research papers
The LaueUtil toolkit for Laue photocrystallography. II. Spot finding and integration
^{a}Chemistry Department, University at Buffalo, State University of New York, Buffalo, NY 142603000, USA, and ^{b}Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
^{*}Correspondence email: jak@kalinowscy.eu, annamaka@buffalo.edu
A spotintegration method is described which does not require prior indexing of the reflections. It is based on statistical analysis of the values from each of the pixels on successive frames, followed for each frame by morphological analysis to identify clusters of high value pixels which form an appropriate mask corresponding to a reflection peak. The method does not require prior assumptions such as fitting of a profile or definition of an integration box. The results are compared with those of the seedskewness method which is based on minimizing the skewness of the intensity distribution within a peak's integration box. Applications in Laue photocrystallography are presented.
Keywords: photocrystallography; Xray diffraction; Xray detection; CCD; Laue method; spot integration; background estimate; statistical analysis; image filtering.
1. Introduction
Treatment of Laue diffraction data has frequently attracted attention, in particular since the development of timeresolved pump–probe diffraction techniques, in which the Laue method has specific advantages (Ren et al., 1999; Anderson et al., 2004; Kamiński et al., 2010). The renewed interest in the Laue technique has resulted in the development of several computer programs. Notable are the Daresbury Laue Software Suite (Helliwell et al., 1989), LaueView and Precognition (Ren, 2010; Šrajer et al., 2000), and LaueGui (Messerschmidt & Tschentscher, 2008).
In the commonly used Laue data processing procedures the reflection's position on the detector is predicted and a box is established that encloses the area where the diffraction signal is expected. Subsequent integration of the identified spots is performed either by twodimensional profile fitting (Helliwell et al., 1989; Šrajer et al., 2000; Moffat, 2001) or by selecting a mask on the detector surface using statistical criteria as in the seedskewness method (Bolotovsky et al., 1995; Bolotovsky & Coppens, 1997). Although the latter does not in principle require information on the predicted spot position, the practical implementation of the method relies heavily on this information (Messerschmidt & Tschentscher, 2008). The above processing sequence cannot be accomplished without a priori knowledge of crystal orientation with respect to a diffractometerbased coordinate system; in other words, without successful indexing of the Laue pattern, which may be timeconsuming.
In Laue crystallography the et al., 2000; Ren & Moffat, 1995).
is replaced by a shell of varying thickness, often referred to as the Ewald region. As a result the diffraction spot profile is affected not only by the sample's mosaic spread and incident beam divergence but also by the Xray bandwidth. Therefore reflection 3D profile reconstruction is not possible with existing software and integration is typically accomplished on a framebyframe basis, followed by appropriate corrections, scaling and wavelength deconvolution procedures (ŠrajerThe RATIO method (Coppens et al., 2009), as used in pump–probe photocrystallography, avoids the spectral deconvolution step. As the interest is in the ratios of the lightON and lightOFF intensities, collected sequentially in consecutive frames, wavelengthdependent effects such as the of the beam and the absorption and detector response effects are essentially eliminated. Small shifts in spot positions may occur if the cell dimensions are affected by the excitation, an effect that may be pronounced when conversion percentages are appreciable. However, for the low conversion percentages of ∼6% or less achieved in many studies including our own (Benedict et al., 2011; Makal et al., 2011; Collet et al., 2012), cell dimension changes are not significant.
In the previous paper on the LaueUtil toolkit we proposed a method for a rapid orientation matrix determination in Laue crystallography (Kalinowski et al., 2011). Here we focus on the task of efficient Laue data integration. The presented method combines identification of the diffraction spots and their integration into a single algorithm. The complete set of images is treated as a single dataset with `frame by frame' or `pixel by pixel' views used in tandem. The method explicitly uses a statistical approach and requires temporal stability of the source and some simple assumptions on the noise distribution in the background. No prior information about the crystal orientation, sample cell parameters or their stability in the course of experiments is required.
2. The method
2.1. Assumptions and outline
Our integration method does not imply uniformity of the background across the surface of a diffraction spot. It requires a series of diffraction images at subsequent values of the φ scan angle. For application to data collected at Xray freeelectron laser sources it would require at least modifications to allow for scaling of subsequent frames.
The method consists of three steps. First, for each of the pixels, intensity values are collected from all frames, leading to onedimensional arrays which are statistically analyzed to estimate the background contributions for each of the individual pixels. The idea is illustrated in Fig. 1. In the second step the mask is defined for each of the frames from the pixel background values previously estimated, and optimized. Finally, the processed masks are analyzed to determine the footprints of each of the reflections which are subsequently integrated to obtain the corresponding intensities.
2.2. Pixel statistics
The measured values of a pixel on all frames can be regarded as a sample of the pixel background intensity which can include an unknown number of outliers corresponding to the presence of spots. The estimate of the background on the frames is then achieved by exclusion of the outliers in perpixel samples of values. The simplest approach is to assume that a certain percentage (usually about 20–30%) of the highest values along a pixel line (i.e. all values of a specified pixel on the successive frames) represent spot contributions. Fig. 2 shows an example of such a pixelsample analysis. The remaining values are used to estimate, for each pixel, the parameters of its background distribution such as the mean, the variance, the median or the interquartile distance, the latter being defined as the difference between the 25thpercentile and the 75thpercentile statistics of a pixel. Then, from the knowledge of the pixelbypixel background characteristics, we use a simple condition to identify which pixels contribute to spots and thereby define a raw mask on each frame. For the jth image, the condition applied to the ith pixel is
where I_{(i,j)} is the ith pixel value in the jth image, 〈I_{i}〉_{background} and σ(I_{i})_{background} are, respectively, the mean value and the standard deviation of the ith pixel background, and c is a positive adjustable parameter with a default value of 3.0.
This simple approach, henceforth referred to as the constantfraction approach, though reasonably successful, suffers from two drawbacks. The first is that the background tends to be underestimated for pixels which contribute to reflections only in a few frames. The second is that some intense reflections located close to the origin of φ scan angle and contributing to the values of some pixels on a significant portion of the frames which shows up as spotlike features in the reconstruction of the background as discussed in §3.1. As all the are available for immediate inspection in the output file, a user can decide on the tradeoff between a global underestimate of the background due to too large a removed fraction versus a local background overestimate due to an incomplete filtering of the signal.
can be present on many adjacent frames, in some cases spanning a range of over 25° in the2.3. Advanced pixel statistics for redundant measurements
We consider a pump–probe experiment in which, for each goniometer setting, a series of repeated measurements is made, both with or without the laser pump pulses (referred to below as the ON and OFF frames). In our experiments, measurements are made ten times in both situations, giving for each pixel a block of 20 frames for a single goniometer setting. This allows an improved statistical analysis. We consider the 20 values in each block as independent statistical samples. We then test whether all block samples are from the same background distribution, or whether some of them, socalled `block outliers', are from distributions with higher median value as a result of the presence of diffraction spots.
We use the nonparametric Kruskal–Wallis (K–W) test (Kruskal & Wallis, 1952; Corder & Foreman, 2009) which checks whether samples originate from the same distribution (see Appendix A). For each pixel, all blocks of 20 values are sorted by their median. Then the one with the highest median value is recursively eliminated until the remaining samples pass the K–W test using the approximation for K–W test critical values, where n is the number of remaining blocks and α is a userselected significance, with a default value of 5%. Fig. 3 illustrates the K–W analysis performed on a given pixel's series of blocks. Like in the constantfraction method, for each pixel the remaining samples are combined and used to estimate the distribution parameters listed earlier. This method allows efficient detection of block outliers. However, it is not designed to identify erratic singular outliers which are related to noise. To avoid any bias in the subsequent mask definition caused by these outliers, we apply a similar criterion as in the constantfraction method but using the estimated median and interquartile distance, which are more robust to erratic outliers.
For the ith pixel the criterion for acceptance as part of a mask is
where I_{(i,j)} is the ith pixel value in the jth image, median(I_{i})_{background} and iq(I_{i})_{background} are, respectively, the estimated median value and interquartile distance of the ith pixel background, and d is a positive adjustable parameter with a default value of 3.0.
2.4. Filtering masks
The procedure described above is carried out on a pixelbypixel basis and is completely insensitive to the spatial relationships of the pixels. We expect, however, that spots have a nonnegligible size and some definite shape though not necessarily as simple as circular or elliptical. The next step is therefore based on a framebyframe analysis and mask filtering applying binary morphological operations in two steps (Pierre, 2003). First, erosion operations are applied to remove isolated pixels or lines from a mask. Then, dilation operations are used to add back some relevant pixels lost during erosion operations and also to add a margin to spot footprints. Assuming no bias in the background estimate, we expect that the only adverse effect of the latter operation on the resulting intensities may be an increase in their variances. The benefit of applying dilations is a possible correction of pixel omissions from the previous phase, which may occur at the edge of the spots where the increase of intensity is below the threshold for the pixelbypixel outlier detection (Fig. 4). The numbers of erosion and dilation operations are adjustable parameters with default values of 1 and 2, respectively.
The erosiondilation method produces an improved mask for each frame. These masks can be used directly in the integration step, or can be merged per ON/OFF pair or per block by performing a logical OR operation to obtain a mask shared by the corresponding pair or block of frames. This prevents possible bias due to differences in spot footprints in calculating the ON/OFF intensity ratios. Masks resulting from both types of merging are compared in Fig. 5.
The number of spot candidates produced by the method is generally larger than the number of Bragg reflections that could occur on a given frame. This results from highintensity noise that may be indistinguishable from spots or from possible splitting of very elongated spots by the erosion operation. We do not consider an excessive number of spots a significant issue as they are effectively filtered by subsequent application of the LaueUtil indexing routine. It should also be noted that in some cases too many dilations can cause merging of particularly extended spots; therefore it is advisable to visually inspect the resulting masks to verify the choice of morphological operations.
2.5. Final integration and implementation details
The resulting frame masks are scanned to identify reflection footprints which are assigned a unique numerical label l. Their integrated intensities are calculated as
where M_{l} is a list of pixels belonging to spot l, I(i, j) is the intensity of pixel (i, j) and B(i, j) is the estimated mean, or median in the K–W method, of the background intensity distribution.
The presented method does not directly provide an estimate of intensity errors (`σI') for each of the reflection intensities. Instead, sample statistics based on the redundant measurements are used to estimate the errors.
Following the general design of the LaueUtil suite, integration results are stored in HDF5 files together with masks and collected statistics. This choice allows for easy inspection and analysis of the data using general purpose HDF5 data visualization programs like HDFView (The HDF Group, 2010) or advanced statistical toolkits like R (R Development Core Team, 2010).
The automatic data integration procedure, including retrieval of the experimental data from frames and storing them in HDF5 format (with compression), takes from 5 min for 90 frames, 0.7 GB standard data collection, 20 min for a shortdiagnostic 20° scan with ten ON/OFF frame pairs per angle (420 frames, 3.3 GB), up to about 90 min for a full 90° photocrystallographic dataset with ten ON/OFF frame pairs per angle (1800 frames, 14.5 GB). Tests were performed on a standard desktop computer, i.e. a Linux machine using a single core of an AMD Phenom II X6 1090T processor. The running times should be compared with the time required to perform sole data compression with standard system tools (tar z) which are, respectively, 1 min, 4 min, 21 min and network transfer times: 19.3 min for 14.5 GB at a theoretical maximum speed of 100 Mbps Ethernet connection.
The algorithm is suitable for reimplementation in a highthroughput processing system, with several optimizations envisageable, including parallelization, direct connection with data collection software, inmemory processing of whole datasets, limited usage of compression and storage of intermediate data. However, such developments would exceed our current needs and would require sufficient support from the computing infrastructure used for data collection.
3. Results and discussion
Application of the method is illustrated with two datasets, representative of the data collected in our photocrystallographic experiments.
Dataset 1 is a set of five short 21° scans (subsets 1–5) consisting of 21 blocks of ten ON/OFF frame pairs collected at 1° φ spacing. All five scans have the exact same φ range. Such scans are typically acquired in order to test crystal response to various laser powers or pump–probe delay times. The former was tested in the case presented here.
Dataset 2 consists of 90 OFF frames, routinely collected at 1° φ spacing in order to assess general crystal quality. It is also a typical example of a standard crystallographic dataset.
The datasets were collected for crystals of two solvates of Cu(I) organometallic complexes listed in Table 1. All data were collected at the 14ID beamline of the BioCARS station at the Advanced Photon Source with an undulator setting of 15 keV and a MARCCD165 CCD detector at fixed position, used in several of our photocrystallographic Laue experiments.

Datasets were processed in parallel with the LaueUtil and LaueGui integration software, the latter utilizing the seedskewness method (Bolotovsky et al., 1995; Bolotovsky & Coppens, 1997; Coppens et al., 2010).
3.1. Background reconstruction for a complete dataset
Background mean values obtained for dataset 2 with the constantfraction method are presented in Fig. 6. These values correctly reconstruct the beamstop shadow as well as the increased background at low scattering angles and a slight conical shadow of the copper mount, which partly obstructs the Xray beam. The background decreases at higher scattering angles. Also visible are differences between the background on the four quadrants of the detector. The background estimate even in the constantfraction approach will correct for variations in noise intensity of the different detector regions and can serve as a diagnostic tool for the characteristics of the detector.
The background variance for each pixel is similarly presented in Fig. 6. In addition to the radial dependence observed for the mean or median and the difference between separate CCD quadrants, the statistic reproduces the Moiré pattern predicted for the variance of the background pixel values for CCD detectors with optical taper, as described by Waterman & Evans (2010). As noted in their work, this effect can significantly bias integration routines utilizing profile fitting.
Comparison of the background estimates with the constantfraction and K–W approaches can be best illustrated for any subset of dataset 1. These subsets have a relatively short angular range and a block of repetitive frames at each φ angle. As a result, the contributions from the reflections constitute a significant part of the counts for certain pixels. Application of the constantfraction approach in such a case leads to `spotlike' contamination on the reconstructed background, as evident in Fig. 7(a). The more sophisticated K–W method effectively reconstructs the proper background (Fig. 7b). The difference in detection of outliers is shown for a pixel located in the spotlike feature [Figs. 7(c) and 7(d)].
3.2. Statistical distribution of the background intensities
A et al., 1995). The sources of background noise result from specifics of the detector design, as well as external sources such as diffuse scattering by air. Assuming that the fluctuations of the noise are independent of time and there is no Bragg signal, the average event rate for each pixel unambiguously belonging to the background will be constant, and simple counting statistics should be applicable. In order to verify the nature of the background signal distribution in the current case, the statistical distributions of the pixel background values in dataset 2 were examined after the LaueUtil processing with the constantfraction approach. The background samples tend to exhibit a symmetrical almostGaussian distribution (Fig. 8). The assumption of a can be tested by examining the relation between the mean values of the background intensities and the corresponding variances, selecting only pixels which are not part of spots on any of the frames. Fig. 9 shows that for almost all of these pixels the estimated intensity variances are smaller than their estimated intensity means. There is no obvious dependence of the variances on the average intensities, whereas the Poisson law implies the definite relationship of σ^{2}(I_{background}) = 〈I〉_{background}. A possible explanation is that the numerical data collected on raw frames are not necessarily an exact representation of events at the detector surface (Waterman & Evans, 2010). The information from the detector is processed to convert optical signals into electronic ones and enhanced prior to storage. This preliminary step can lead to nonPoissondistributed numerical data, depending on the detector specifications and setting.
of the background signal is often assumed in Xray data integration algorithms to calculate standard deviations (Bolotovsky3.3. Comparison with the seedskewness integration
Reflection intensities resulting from LaueGui software (Messerschmidt & Tschentscher, 2008; Peters, 2003) were compared with the outcome of the constantfraction LaueUtil results on dataset 2 and K–W results for dataset 1. In both instances a simple linear relationship exists between the intensities processed by the two methods, which therefore in principle should not affect the response ratios. The intensities from the LaueGui method are systematically lower than those obtained with the current approach. Fig. 10 illustrates the correlation of intensities integrated using LaueUtil and LaueGui. A magnification of the lowintensity range (I ≤ 10000) is also plotted and confirms the linear relation between the LaueUtil and LaueGui intensities. A major source of the discrepancy between the two groups of intensities is due to the difference in the background calculations. Fig. 11 illustrates the discrepancies between the two methods. Each reflection is represented by a mark colored as a function of the differences between the average background intensities (normalized perpixel) estimated by the LaueUtil and LaueGui programs. In a first approximation, the closer the reflection to the beam center the more significant the difference between the two methods. The LaueUtil algorithm provides a lower estimate of the background in the majority of cases. This difference of average background can be significant for reflections less than half way from the beam center to the edge of the frame. The maximum difference, excluding the edge of the detector, is about 79%.
The discrepancy likely results from the lack of accuracy of the footprint definition in LaueGui. Part of the reflection tail is sometimes included in the background count, as illustrated in Fig. 12, which shows the relation between the normalized background difference and the surface of the reflection footprint. Almost all spots with background differences larger than 20 are very strong and located in the vicinity of the beam center, with the LaueGui background being larger. Only a few reflections on the edge of the detector have opposite background differences. The background estimate method used in LaueGui allows inclusion of pixels located outside the actual active detector area in the background calculation, which explains this background underestimate relative to LaueUtil. In all instances the background estimate in the LaueUtil method appears more reliable.
3.4. Prompt signal analysis during pump–probe experiments
No information on the unitcell parameters or crystal orientation is required for data integration with the LaueUtil tool. As a result, data can be integrated without the timeconsuming indexing of the Laue pattern. It allows prompt evaluation of the lightinduced signal. The intensities for any spot on any frame can be analyzed (plotted or otherwise processed) to ascertain whether or not there are systematic ON versus OFF differences. The method is especially advantageous when unitcell parameters are not known, or when the sample is twinned. The only necessary condition is that cell parameters do not change significantly upon laser exposure. This can be immediately verified by analysis of the spots positions on consecutive ON and OFF frames. Table 2 presents the experimental ON to OFF ratios for selected spots in the five subsets of dataset 1, which differ in applied laser power. Data were integrated using a common mask for each 20 frames in a block. Ratios of intensities were obtained for all ON/OFF frame pairs, their averages calculated and standard deviations estimated from the ten repeated measurements. Fig. 13 shows a reasonable agreement between ratios obtained applying LaueUtil and LaueGui software on dataset 1 (subset 1).

4. Conclusions
A new approach to Laue Xray data integration from CCD detectors is presented. The method uses simple statistical tools for identification of the background values for a given pixel on all frames in a scan.
Two particular approaches are described, their applicability depending on the Xray measurement strategy. The constantfraction approach is best suited for conventional data collection strategies, in which the crystal orientation and the resulting pattern are being changed from frame to frame. In the photocrystallographic experiments, the total range of crystal orientations may be limited. However, at each crystal orientation a batch of frames can be measured. In such case the K–W method is more suitable and yields superior results in terms of identification of the pixels belonging to the background and therefore leads to more reliable values of the Bragg intensities.
As the method is strictly based on statistical analysis of the pixel values, it does not depend on a data indexing routine, and thus allows monitoring Xray intensities and lightinduced changes even when cell parameters are not known. In the case of ultrafast photocrystallographic experiments in which cell parameters, and hence reflection positions, do not vary significantly between ON and OFF exposures, the response ratios can be calculated promptly. As no integration box is to be defined or profile is to be fitted, the method is also more suitable for dealing with reflections of elongated shape, as often observed in such experiments.
In its current implementation the method presented here is applicable mainly to data from photocrystallographic synchrotron experiments, although conventional data can also be processed with the constantfraction approach. The only limitation of the method is the stability of the background levels during the experiments, depending on the Xray source stability and diffuse scattering from the crystal support. When the stability criterion is fulfilled, the method yields prompt response ratios for subsequent photocrystallographic analysis.
APPENDIX A
The K–W test
The K–W test is a nonparametric method for testing whether two or more independent samples of values share a similar population distribution. In contrast to the oneway analysis of variance, ANOVA, the K–W test does not require an assumption about the nature of sample distribution, such as normality. The null hypothesis of this test is that the populations from which the samples originate share the same probability distribution. Like many other nonparametric tests the K–W test is based on the calculation of sample ranks. The test consists of five steps.
Let us assume a series of M samples with N_{i} the number of values in the ith sample and N the total number of values in this set:
(a) Sort all values from all samples together, and assign a rank to each value from 1 to N. If there are subsets of tied values, the average value of their ranks must be calculated and assigned to all of them.
(b) For each ith sample, calculate its average rank ,
where r_{(i,k)} is the rank of the kth value of the ith sample.
(c) Deduce the statistical test coefficient K defined as
with = (N + 1)/2 the average rank of the full set.
K can be rewritten as follows,
(d) If there are some tied values in the total set, divide the K value by a tiecorrection factor T to obtain K_{corrected}. This factor is given by
where L is the number of tied value sequences and, for each sequence k, p_{k} is its size.
(e) Approximate K_{corrected} by a χ^{2} distribution with M − 1 and calculate the corresponding probability pvalue . This approximation is reasonable if the samples are larger than 5. Depending on the desired α level, the null hypothesis is rejected if the pvalue ≤ α.
Acknowledgements
Support of this work by the National Science Foundation (CHE0843922) is gratefully acknowledged. Use of the BioCARS Sector 14 was supported by the National Institutes of Health, National Center for Research Resources, under grant RR007707. The Advanced Photon Source is supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No.W31109ENG38.
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