It is important in the analysis of images of periodic specimens by Fourier techniques to extract the best values for the amplitudes and phases of the transform maxima. When many images are to be analysed, as, for example, in a 3-dimensional reconstruction, it may also be important to have a procedure that is computationally efficient. A peakprofile fitting method is described here, and is compared to 2 other methods that have been published by applying all 3 methods to negative-stain images of the cell wall of Lobomonas piriformis. This analysis would be out of place in the main text, but is necessary for a complete explanation of the methods we have used.
For ideal Fourier filtering of a perfectly ordered periodic image, just those spatial frequency components that are integrally related to the unit-cell repeats should be extracted from the full frequency spectrum of the image. An image reconstructed by back-transformation of these values will contain only the features of the original image that obey the crystal repeat symmetry. However, the mathematics of discrete Fourier transforms, as used in the fast Fourier transform (FFT) algorithm, yield for an interval of N sample points frequency components at values f/N, f = 0,1, N-1 (Cooley & Tukey, 1965). This means that the periodicities of a crystalline image can only be exactly represented if the N points include an integral number of unit-cell repeats. Since all available microdensitometers use an orthogonal sampling lattice, this condition cannot be met for an image of a 2-dimensional crystal with non-orthogonal axes (and, in practice, only rarely for one with orthogonal axes). The frequency components due to the crystalline repeats are thus ‘aliased ‘in the sampled transform by adjacent frequencies.
The points around each spot thus have complex amplitudes, which are proportional to a sine function centred on the lattice point and sampled at the transform points. When an integral number of unit-cell repeats is contained in each direction in the image the sine function is sampled at its maximum and zeros. In general, however, the function is sampled at an arbitrary series of points separated by one transform unit.
The complex values FST for each reciprocal lattice point may be extracted, as suggested by Saxton (1978), by a least-squares fit over a small window of the transform in the neighbourhood of each spot position. Alternatively, the values may be obtained by direct inspection of the transform (Unwin & Henderson, 1975); applying Parseval’s theorem (e.g. see Saxton, 1978) shows that the summed intensity of the surrounding points should be equal to the spot intensity, and consideration of the form of the peak profile shows that the phase should be constant across the 4 points surrounding the spot maximum. The values for FST produced by either of the last 2 methods will not in general be theoretically correct. This is because there will be a non-integral number of unit cells in each direction, and the transform will therefore contain contributions from the fractional unit cells included. However, the error will depend on the ratio of the contribution of the fractional unit cells to the total, which will decrease with increasing number of unit cells in the image. If a large number of unit cells are present in the image this error would be expected to be fairly small.
Although the above method gave reasonable values for the cell parameters quite rapidly, it was considered inadequate for 2 reasons. Firstly, it was tedious to examine every spot interactively and required a subjective judgement to include or exclude each one. Secondly, the centre-of-gravity of the intensity does not make full use of the available data, as the phase is ignored, and does not generally give the correct position for the centre of an ideal spot. This is illustrated in Fig. A1 by the results of a numerical calculation, where the position of the centre of a sine function is plotted against the result of the centre-of-gravity estimate. (Provided the window size is at least 3 points, it has little effect on the form of the function.) The r.m.s. deviation of the centre-of-gravity calculation from the correct value for the centre corresponds to about 0·11 of a transform unit.
More accurate unit-cell parameters were therefore determined by the following procedure. The partially refined unit-cell parameters produced by the method described above were used to predict the position of every lattice point within a given resolution limit in the transform. For each predicted peak position the intensity maximum was found by searching a 9 by 9 point window around the spot for the 2 by 2 point box of maximum summed intensity. Spots were included in the subsequent refinement if this summed intensity was greater than 10 times the background intensity (estimated as the mean intensity around the perimeter of the 9 by 9 point window). The centre of gravity of this maximum box was then used as a starting point for a peak-analysis search. The intensity was calculated by the least-squares profilefitting procedure described below, taking the spot centre at successive positions o-i of a transform unit apart over a range of 1 transform unit either side of the starting position in both dimensions. This calculation gave a sharp peak for all spots meeting,the intensity criterion, whose position agreed well with estimates of the spot centres made by inspection of the intensities and phases around the spots. All the refined spot positions, generally about half those eventually included in reconstruction, were then included in the lattice calculation, and typically gave an unweighted r.m.s. deviation of between 0 ·1 and 0·2 of a transform unit in distance.
Calculation of amplitudes and phases
Method 1 (interpolation and trimming to an integral number of unit cells). An initial Fourier transform was calculated and used to determine accurate unit-cell parameters, from which was derived an interpolation matrix such that the interpolated image was sampled at an integral number of points in each direction of the unit cell. A simple bilinear interpolation was used (Smith & Aebi, 1973), followed by trimming to an integral number of unit cells. Amplitudes and phases were taken directly from the reciprocal lattice points in the Fourier transforms of the interpolated images. No correction for the background intensity was made but the ratio of peak to background intensity was stored along with the amplitudes and phases of each spot.
The peak to background ratio was retained as for method 1.
Analysis of the spot data
RESULTS AND DISCUSSION
The images used for this study were recorded and digitized as described in the main text. The Fourier transforms were calculated and the lattice parameters were refined; then amplitudes and phases were calculated for each image by each of the 3 methods outlined above. Spots were judged to be significant and were then included in scaling if the peak to background ratio, or the equivalent quantity given for the profile method, was at least 2· 0. However, in the case of the interpolated data (method 1) this gave about 80 significant spots out of 160 possible peaks to 2· 0 nm resolution, whereas the other 2 methods each gave about 130 spots. For the purposes of comparison, therefore, only those spots judged as significant by all 3 methods were included in the statistics given. In fact, omitting the extra spots from the data sets produced by methods 2 and 3 did not significantly change the unweighted R factors, so these spots represent reproducible data, mainly at higher resolution, which are lost by the first method. The reason for this may be the errors introduced by the interpolation. Also, significantly less of the image is finally used by the interpolation method as a large portion is discarded when trimming. This method would make better use of the scanned data with a unit cell having an inter-axial angle nearer to 90°.
Table A1 shows overall statistics on the agreement between the 3 films as determined by each method. Table A2 shows the agreement between the averaged data sets from each method. All 3 methods reassuringly produce very similar results as judged by the values in Table A2. This confirms that in this case the effect of the fractional unit cells included for methods 2 and 3 is negligible. (About 300–400 unit cells were present in the scanned areas.) It would be expected to be more significant, however, if only a small number of unit cells were included in the image, and in such a case the interpolation method might be the safest choice. According to Table A2, all 3 methods give fairly good agreement between the 3 films, with the profile method (method 3) giving somewhat better values than the other two.
A factor that has not been examined in this study is the relative sensitivity of different techniques to small amounts of incoherence in the crystal lattice of the image. The profile-analysis approach does, however, allow a quantitative assessment of the coherence of the lattice by analysing how well the spots fit the sine profile. This may be done by comparing the residual in the least-squares calculation with the background intensity for each spot. These should be approximately equal if the peaks conform to the theoretical profile, but will be very different if not, or if the unit-cell parameters are not correct. Both the cell wall of Lobomonas and several other specimens that have been examined so far (unpublished work) all show good agreement with the sine profile, as judged by the lack of closure to background ratio. However, it is necessary to examine the Fourier transform of any image critically before applying Fourier filtering, whatever method is used.
In conclusion, the profile analysis method is computationally rapid and efficient in its use of the data. The ratios of the profile peak intensity to the least-squares residual and of the residual to the surrounding background give objective criteria for judging the significance of the spot and for deciding whether the image is sufficiently coherent to justify this type of Fourier analysis. Since we obtain what appear to be more reliable results, as judged by the agreement between different images, by this method than by any other method we have tried, we have used the profile analysis for all the reconstructions described in this paper. We are currently using it on other specimens and on the large number of images necessary for a 3-dimensional reconstruction of the cell wall of Lobomonas.
We wish to thank Dr W. O. Saxton of the Cavendish Laboratory, Cambridge, U.K. for useful discussion and for provision of some of the film-scanning facilities used, and Drs R. Henderson, L. Amos and J. Deatherage of the M.R.C. Laboratory of Molecular Biology, Cambridge, U.K. for advice and discussion and for providing source listings of their computer programs.