The problem of aligning a set of single-particle projections has a number of possible solutions. Usually it is assumed that a reference image is available, to which each particle is matched using the maximum of the cross-correlation function to determine the translation and rotation parameters. The disadvantage of this approach is that the resulting parameters are biased by the choice of the reference image. As a remedy, the alignment is normally repeated once or twice, using the average obtained in the previous step as new reference. However, this measure fails when the images are extremely noisy or when the input images show particles in more than one view : features of the original reference still tend to dominate the final average. In addition, certain steps in the preparation of a reference image, such as masking and filtration, require choices of parameters that are more or less intuitive. From this brief outline of the problem is is clear that, to obtain a result that does not depend on subjective decisions, the use of a reference image must be entirely eliminated. This is particularly important for images of ice-embedded particles.

In devising a reference-free alignment scheme, it is advantageous to use a concept of alignment as a feature of the entire image set. We say that the set of images is aligned if certain function calculated for the entire set reaches maximum. The value of this function is calculated in the following way: we sum up all the images (taking into account their current position parameters), we square each pixel and finally we add all the values of these squared pixels. In this way we arrive at one number which describes the 'quality' of alignment. It can be shown that such 'measure of alignment' is well defined: it agrees with our intuitive notion about the 'best' relative orientation of the series of images. It can be also shown that this definition is equivalent to the following definition: the set of images is aligned if all the possible pairs of images from this set are in the 'best' relative orientation as determined by the maximum of cross-correlation function (CCF). This second definition suggests calculation of CCFs between pairs of 'raw' images, which is not recommended because the usually low signal- to-noise ratio (SNR). The algorithm implemented in the Alignment Operations (AP **) therefore uses the first definition.

More details and test results are given in: "Three- dimensional reconstruction of single particles embedded in ice", P.Penczek et al., Ultramicroscopy 40 (1992) 33-53.

The reference-free alignment algorithm consists of two parts: in the first, a "random approximation" of the global average is found and in the second, this average is iteratively refined until convergence is achieved.

The purpose of the first part is to find a good approximation of the global average without using a reference image. At the beginning two images are randomly picked from the whole set and brought into mutual register. These two images are added according to the orientation parameters found, thereby providing the first approximation of the average with improved SNR compared with a single image. Then this average can be used to find the position of the third image (again picked randomly), which again is added to the sum of the two initial images. This procedure continues using in each step an 'improved' average, until all images from the whole set are included.

From this description one can easily see that the result of this procedure is not the 'optimal' one as it depends on the particular random sequence used. To mitigate this bias the whole first part is repeated using another random order of image inclusions, resulting in a new initial average. Then the two averages are brought into register and added, resulting in the initial "randomly" estimated average.

The second part of the algorithm is the iterative refinement of the initial average. In each step one image is removed from the set and the subaverage is created from the remaining images. (It should be noted that the exclusion of images is now done in sequential order.) The removed image is brought into register with the subaverage and a new average is created. This step is repeated for each image in the set. The whole procedure is repeated until a 'stable' position of each image is reached. This means that the algorithm is terminated if no single image changes its position.

In summary, the main features of the algorithm described are as follows:

a) a reference image is not used; thus, the problem of its selection and preparation is eliminated, and the result of the alignment no longer depends on the choice of any particular reference image;

b) the assumption of similarity among the input images is not used; thus the algorithm is able to align even images that share only a small number of common features;

c) the algorithm is suboptimal, which means that the scheme proposed does not lead to such positions of images for which the 'measure of alignment' used reaches maximum. It also means that by using some more elaborated (and much more time-consuming) scheme, further improvement of alignment can be achieved. In particular, the result depends on the initial estimate of the average. Since the calculation of this estimate is based on the random order of image inclusion, the overall result will differ for different runs of the procedure on the same data set.

There are two implementations of the alignment-free algorithm in SPIDER: the first (operation 'AP SR') performs both shift and rotational alignment with additional placement of the center of gravity of the particles at the image center and a second one, which separates the shift from the rotational alignment (operations 'AP SA' and 'AP RA', respectively), giving the user more freedom in the choice of the procedure parameters and allowing either of the alignment procedures to be performed separately.

Combined Alignment ('AP SR')

Operation 'AP SR' is recommended in most cases - it works faster and doesn't require writing or modifying lengthy procedures. Nor does it have any memory requirements, as it is able to switch automatically between "in-core" and "on-disk" versions. The strategy implemented closely follows the description given in the latter part of this text. The main part of the procedure brings two objects into register. This is achieved by alternating shift and rotational alignment until the relative change of position is smaller than 0.5 pixels (tests show that smaller numbers result in numerical instabilities due to interpolation errors). After alignment of all the objects (which constitutes one iteration step of the procedure) the global average is centered using the approximated location of its center of gravity (see the 'CG PH' manual page) and alignment parameters for all the images are modified accordingly. Iterations are repeated until stable alignment is reached (see above). For each iteration the resulting alignment parameters are stored in a document file and the corresponding average is produced.

Strategy for 'AP SR'

a) All the input images should have positive contrast, i.e. objects should be bright on the dark background.

b) Decide the limits of integration for the rotational search (in terms of inside and outside rings). The outer radius has to correspond to the particle size. To avoid oversampling (and thus the relatively large error of interpolation) an inner radius of 5 was found to be sufficient for all practical purposes.

c) Specify expected size of the objects to be aligned. The number given restricts the shift allowed, thus a fragmentation of the objects (due to shift beyond image frame) is avoided.

d) Perform 'AP SR' on the input image series.

e) Use alignment parameters stored in the document file and operation 'RT SF' to create the aligned image series and, optionally, verify resolution achieved using operation RF M. Should the results be unsatisfactorily: (i) modify parameters specified in (b) and (c) and repeat alignment; (ii) increase number of input images to improve SNR and repeat alignment; (iii) use separate alignment operations described in next part and control each step carefully.

Further reading is suggested only for users wishing to apply operations implementing separate shift and/or rotational alignments.

Separate Alignment Operations ('AP SA' and 'AP RA')

Two operations implement the idea of reference-free alignment separately: first to find the shift alignment ('AP SA') and a second to find the rotational alignment ('AP RA').

Both operations were implemented in a way that minimizes the time of calculations. The shift alignment operation reads each image into the memory and calculates its Fourier transform using the mixed-radix FFT (Fast Fourier Transform) algorithm. This makes possible the use of images with almost any size (except for dimensions involving large prime numbers), in particular with the size very close to the actual size of the particles. The limitation of the surrounding background noise achieved in this way improves the SNR of the cross-correlation functions calculated. All following operations (shifts, averaging, calculation of CCF) are performed in Fourier space on the transformed images stored in memory. No padding is done, which means that images are treated as circularly continuous (i.e., the left side of the image borders the right and the top borders the bottom). An additional option allows checking for the 180-degree rotation of images (also readily implemented in Fourier space), which further speeds up the whole alignment process. The output of the operation contains the shift parameters found for each image. An important feature in the design of this opeeration is that it does not enforce any particular position of the final average.

The next step, the rotational alignment, requires all particles to be placed in such a way that their centers coincide with the origin of the coordinate system used for rotation. Since such centers are, for obvious reasons, difficult to define or find for raw data, it is recommended that the average obtained from the shift alignment step be used for the estimation of necessary centering parameters. This average has usually high SNR and can be easily centered using, for example, a low-pass filtered disc as reference or using 'CG PH' to find its approximate center of gravity. The parameters found by the shift alignment operation must be subsequently corrected by this additional shift so that all particles will be moved to the central position.

The rotational alignment operation is used as follows: First, the limits of integration (in terms of inside and outside rings) must be decided. The outer radius must correspond to the particle size. To avoid oversampling (and thus the relatively large error of interpolation) an inner radius of 5 was found to be sufficient for all practical purposes. With these limits set each image is read into the memory and its quadratic interpolation is calculated. This interpolation is used to re-sample the image in polar coordinates, with the number of points in each ring equal to the power-of-two number which is closest to 2*pi*l, where l is the ring radius. Then the Fourier transform of each ring is calculated, and this initialization procedure is repeated for each image. The "reference-free" algorithm described in the previous section is implemented entirely in the Fourier domain using the CCF for the search of the optimum rotation parameter.

Strategy for 'AP SA' and 'AP RA'

The recommended strategy of alignment using two separate operations is as follows:

a) Use shift alignment operation ('AP SA') for the whole data set with option "180-degree check" turned on (to speed up the alignment process). In this way the entire input data set will be 'centered', which means that the majority of particles will have the same position with respect to the image center.

b) Center the average obtained from step a) using some kind of neutral rotationally symmetric shape (for example, a disk) as the reference, or using the operation 'CG PH' to find its center of gravity.

c) Calculate the resulting position parameters for each image and create the new, 'centered' data set.

d) Apply rotational alignment ('AP RA') to the image series calculated in step c).

e) Apply shift alignment ('AP SA') to the image series calculated in step c) using angles found during step d) and stored in the document file.

f) Calculate the resulting position parameters for each image, taking into account shifts and rotations found during steps d) and e) using the operation 'SA P', and create the new data set. Store the alignment parameters found in the document file.

g) Repeat steps d-e-f as many times as needed (usually 3-5), each time calculating resulting alignment parameters and storing them in the document file. After each step the resolution of the data set obtained can be checked, and if there is no substantial improvement the procedure can be terminated.

The strategy described is implemented in the Spider procedure included as an example in the 'AP SA' and 'AP RA' manual chapters.

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