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.

Various approaches have been used to solve this problem. One conceptually simple idea is to abandon the FFT algorithm and to calculate directly the transform values at the non-integral frequencies required (Shelley, Hillman & McPherson, 1980). This requires a preliminary transform to measure the reciprocal lattice parameters accurately and thus determine which frequencies are needed. A second possibility, which has been used by Aebi et al. (1973), is to resample the image by interpolation such that the sample intervals are integral divisors of the unit-cell repeats, and to trim the interpolated image to contain an integral number of unit cells in each dimension. The required frequencies are then exactly contained in the discrete transform. This procedure again requires an initial transformation to determine the unit-cell parameters followed by a 2-dimensional interpolation and a second Fourier transformation of the interpolated image. It is therefore fairly time-consuming. The errors introduced by the interpolation have been discussed by Smith & Aebi (1973). The third possibility is to deduce the values for the exact, non-integral transform spot positions from the values at the surrounding transform points. The following derivation of the form of the sampled spot peak at the nearby points follows that given by Saxton (1978); for simplicity, stated for the 1-dimensional case. Consider the contribution of a component of spatial frequency 5 with (complex) amplitude Fs to the Appoint sampled function g: (lower case letters denote real-space and upper case Fourier-space quantities)
formula
Thus at a point K in the transform:
formula
This geometric series may be summed to give:
formula
The form of P(K—S) may be more easily seen by transforming the origin to the centre of the sampled interval by applying a phase shift of exp (— πi (K— S)). (It should be noted that practical FFT programs take the origin at the beginning of the sampled interval, and this phase shift must be applied subsequently.)
formula
Similarly in 2 dimensions:
formula
for a spot position (S, T).

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.

Lattice refinement

It is necessary to determine the reciprocal lattice vectors in the transforms as accurately as possible for all of these methods. This is also important for the calculation of tilt angles from images of tilted crystals that are to be used in 3-dimensional reconstruction. The basic scheme used for refinement was to obtain a list of coordinates for the centres of a number of transform spots and to use these to calculate the unit-cell vectors that minimize the sum of the squared discrepancies between the measured and predicted positions. Initially, spot centres were refined by taking the ‘centre of gravity ‘of the intensity (IJK) around the spot:
formula
where f and K were taken over a small window, typically 5 by 5 points, around the approximate spot position. An interactive program was written to select transform spots and carry out the centre-of-gravity refinement; for each selected spot the initial and refined parameters, and the windowed intensity data were displayed. The spots could then be added to the list for the unit-cell vector calculation or rejected if a suitably centred large peak was not apparent. When this method was applied to the transforms of images of the cell wall of L. piriformis using all the reflexions observable in grey-scale pictures, usually about 50 spots, root-mean-square (r.m.s.) residuals, in a distance of about 0·3 of a transform unit were obtained. (One transform unit being defined as 1 /N.)

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.

Figure A1.

Position of the calculated centre of gravity of a sampled sine function plotted against the true position of its maximum value. The sine function was sampled at a series of points separated by a distance of I unit. Provided the number of sampled points included in the window is at least 3, the actual number has little effect on the form of the function.

Figure A1.

Position of the calculated centre of gravity of a sampled sine function plotted against the true position of its maximum value. The sine function was sampled at a series of points separated by a distance of I unit. Provided the number of sampled points included in the window is at least 3, the actual number has little effect on the form of the function.

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.

Method 2 (summed intensity, average phase). The peak intensities were calculated from the transforms as the summed intensity of a 3 by 3 point window centered on each spot. Background correction was made by subtracting 9 times the mean intensity of the perimeter of a 9 by 9 point window around each spot. The phases were calculated as the weighted mean of the phaseJK) of the 4 points surrounding the spot:
formula

The peak to background ratio was retained as for method 1.

Method 3 (peak profile analysis). Amplitudes and phases were calculated from the integrated complex amplitudes (A + i B) determined by a least-squares fit to the sine profiles:
formula
where f and K were taken over a small window around the spot position (S, T), typically 5 by 5 points. The ratio of the peak intensity to the sum of the residuals from the least-squares fit, normalized by dividing by the number of points in the window, was retained as an analogous quantity to the peak to background ratio.

Analysis of the spot data

Data from 3 images of the cell wall of L. piriformis were calculated by each of the 3 methods given above. For each method the 3 data sets were scaled together and statistics were calculated on the agreement of the phases and amplitudes. For ampli-tude scaling a simple unweighted least-squares scaling was used between the first and each subsequent data set:
formula
summed over all common reflexions (h, k) between data sets i andj. The relative phase origin was refined by searching for the origin shift (Δx, Δy) that minimized the quantity
formula
between the first and each subsequent data set (see Amos, 1975).
The agreement of the data sets was assessed by unweighted R factors calculated from the mean values hk and Fhk) summed over all the common reflexions.
formula
Finally, since we believe this structure to have 2-fold symmetry, the origin of the averaged data set was refined to the best 2-fold position by minimizing the summed discrepancy between the phases and 0° or 180°
formula
where ϕhk is the phase of the (A, k) spot shifted to the best phase origin and ϕ0hk. Is the symmetry-constrained phase nearest to this value (0° or 180°). The value of this quantity should provide a further test of the quality of the data.

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.

Table A1.

Phase and amplitude agreement between the three films for each method

Phase and amplitude agreement between the three films for each method
Phase and amplitude agreement between the three films for each method
Table A2.

Phase and amplitude agreement between the data sets from the three methods

Phase and amplitude agreement between the data sets from the three methods
Phase and amplitude agreement between the data sets from the three methods

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.

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