Section 7.2.3. Calibration and correction

It is important to remove both the electronic offset and the accumulated dark charge from an X-ray image. Since the integrated dark current varies from pixel to pixel and with time, a set of images needs to be taken (with no X-rays), matched in integration time to the X-ray exposures. With a properly temperature-stabilized detector, the background images may be acquired in advance and used throughout an experiment. Because the background image has noise, it is common to average a number of separate backgrounds to minimize the noise.

Cosmic rays and radioactive decay of actinides in the fibre-optic glass produce large-amplitude isolated signals (`zingers') within an X-ray image. These accumulate randomly in position and in time. For the short exposures typical at synchrotron sources and for data sets with highly redundant information, the few diffraction spots affected by these events can be discarded with statistical analysis. For longer exposures, or for data with less redundancy, two (or more) nominally identical exposures can be taken and then compared pixel by pixel to remove spurious events (see Barna et al., 1999).

Geometric distortions arise in the optical coupling of the system. If they are stable, the distortions can be mapped and corrected. Long-term stability is possible for a phosphor fibre-optically coupled directly to a CCD, since all distortions are mechanically fixed. Intensifier-based systems are subject to changes in magnetic and electric fields, hence stability is more of a problem.

Geometric distortions may be either continuous or discontinuous. Fibre optics often have shear between adjacent bundles of fibres. In this defect, one group of fibres will not run parallel to a neighbouring group, causing a discontinuity in the image. Rather than dealing with such discontinuities, tapers with low shear (less than one pixel maximum) are usually specified. Even with low shear, there is a continuous distortion (several per cent), which varies slowly over the face of the detector. Such distortion is inevitable, as the temperature profile cannot be precisely controlled in the large block of glass comprising the fibre optic as it is processed. To map this distortion, an image is taken of a regular array of spots. Such an image can be made by illuminating a shadow mask of equally spaced holes with a flood field of X-rays. Holes 75 µm in diameter spaced on a 1 × 1 mm square grid are adequate for mapping the distortions present in most fibre-optic tapers. Such masks have been lithographically fabricated in an X-ray opaque material, such as 50 µm tungsten foil (Barna et al., 1999).

Given an image produced with this X-ray mask, the displacement map for every pixel in the original image can be computed as follows: find the centroid of each mask spot and its displacement relative to an ideal lattice. The array of spot positions and associated displacements can then be interpolated to find the displacement for each pixel in the original distorted image.

The displacement of a pixel from original to corrected image will not in general be a whole number. Rather, the intensity of a pixel will be distributed in a local neighbourhood of pixels in the corrected image centred about the position given in the displacement map. The size of the neighbourhood depends on the local dilation or contraction of the image; typically the intensity will be distributed in one to nine pixels. This distribution procedure yields a smooth intensity mapping. Applying corrections to mask images that have been arbitrarily displaced shows that the distortion correction algorithm is good to better than 0.25 pixels (Barna et al., 1999).

The geometrical distortion is tied intimately to the correction of the response of the system (see below). Since distortions produce local regions of dilation or contraction of the image, pixels will, in general, correspond to varying sizes. This variation must be included in the flat-field calibration.

Variation in response arises from nonuniformity in the phosphor, defects in the fibre optics and pixel-to-pixel variation in the sensitivity of the CCD itself. Of these, variations in the fibre optics are usually most pronounced, often with decreased transmission at the bundle interfaces, resulting in the characteristic `chicken-wire' pattern. The response of the system is generally stable with time, although exposure to the direct beam at synchrotron sources will cause colour centres to form in the glass of the fibre optic. Recalibration of the system will correct for the reduced transmission in the exposed spot.

Light output from the phosphor depends on a number of factors: phosphor thickness, X-ray energy, angle of incidence and depth of conversion within the screen. X-ray photons are absorbed within phosphor grains with an exponentially decaying distribution as they travel deeper into the phosphor layer. Making the phosphor layer thicker increases the probability that the X-ray photons are absorbed, yielding an increase in the signal produced. But thicker phosphor layers also scatter the visible photons which one desires to collect in the CCD. At some point, the loss of light will be greater than the increase in X-ray stopping power and the net response will decrease. For one particular phosphor preparation, this appears to happen at circa 85% stopping power. The phosphor-screen resolution also falls with increasing screen thickness, although surprisingly slowly owing to the diffusive nature of the light scatter. These effects are discussed in Gruner et al. (1993).

Consider a given phosphor layer with thickness nonuniformity. For X-ray energies where the stopping power of the phosphor is low, regions of the phosphor with a thickness larger than the mean will be brighter owing to the increased X-ray stopping power. For energies at which X-rays are strongly absorbed, the increased opacity of the thicker regions of the phosphor will cause the response to go down. This illustrates the importance of calibrating the response of the system to the particular X-ray energy of interest.

In like manner, X-rays impinging on the phosphor at an angle away from the normal are presented with a longer path length and hence an increased stopping power. Also, because of the oblique angle, the distribution of visible-light production will be shifted toward the phosphor surface. Again, for strongly absorbed X-ray energies, the increase in the optical path for the light will cause the recorded signal to fall, whereas for X-rays not as strongly absorbed, the signal will increase.

To map the nonuniformity in response, one would ideally use a uniform source of X-rays of the proper energy placed at the position of the sample. This would calibrate the detector with the proper energy and angle of incidence for the diffraction data to be corrected. Correction factors are computed from a series of images taken of this uniform source. Sufficient numbers of X-rays per pixel must be collected to reduce the shot noise in the X-ray measurement to the required level (e.g. 40 000 X-rays per pixel must be acquired to correct to 0.5%).

Providing a truly uniform source with an arbitrary X-ray energy and angular distribution is difficult at best. Other sources can be used, however, with good results. Amorphous samples containing a variety of elements can be fabricated which produce X-ray fluorescence at various wavelengths when excited by a synchrotron beam (Moy et al., 1996). These can be placed at the position of the sample, thereby mimicking the angular distribution of X-rays from the experiment. The fluorescence is not uniform in space, however, so that the actual distribution must be mapped by some means. Once mapped, however, these samples provide a stable calibration source.

Another alternative is to separate the calibration procedure into several parts, mapping the dependence at normal incidence and treating the angular dependence as a higher-order correction. By moving an X-ray source sufficiently far away, the detector can be illuminated at near-normal incidence with excellent uniformity. For example, an X-ray tube at 1 m distance can produce a field with uniformity better than 0.5% over a 10 × 10 cm area. A sum of images with sufficient X-ray statistics, taken with the area dilation of each pixel computed through the geometric distortion calibration, can be used to compute a pixel-by-pixel normalization factor. Again, this should be determined for the X-ray energy of interest. In practice, the appropriate energy may be approximated by a linear combination of several energies. However, the proper coefficients will need to be determined empirically. This can be judged in diffraction having broad diffuse features, because imperfections in the phosphor stand out clearly when the data are corrected using factors derived from the wrong X-ray energy.

It has been found empirically that the light output for a given X-ray energy has a quadratic variation with the angle of incidence. The quadratic coefficient varies with X-ray energy and may be either positive or negative. The angular dependence may be measured by illuminating the detector with a small stable spot of X-rays and recording the integrated dose at various angles of incidence. Given the placement of the detector in relation to the beam and sample, the angle of incidence at a particular pixel can be computed and can be used to find the correction factor needed. With this method, a change in the experimental setup does not require a new calibration, just the computation of a new set of coefficients. The combination of energy and obliquity sensitivity varies slowly and may be approximated by a quadratic or cubic fit to a surface as a function of X-ray energy and angle. The few coefficients defining this surface allow quick computation of the combined energy and obliquity factor with which to multiply the local flat-field correction for X-rays of known incident energy and angle.

The obliquity correction is often ignored, since the solution of structures from X-ray diffraction typically includes a temperature factor which also varies with angular position. Uncorrected angular dependence of detector response will be convoluted with the true temperature factor and often does not impede the solution of the structure.

These procedures do not allow correction to arbitrary accuracy, however. The calibration data are taken with uniform illumination, whereas diffraction spots are localized. Given a nonzero point spread in the detector, the computed correction factor arises from a weighted average of many illuminated pixels. The signal from a diffraction spot only illuminates a few pixels, so the true factor might well be different. This should be less of a problem as diffraction spots become larger, becoming more like a uniform illumination. Measurement for one detector showed 75 µm spots could be measured to 1% accuracy, whereas 300 µm spots could be measured to 0.3% accuracy (Tate et al., 1995).

The size of available fibre optics and CCDs and the inefficiencies of image reduction limit the practical imaging area of a single CCD system. Closely stacked fibre-optic taper CCD modules can be used to cover a larger area. Although the image recorded from each module could be treated as independent in the analysis of the X-ray data, merging the sub-images into one seamless image facilitates data processing. Each module will have its own distortion and intensity calibration. It is no longer possible to choose an arbitrary lattice onto which each distorted image will be mapped: the displacement and scaling must be consistent between the modules. This would be accomplished most easily by having a distortion mask large enough to calibrate all modules together, although it is possible to map the intermodule spacing with a series of mask displacements.

Flat-field correction proceeds as in the case of a single module detector after proper scaling of the gain of each unit is performed. There can potentially be a change in the relative scale factors between modules, since each is read though an independent amplifier chain. Multimodule systems emphasize the need for enhanced stability and ease of recalibration.