Calibration¶
The determination of the geometry of the experimental setup for the diffraction pattern of a reference sample is called calibration in pyFAI. A geometry setup is composed of a detector, the six refined parameters like the distance and fixed parameters like the wavelength (or the energy of the beam), they are all saved together into a text files named “.poni” (as a reference to the point of normal incidence) which is subsequently used for processing the experiment.
The program pyFAI-calib is used for calibrating the experimental setup using a constrained least squares optimization on the Debye-Scherrer rings of a reference sample (\(LaB_6\), \(CeO_2\), silver behenate \(AgBh\), …) and saves the results into a .poni file. Alternatively, geometries calibrated using Fit2D can be imported into pyFAI, including geometric distortions (i.e. optical-fiber tapers distortion) described as spline-files. For Fit2D compatibility, please refer to the tutorial on basic usage of pyFAI.
By storing all parameters together in a single small file, the risk of mixing two parameters is highly reduced and we believe this helps performing better science with fewer mistakes.
While entering the geometry of the experiment in a poni-file is possible, it is easier to perform a calibration, using the Debye-Sherrer rings of a reference sample called calibrant. About 30 calibrants are provided by pyFAI like \(LaB_6\), ceria \(CeO_2\), silicon, corrundum or silver behenate. Among other simple compound, all of the NIST Standard Reference Materials have been are tabulated and are directly available as calibrant. One can alternatively provide its own calibrant description files which is a simple text-file containing the largest d-spacing (in Angstrom) for a set of Miller plans. A useful reference to generate this file is the American Mineralogist database [AMD] or the Crystallographic Open database [COD].
The calibration procedure is divided into 4 major steps:
Pre-processing of images:¶
The typical pre-processing consists of the averaging (or better median filtering) of darks images. Dark current images are then subtracted from data and corrected for flat. The pre-processing is best performed using the pyFAI-average tool, which documentation is available in the Application manuals.
If saturated pixels exists, the are likely to be treated like peaks but their positions will be wrong. It is advised to either mask them out or to desaturate them (pyFAI provides an option, but it is expensive in calculation time). A Mask drawing tool, called pyFAI-drawmask, is installed together with pyFAI and its documentation available in the Application manuals.
To start the calibration the pyFAI-calib tool will need:
an image with Debye-Sherrer rings
the energy or the wavelength
the calibrant name or the d-spacing file of the calibrant
the detector description.
Peak-picking¶
Once started, pyFAI-calib will ask you to select rings. The Peak-picking consists in the identification of peaks and groups of peaks belonging to same ring. It can be performed by 4 methods described hereafter.
Massif detection¶
This method consists in making the difference of the original image and a blurred image. Then look for a large contiguous region of positives values, corresponding to a single group of peak. The blurring parameter can be adjusted using the “-g” option in pyFAI-calib.
Blob detection¶
The approach is based on difference of gaussians (DoGs) as described in the blob_detection article of wikipedia.
It consists in blurring the image by convolution with a 2D gaussian kernel and making differences between two successive blurs (called Difference Of Gaussian or DoGs). In theses DoGs, keypoints are defined as the maxima in the 3D space (y,x,size of the gaussian). After their localization, keypoints are refined by Savitzky Golay algorithm or by an interpolation at the second order which is equivalent but uses less points. At this step, if the estimation of the maximum is too far from the maximum, the keypoint will be considered as a fake maximum and removed.
Steepest ascent¶
This is very naive implementation which looks for the nearest local maximum. Subsequently a sub-pixel optimization is performed based on a second order expansion using the local gradient and hessian.
Monte-Carlo sampling¶
Series of peaks can be extracted using the Steepest Ascent on randomly selected seeds. This method can be biased towards an already known geometry by starting from points which are supposed to be on the ring.
Refinement of the parameters¶
After selecting groups of peaks, each of them is assigned to a Debye-Scherrer ring number (0-based numbering in python) and associated to a d-spacing value hence a theoretical 2theta value. A supervised least-squares refinement, performed on the difference of peak position’s 2-theta values versus the expected ones from calibrant provides the 6-geometry parameters fitted.
The default optimization procedure is the Sequential Least SQuares Programming implemented in scipy.optimize.slsqp. The cost function is the sum of the square of the difference between the expected and calculated 2theta values for the various peaks. This sum is dependent on the number of control-points.
Validation of the calibration¶
Validation by an human being of the geometry is an essential step: pyFAI will overlaps to the diffraction image, the lines corresponding to the various diffraction rings expected from the calibrant. Those lines should be in pretty good agreement with the rings of the scattering image. The average error per control point (delta 2theta error in radian) is printed out and offers a quantitative measurement of the relative quality of the fit for similar setups/experiment. Nevertheless its absolute value has no meaning, except the lower, the better.
Subsequently, pyFAI offers some validation options in to check the quality of the fit. some of them global, some of them limited to given rings.