Determination of the best background plane

Leslie, A. G. W.

doi:10.1107/97809553602060000675

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. 11.2, p. 213 | 1 | 2 |

Section 11.2.5.1. Determination of the best background plane

A. G. W. Leslie^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: andrew@mrc-lmb.cam.ac.uk

11.2.5.1. Determination of the best background plane

| top | pdf |

The background plane constants a, b, c are determined by minimizing $[R_{1} = {\textstyle\sum\limits_{i = 1}^{n}} w_{i} \left(\rho_{i} - ap_{i} - bq_{i} - c \right)^{2}, \eqno(11.2.5.1)]$ where $[\rho_{i}]$ is the total counts at the pixel with coordinates $[(p_{i}q_{i})]$ with respect to the centre of the measurement box, and the summation is over the n background pixels. $[w_{i}]$ is a weight which should ideally be the inverse of the variance of $[\rho_{i}]$ . Assuming that the variance is determined by counting statistics, this gives $[w_{i} = 1\big/GE\left(\rho_{i}\right), \eqno(11.2.5.2)]$ where G is the gain of detector, which converts pixel counts to equivalent X-ray photons, and $[E(\rho_{i})]$ is the expectation value of the background counts $[\rho_{i}]$ . In practice, the variation in background across the measurement box is usually sufficiently small that all weights can be considered to be equal.

This gives the following equations for a, b and c, as given in Rossmann (1979), $[\pmatrix{{\textstyle\sum} p^{2} &{\textstyle\sum} pq &{\textstyle\sum} p \cr {\textstyle\sum} pq &{\textstyle\sum} q^{2} &{\textstyle\sum} q\hfill \cr {\textstyle\sum} p\hfill &{\textstyle\sum} q\hfill &\hfill n\cr} \pmatrix{a \cr b \cr c \cr} = \pmatrix{{\textstyle\sum} p\rho\cr {\textstyle\sum} q\rho\cr {\textstyle\sum} \rho\hfill\cr}, \eqno(11.2.5.3)]$ where all summations are over the n background pixels.

11.2.5.1.1. Outlier rejection

| top | pdf |

It is not unusual for the diffraction pattern to display features other than the Bragg diffraction spots from the crystal of interest. Possible causes are the presence of a satellite crystal or twin component, white-radiation streaks, cosmic rays or zingers. In order to minimize their effect on the determination of the background plane constants, the following outlier rejection algorithm is employed:

(1) Determine the background plane constants using a fraction (say 80%) of the background pixels, selecting those with the lowest pixel values.
(2) Evaluate the fit of all background pixels to this plane, rejecting those that deviate by more than three standard deviations.
(3) Re-determine the background plane using all accepted pixels.
(4) Re-evaluate the fit of all accepted pixels and reject outliers. If any new outliers are found, re-determine the plane constants.

The rationale for using a subset of the pixels with the lowest pixel values in step (1) is that the presence of zingers or cosmic rays, or a strongly diffracting satellite crystal, can distort the initial calculation of the background plane so much that it becomes difficult to identify the true outliers. Such features will normally only affect a small percentage of the background pixels and will invariably give higher than expected pixel counts. Selecting a subset with the lowest pixel values will facilitate identification of the true outliers. The initial bias in the resulting plane constant c due to this procedure will be corrected in step (3). Poisson statistics are used to evaluate the standard deviations used in outlier rejection, and the standard deviation used in step (2) is increased to allow for the choice of background pixels in step (1).

References

Rossmann, M. G. (1979). Processing oscillation diffraction data for very large unit cells with an automatic convolution technique and profile fitting. J. Appl. Cryst. 12, 225–238.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 11.2, p. 213