Robust Pixel Gain Calibration with Limited Statistics

TL;DR

This paper introduces a stable pixel gain calibration method for detectors that requires significantly less statistical data than traditional approaches, improving spectroscopic accuracy.

Contribution

A novel calibration technique using histogram cross-correlation that achieves accurate pixel gain maps with minimal statistical data.

Findings

Achieves stable gain calibration with an order of magnitude less data

Provides accurate gain maps comparable to high-statistics methods

Validated with synchrotron monochromatic radiation measurements

Abstract

Pixel detectors typically display pixel-to-pixel gain variation of a few percent which result in reduced spectroscopic performance. We have developed a calibration method which relies on cross-correlating histograms of many pixel pairs and obtaining large sets of relative shifts. These were subsequently used to calculate absolute pixel shifts and corresponding pixel gains. We demonstrate that this method yields stable gain calibration maps with an order of magnitude less statistics than required by typical approaches. Finally, we demonstrate the accuracy of the method by comparing with gain maps obtained with good statistics and monochromatic radiation at a synchrotron beamline.

Full text

Robust Pixel Gain Calibration with Limited Statistics

G. Blaj*, , G. Haller, , C. Kenney G. Blaj, G. Haller, and C. Kenney are with SLAC National Accelerator Laboratory, Menlo Park, CA 94025.* Corresponding author: blaj@slac.stanford.edu

I Introduction

Pixel detectors typically display pixel-to-pixel gain variation of a few percent which result in reduced spectroscopic performance [1]. Integrating pixel detectors provide spectroscopic information directly, while photon counting detectors through threshold scans. Gain maps are used to rectify pixel-to-pixel gain variation.

Ideally, the pixel gain is calibrated with monochromatic radiation (e.g., from a synchrotron or free electron laser beamline, or laboratory x-ray sources with monochromators). However, beam time is usually valuable, and monochromatic sources are not always available [2], while x-ray tubes and radioactive sources produce complex spectra.

For calibration of ePix100a cameras, we used a Mo x-ray tube with Zr and Al filters (to optimize the Mo K\textalpha line) and acquired limited statistics ($50250$ photons), sampled from a complex spectrum (where accurate fitting of a peak with charge sharing would require \approx$$1000 photons per pixel [3]).

We have developed a calibration method which relies on cross-correlating histograms of many pixel pairs and obtaining large sets of relative shifts. These were subsequently used to calculate absolute pixel shifts and corresponding pixel gains.

We demonstrate that this method yields stable gain calibration maps with an order of magnitude less statistics than required by typical approaches. Finally, we demonstrate the accuracy of the method by comparing with gain maps obtained with good statistics and monochromatic radiation at a synchrotron beamline.

The robust gain calibration method presented here can be used for any pixel detector with minimal effort and assumptions on calibration spectrum. It is particularly useful when the quantity and/or quality of calibration data is limited and repeated measurements could be difficult (e.g., previous experiments at synchrotron or free electron laser sources, FEL).

II Methods

We used an x-ray tube with Mo anode and Zr and Al filters to emphasize the Mo K\textalpha peak [3]. We acquired two data sets of \approx$$33\,000 frames each, with the x-ray tube operated at $45\text{\,}\mathrm{kV}$ and $50\text{\,}\mathrm{kV}$, respectively. We collected histograms of “single pixel events” for each pixel (i.e., all 8 neighboring pixels within noise, $\leq 4\sigma$), yielding an average of $240$ photons per pixel. Fig. 1(a) shows an example of pixel spectrum with limited statistics. The peak position obtained through fitting or centroiding is relatively noisy [3].

A change in pixel gain results in an apparent shift of (calibration) peak position. While determining the peak shift is nontrivial with limited statistics, we developed a robust approach for spectra with a sharp transition (i.e., from a quasi-monochromatic source and/or an absorption edge filter). For each pair of pixel spectra we estimate the relative peak shift by calculating the cross correlation, finding its maximum value and corresponding delay as in Fig. 1(b). This delay is the relative shift of the two pixel spectra.

The relative shift pairs are saved in a “shift matrix” of size $N\times N$ (with $N$ the number of pixels), see Fig. 2. With limited statistics, relative shifts of individual pairs could be imprecise, however, the column average of the shift matrix yields a robust and accurate estimate of the absolute pixel shift111Demonstration omitted for brevity..

The algorithm complexity is $O(N^{2})$; for large numbers of pixels we can optimize the run time by dividing the problem and calculating (1) shifts for pixels within each column, and (2) shifts of individual columns; their sum yields the absolute pixel $i$ shift $\delta_{i}$ and the pixel gain as $g_{i}=1+\delta_{i}/E_{0}$ where $E_{0}$ is the peak position.

III Results

Fig. 3 shows the robust gain map obtained for an ePix100a camera, showing good uniformity ($99.2\text{\,}\mathrm{\char 37\relax}$ of the pixel gains in the $0.951.05$ range). Fig. 4 shows the aggregated spectrum of all pixels, either raw (thin red line) or after gain calibration (thick black line). The spectroscopic performance is significantly improved by using a gain map [4, 5].

To test the robustness of this method, we applied it to two independent data sets obtained in different conditions. The results are shown in Fig. 5, with each black point indicating two independent gain estimations of one pixel. A linear regression yields a slope $a=$1.000$~{}\pm~{}$3\text{\times}{10}^{-6}, with $R^{2}=$0.987 and residual gain noise $0.16\text{\,}\mathrm{\char 37\relax}$. This corresponds to an error due to gain calibration of $0.51$ adu at the Mo K\textalpha peak, demonstrating good repeatability of the robust gain calibration with different statistics limited data sets. Finally, reducing the calibration data from $240$ to $60$ photons per pixel results in a relatively small increase of gain noise to $0.34\text{\,}\mathrm{\char 37\relax}$.

IV Conclusion

We presented a robust pixel gain calibration, demonstrating good performance and reproducibility with limited spectrum quality and statistics (an order of magnitude less than required by other methods). This method is relatively fast, with a Python program calibrating $0.5$ megapixel ePix100a cameras in about $16\text{\,}\mathrm{min}$ on a laptop with $2.9\text{\,}\mathrm{GHz}$ CPU. It is also simple to implement, requiring minimal assumptions on spectrum shape.

The robust gain calibration method could be particularly useful when the quantity and/or quality of calibration data is limited and repeated measurements could be difficult (e.g., previous experimental data at synchrotron or FEL sources).

Acknowledgement

Use of the Linac Coherent Light Source (LCLS) and Stanford Synchrotron Radiation Lightsource (SSRL), SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. Publication number SLAC-PUB-17459.