Intensity estimation

Kabsch, W.

doi:10.1107/97809553602060000676

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 222 | 1 | 2 |

Section 11.3.3.4. Intensity estimation

W. Kabsch^a ^*

^a Max-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail: kabsch@mpimf-heidelberg.mpg.de

11.3.3.4. Intensity estimation

| top | pdf |

If an expected intensity distribution $[\{p_{i} | i\in D_{0}\}]$ of the observed profile is given in a domain $[D_{0}]$ , the reflection intensity I can be estimated as $[I = {{\textstyle\sum\limits_{i\in D}}(c_{i}-b_{i})p_{i}/v_{i}}\bigg/{{\textstyle\sum\limits_{i\in D}}p_{i}^2/v_{i}},]$ which minimizes the function $[\psi(I) = {\textstyle\sum\limits_{i\in D}}(c_{i} - I\cdot p_{i} - b_{i})^2/v_{i} , \qquad {\textstyle\sum\limits_{i\in D_0}} p_{i} = 1.]$ $[b_{i},c_{i},v_{i}\ (i\in D)]$ are background, contents and variance of pixels observed in a subdomain $[D\subseteq D_{0}]$ of the expected distribution. The background $[b_{i}]$ underneath a diffraction spot is often assumed to be a constant which is estimated from the neighbourhood around the reflection. Determination of reflection intensities by profile fitting has a long tradition (Diamond, 1969; Ford, 1974; Kabsch, 1988b; Otwinowski, 1993). Implementations of the method differ mainly in their assumptions about the variances $[v_{i}]$ . Ford uses constant variances, which works well for films, which have a high intrinsic background. In XDS, which was originally designed for a multiwire detector, $[v_{i}\propto p_{i}]$ was assumed, which results in a straight summation of background-subtracted counts within the expected profile region, $[I=\sum\nolimits_{i\in D}(c_{i}-b_{i})/\sum\nolimits_{i\in D}p_{i}]$ . This particular simple formula is very satisfactory for the low background typical of these detectors. For the general case, however, better results can be obtained by using $[v_{i} = b_{i} + Ip_{i}]$ for the pixel variances as shown by Otwinowski and implemented in DENZO and in the later version of XDS. Starting with $[v_{i} = b_{i}]$ , the intensity is now found by an iterative process which is terminated if the new intensity estimate becomes negative or does not change within a small tolerance, which is usually reached after three cycles. It can be shown that the solution thus obtained is unique.

References

Diamond, R. (1969). Profile analysis in single crystal diffractometry. Acta Cryst. A25, 43–55.Google Scholar

Ford, G. C. (1974). Intensity determination by profile fitting applied to precession photographs. J. Appl. Cryst. 7, 555–564.Google Scholar

Kabsch, W. (1988b). Evaluation of single-crystal X-ray diffraction data from a position-sensitive detector. J. Appl. Cryst. 21, 916–924.Google Scholar

Otwinowski, Z. (1993). Oscillation data reduction program. In Proceedings of the CCP4 study weekend. Data collection and processing, edited by L. Sawyer, N. Isaacs & S. Bailey, pp. 56–62. Warrington: Daresbury Laboratory.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 222