4 mm 2 , operated constantly at 40 kVp/30 mA throughout the whole study.Exploiting the full length of the x-ray hutch, the detector was placed at a distance of 2.1 m from the source and the sample could be placed 15 mm in front of the detector.This set-up resulted in an unsharpness of ∆ = 2.9 µm, corresponding to 5.3% of the pixel size.
To summarize, the proposed method comprises the following steps: 1. Acquire n × n images with 2D sub-pixel rastering of the sample 2. Re-ordering of the data according to Eq. ( 1) to obtain the raster-scanned image 3. Richardson -Lucy deconvolution to restore the latent image

CT reconstruction
In our implementation, a standard FBP algorithm with a Ram-Lak filter was used to reconstruct the CT images.FBP was chosen over state-of-the-art iterative reconstruction methods in order to directly show the performance of the proposed XDM method regardless of the choice of regularization parameters that can additionally affect the obtained resolution.The Ram-Lak filter was selected because its wide-spread use and tendency to preserve high-frequency image content.

Experimental results
To demonstrate the potential benefit of the XDM method, results from the measurements of different objects are shown: First, a slanted edge was measured with increasing scanning parameter n to quantify the gain in resolution.
Second, a Siemens-star phantom with radially distributed spacial frequencies in the range between 0.5 lp/mm and 11.5 lp/mm was used to directly demonstrate the image quality improvement and to assess the influence of the deconvolution.The star was measured in projection using 2D raster-scanning.
Third was the joint head of a chicken tibial bone, measured in CT and 1D raster-scanning.

Slanted edge measurements
As edge device we used a 0.5 mm thick Gd foil placed directly in front of the detector at an angle of approx. 2 • with respect to the detector columns.The edge images were processed following the procedure explained in [17] to obtain the pre-sampling MTF.XDM images of the edge were acquired for n = 1, 2, ... , 5 with constant exposure time in each scanning step.To ensure that the edge covers enough positions across the detector pixels to accurately determine the presampling MTF, its physical length was 25 mm and a region of 256 × 256 pixels corresponding to one readout ASIC was selected for MTF determination.In Fig. 3(a), the measured MTF is shown for n = 1, 2, ... , 5.Only the MTF of the reference measurement n = 1 shows the typical behaviour of a MTF derived from a box-like PSF with several local maxima following a sinc(p f ) curve with the pixel size p and the spatial frequency f .The MTF from XDM images has a faster decay arising from the fact that the deconvolution step cannot perfectly restore the sub-pixel image due to the noise present in the recorded images.This is due to the well known noise amplification inherent to most maximum-likelihood algorithms which in the case of Richardson-Lucy leads to mid-frequency noise (speckles) that degrade the MTF in the deconvolved images.
To quantify the resolution in terms of the smallest visible structure size R lim in the XDM images, the 5% MTF criterion was applied to the measured curves since the MTF should be- come zero close to the frequency that corresponds to one pixel pitch.The resulting values are plotted in Fig. 3(b).Fitting the data with a hyperbolic function proves that in the given range, the parameter n clearly determines the achievable resolution R lim : where p is the original pixel size.We expect this result to hold true even for larger n.However, we could not measure n > 5 with the present set-up, since the used linear stages (LTM 120-400, OWIS GmbH) only guarantee a reproducible positioning in the rage of 12 µm.Note that in the case of the 5% MTF criterion, the determined resolution limit typically lies below the Nyquist frequency for the given pixel size.Hence, aliasing might already be seen in this frequency range, depending on the orientation and shape of the imaged pattern.
Additionally, the amount of noise present in the image was determined.For each n, an image region in the background excluding any objects was selected and the standard deviation of the intensities relative to the mean value in this region was measured, see dash-dotted line in Fig. 3(b).In our study noise did not depend on the XDM parameter but stayed constant at approx.0.71% of the background image intensity.Due to keeping the exposure time constant at each of the n × n scanning steps, the photon statistics in each 'pixel´of the XDM images did not change compared to the reference image.At the same time the total exposure time and x-ray dose to obtain a full image increases with the same factor n × n.The deconvolution step evidently does not introduce a bias to the noise in flat image regions.Together with the results from Fig. 3(a), these findings exactly reflect the behaviour of a detector with a physical pixel size of p/n, where also an increase in dose by a factor of n × n is required to keep the photon noise at a constant level.

Siemens-star
A photograph of the used Siemens star can be seen in Fig. 4(a).The structure consists of a cut Pb foil with a thickness of 50 µm, embedded between two 1 mm acrylic plates.To resolve the line pattern down to the lowest present spacial frequency, the star was imaged with scanning factor n = 3. Figure 4(b) -(d) provides a comparison between a standard radiography image (pixel size 55 µm, n = 1), a raster-scanned image without deconvolution and the XDM approach (pixel size 18.3 µm), respectively.Comparing the line patterns near the center of the star, these can easily be resolved in the XDM image while aliasing of the structure is clearly visible in the standard image.
To illustrate the effect of the deconvolution step, Fig. 4(e) shows a comparison of the obtained patterns along the marked line segments in the three images with increasing spacial frequency towards the center of the line.The XDM image shows significantly improved contrast at lower frequencies and therefore has better resolution limit than the corresponding image without deconvolution.

Chicken bone
The chicken bone sample was measured in tomographic mode with a 1D scanning factor of n = 5 perpendicular to the rotation axis.Hereby, the resolution within a given CT slice is increased while the slice thickness is kept at the original pixel size.The total measurement time was approximately 12 hours and 1400 projection images were recorded.The measurement time was fractioned between the raster-scanning steps and a reference image was reconstructed with the original pixel size of p = 55 µm, but 5 times better counting statistics by re-binning the raster-scanned images.As Fig. 5 shows, the XDM image is again superior in terms of contrastto-noise ratio and resolution regardless of the lower counting statistics in each scan step that contributed to the image.

Discussion and conclusion
In this work we have successfully demonstrated the high-resolution potential of the XDM method without making use of highly expensive micro-or nano-focus x-ray sources.The proposed method is straight forward to implement in every laboratory set-up that utilizes detectors with a nearly box-shaped PSF.The smallest feature size that can be resolved in the XDM images was shown to decrease hyperbolically with the number of sub-pixel rastering steps.Therefore, the deconvolution of the raster-scanned images with a previously known PSF is crucial for the quality of the obtained images.
Previous attempts to obtain sub-pixel resolution in x-ray imaging utilizing photon-counting or integrating detectors yielded a much smaller increase in resolution with saturation at higher raster-scanning parameters.This stems mostly from the fact that previous investigations on this topic did not model the spatial detection system response and incorporate an appropriate deconvolution step [20,21].Most detectors available so far have a much wider gaussian-shaped PSF arising from charge sharing in directly converting sensors or lateral propagation of light in scintillators.Although the resulting PSF after the raster-scanning can be deduced for such systems, the deconvolution becomes more prone to noise amplification due to the higher number of iterations necessary for the algorithm to converge.Furthermore, the MTF associated with a box-like PSF drops off much slower compared to gaussian-PSF systems.Therefore, information about high-frequency features in the object is preserved much better by detectors with a box-like PSF.Although the high-frequency structures can also be enhanced by deconvolution for gaussian-PSF detectors, the x-ray dose required to resolve these features in terms of image CNR will increase strongly in such applications.This results in practical limitations when applying these detectors in biomedical radiography.Hence, the nearly box-like PSF of the MEDIPIX3RX operated in CSM is one key feature for the XDM method since access to higher spatial frequencies becomes much more efficient.Other studies investigating the imaging characteristics of the MEDIPIX3RX with a GaAs x-ray sensor have reported a MTF that appears to be related to a less box-like PSF [22].This result however can be explained partly by the fact that GaAs sensors produce x-ray fluorescence which also degrades the spatial resolution and cannot be fully compensated by the CSM.Another issue affecting the obtained PSF is the fact that the authors of the aforementioned work used a Gaussian model to fit the measured PSF data.This imposes a specific behaviour on the resulting MTF curve, particularly a faster decay at higher spatial frequencies.For the work done here, no a-priori model was assumed to fit the PSF and calculate the MTF.
Comparing our method to simple pixel interpolation and de-blurring, it needs to be noted that superresolution based on simple interpolation is always inferior since the interpolation tries to estimate information about image content that is based on neighbouring pixels.In contrast, methods based on sub-pixel sample or detector motion can use additional information about the sample by actual measurements to recover the superresolved image.Hence, these methods will in general always be superior to simple interpolation.
Other advanced superresolution approaches for x-ray imaging have been proposed in the recent past.For example, [23] proposes a joint restoration and de-noising algorithm based on Fourier-space wavelet regularized deconvolution and shows first applications to the detection of micro-calcifications in clinical mammography.By incorporation of a motion estimation step prior to deconvolution, this framework is able to handle arbitrary object shifts without a-priori knowledge of the sample movement which can occur in clinical settings due to patient movement or vibrations and instabilities of the imaging system.Albeit the authors demonstrate a promising improvement of CNR in the restored superresolution images, the question of the quantitative improvement of resolution was not addressed in this work.In our case we have found that using well-defined raster steps leads to a predictable resolution increase.However, #251696 since positioning inaccuracies might occur in any experimental set-up, it is to be expected that also the presented XDM method can benefit from a refinement of the raster stepping positions by motion estimation prior to deconvolution.
For CT applications it was demonstrated that an adapted pixel model in iterative CT reconstruction (detector supersampling) can produce superresolved images at constant radiation dose [24].However, the algorithmic formulation of the superresolution reconstruction in [24] requires prior knowledge or assumptions about the sample composition or density and does not allow for great variations of this value across the sample.Therefore, this method can only be applied to the imaging of binary objects or at least objects consisting of homogeneous materials such as micro-CT of bone.
In the study presented here the achieved resolution was limited particularly due to the positioning accuracy of the used linear stages.However, state-of-the-art positioning equipment can easily provide accuracy in the range of 1 µm.Therefore resolution of only a few micrometers is feasible using the XDM method.
As a generalization of the method, it is possible to combine XDM with geometric magnification of a cone-beam geometry.This could have two application scenarios: First, to reach a given resolution in micro-CT a smaller geometric magnification factor is required in combination with XDM.The scanning FOV would be strongly increased especially in applications where the sample size limits the source-to-sample distance.As an example, a combination of a geometric magnification of M = 3 and an XDM parameter of n = 4 used at a set-up with the MEDIPIX3RX would already be sufficient to obtain a resolution better than 5 µm.In a conventional approach, at least a value of M = 12 is required, which results in an unsharpness in the order of one or more detector pixels due to the source blurring and lowers the overall image contrast.Using XDM instead will result in much sharper images, since the inherent unsharpness is limited to a small fraction of the actual pixel size.Smaller geometric magnification would also relax the requirements on the x-ray source.This allows for the use of more robust and cheaper source technology in micro-CT equipment.
As a second scenario, a combination of a high geometric magnification and subsequent XDM with highly accurate steps using e.g.piezoelectric sample positioning equipment might provide resolution in the nanometer regime.Therefore however, the PSF no longer has a box shape, but the blurring effect from the x-ray source also needs to be considered.However, if the exact form and size of the focal spot is known and the source is sufficiently stable, such effects might be incorporated into a more sophisticated super-resolution reconstruction algorithm.
In principle this method can be extended to medical x-ray tomography systems.Especially for interventional imaging at high resolution, the region-of-interest CBCT concept [25] would open up new clinical opportunities when combined with the XDM method.Despite yielding a much smaller resolution increase using a fixed number of raster images when compared to our method, previous work [20,23] also showed a potential radiation dose reduction by clinical super-resolution x-ray imaging.With respect to full-body CT scanners, the PSF of the detector pixels is box-like as in our experiments.This is due to the large pixel size and additional radiation shielding between individual elements.However, for a clinical translation major mechanical and algorithmic challenges are remaining to be solved.For the near future we see a successful use of the XDM method for lab based x-ray set-ups and micro-CT systems.

Fig. 1 .
Fig. 1.Principle of the XDM method.(a) The sample is raster-scanned in front of the detector with n sub-pixel sized steps.(b) -(d) During the stepping process, the object intensity profile (solid black curve) is sampled at various positions and integrated over one pixel size p.The vertical bars represent the intensity that is recorded in each detector pixel.After sorting of the recorded raster-stepping images an image is obtained with pixel size of p/n that can be described by spatially oversampling the sample structures with respect to the detector pixels.A Richardson-Lucy deconvolution step is performed to restore the latent XDM image (e).

Fig. 3 .
Fig. 3. Resolution and noise obtained with XDM at different scanning parameters n.(a) Shows the measured MTF for different values of n.For larger n an increase in contrast is observed indicating higher resolution.(b) Shows the resolution limit obtained at 5% MTF (left axis) and the amount of noise relative to the background intensity (right axis), depending on n.The minimum feature size decreases hyperbolically with n, whereas the noise does not depend on the XDM parameter.

Fig. 4 .
Fig. 4. Results of the super-resolution imaging of the Siemens-star phantom.Besides a photograph of the used phantom (a), the region around the center of the star is shown in a standard radiography image (n = 1) as reference (b), a raster-scanned image (n = 3) without deconvolution (c) and with the XDM method (d).In (e), a comparison between the obtained amplitudes of the marked line patterns from (b) -(d) is shown.One feature of the XDM images is the superior contrast.In the standard image, structure aliasing is clearly visible, as the amplitude drops to zero.

Fig. 5 .
Fig. 5. CT images of a chicken bone sample taken with 40kVp.(a) Standard image (n = 1), (b) raster-scanned image (n = 5), (c) XDM image.All three images are shown in a window from -400 to 6000 HU.(d) displays the pixel values along the marked lines in (a)-(c).Despite a much lower photon statistics per pixel, the XDM image shows superior feature resolution and contrast.The pixel coordinates of the lines are given in the XDM image units.