8 Beyond spectral imaging: X-ray deconvolution microscopy using
8.2 Determination of the PSF in oversampled images
8.2 Determination of the PSF in oversampled images
Considering the oversampling and the following deconvolution steps in XDM one needs to exactly know the underlying PSF of the imaging system. Despite the small physical pixel sizep = 55 µm the PSF of the usedMEDIPIX3RX-based detector system can be shown to be sufficiently close to a box-function when operated in CSM. Due to the spatial symmetry of the detector pixels, the required PSF can be characterized by a single one-dimensional (1D) line-spread function (LSF), extending the along the detector rows and columns:
LSF(x)∼=
wherexis the pixel coordinate on the detector. The resulting PSF is then described by a two-dimensional box function with symmetry in the x- and y-direction.
To verify the assumption of a box-like PSF obtained after the PCD-based charge-sharing correction, we have measured the LSF according to the widely used slanted-edge method [Samei1998]. We used a 0.5mm thick Gd foil as edge device, 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 outlined in [Samei1998] to obtain the pre-sampling modulation-transfer function (MTF)3. The left plot in figure 8.2 shows the spatial extent of the observed line spread function along a detector row in units of the pixel sizep. Comparing the experimental data to a fit of a box function, the assumption of a box-like PSF seems justified.
The shape and distribution of image noise in the reciprocal space given by the noise-power spectrum (NPS) can also be used to characterize the sharpness of the detector PSF. Therefore, the NPS was mea-sured in a flat region of the images with no object present according to the method in [Garcia-Molla2011].
Any extent of the PSF across pixel borders would result in an effective blurring of the intensity detected in neighboring pixels. Therefore, short-range correlations in the vicinity of the Nyquist frequency fNy = 1p with pixel sizepwould be introduced that lead to a reduced noise density at the associated spatial frequencies. Accordingly, detectors with a broad PSF exhibit a drop-off of the NPS towardsfNy.
3Pre-sampling refers to the theoretical MTF of the image prior to the sampling of the object with the physical detector pixels with finite pixel size. Thus, the obtained MTF is treated as continuous function and extends beyond the spatial frequencies associated with the pixel size.
8 X-ray deconvolution microscopy using photon-counting detectors 1.00
0.10 0.25 0.50 0.75
Figure 8.2:Experimentally measured LSF and NPS of aMEDIPIX3RX-based PCD.The left plot shows the determined LSF along the pixel columns of theMEDIPIX3RX-basedLAMBDA detector in charge summing mode, assembled with a300µm thick Si sensor biased with 100V (solid line). The LSF was obtained using the slanted-edge method. As a reference, the fit of the data to a box-like LSF with a width of one pixel is also shown (dashed line).
Furthermore, the flat NPS shown in the right panel also indicates a box-like PSF since detector-caused blurring leads to an effective averaging between neighboring pixels. This would result in a drop-off of the NPS at higher spatial frequencies close to the Nyquist frequencyfNy.
In the case of the used PCD with charge-sharing correction however, the obtained NPS is flat over the complete range of spatial frequencies. This behavior underlines again the box-like characteristics of the PSF.
Other studies have also been presented on the resolution characteristics of aMEDIPIX3RX-based PCD equipped with a GaAs x-ray sensor. There the authors have reported a MTF that appears not to be related to a box-like PSF [Hamann2015]. However, this result can be explained in part 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 in the aforementioned publications is the fact that a Gaussian model was employed to fit the measured PSF data to mitigate noise effects.
This imposes a specific behavior on the resulting MTF curve, particularly a faster decay at higher spatial frequencies. For the investigations presented here, no a-priori assumptions were made and the PSF together with the resulting MTF have been directly obtained from the measured data.
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8.2 Determination of the PSF in oversampled images To guarantee validity of these results the influence of the finite x-ray source size must be negligible.
This influence can be described by the geometrical unsharpness∆given by
∆ = s·d
o = (M−1)·s , (8.4)
wheresis the source size,othe source-to-object distance,dthe object-to-detector distance and M is the geometric magnification achieved by the set-up.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 maximizingoand minimizingd. In the studies presented here, an x-ray tube with a geometrical source size of0.4×0.4mm2was used, operated constantly at40kV/30mA.
The full length of the x-ray hutch was exploited placing the detector at a distance of2.1m from the source and the object15mm in front of the PCD. This geometry resulted in an unsharpness of∆ = 2.9µm or 5.3% of the pixel size and could therefore safely be neglected.
Now, the full PSF imposed on the raster-scanned super-resolution images can be determined. After n-fold raster-scanning, the pixel size in the XDM image becomespXDM=p/nand the LSF imposed on the latent XDM image is estimated to be again a box-function but with a width ofnpixels in the XDM image space:
LSFXDM(x)∼=
1 if|x| ≤npXDM/2 0 otherwise.
(8.5)
To obtain a PSF in the raster-scanned images with the spatial extent of exactly npXDM in both the x- and y-direction, it is crucial to ensure a sufficient positioning accuracy of the linear stages used to perform the raster-scanning. Equation 8.5 relies on a equidistant and reproducible raster-scanning of the sample. Any uncertainty in the positioning of the sample will result in a un-isotropic PSF and will therefore decrease the achievable resolution. If not properly accounted for, such inaccuracies may cause the deconvolution process to fail.
8 X-ray deconvolution microscopy using photon-counting detectors
B A
5 % MTF
Figure 8.3:MTF and NPS obtained with XDM at different scanning parametersn.(A) Shows the measured MTF for different values ofn. For largernan increased contrast is observed for larger spatial frequencies indicating higher resolution. The dashed black line marks the limit of5% of the MTF used to quantify the resolution. The NPS associated with the images is shown in plot (B). Forn ≥2the curves show oscillations that are imprinted to the noise spectrum during deconvolution by the box-like PSF.