# Aligning Particles

## Centering the Particles Figure 1. Averages showing the stepwise centering of several ring particles. Notice how the ring becomes less diffuse with each iteration, indicating better centering of all the particles. In each image, the particle is white on a dark background.

Usually, most picking algorithms (whether automated or manual) do a fairly good job of placing the picked particles near the middle of the boxed out image. However, fine adjustments still need to be done to make the later steps all that more efficient. A common method of centering searches for the so-called center of gravity. In this method, the image is scanned and the intensity peaks calculated to find the center of the particle 'mass'. The image is then shifted by the correct amounts to maximize the number of peaks near the center.

This method, however, does not work very well if the particle is very asymmetric, since one end might be very heavy compared to the other. In this case, the heavy end will be centered, and the rest of the particle might not necessarily align correctly. Also, particles that have a hollow, such as in a ring, can be miscentered, since some center of gravity algorithms break down when there is a lack of mass in the middle of the object. In these cases, the ring particles are kicked off center so that part of the mass of the ring is in the image center.

Another centering method averages all the picked particles together, and then crosscorrelates each individual particle to the average. The crosscorrelation function is used to determine by how much to shift each particle when trying to center it. When all the particles have been crosscorrelated to the average and shifted, a new average is generated. Once again, all the particles are compared to the new average, and shifted as necessary to center them as best as possible. This iterative process is repeated until no significant shift is necessary for all the particles. Figure 1 shows three averages produced during the iterative process of centering 590 ring-shaped particles. One can see how the averaged ring goes from very fuzzy to having well defined edges, indicating that the data set is nearly centered.

Crosscorrelating to a global average is but one variation on this theme. Similar methods also use an external model or a rotational average of the particle itself as the centering reference. Unfortunately, it can be difficult to obtain a reasonable external refernce, so a global average or a rotational average are most often used.

## Aligning the Particles  Figure 2. An unrotated (left) and rotated plus shifted (right) particle pair.

Once particles have been picked and centered, they need to be aligned, ordered, and classified for the reconstruction. These steps provide information on the relationships between different particles in a data set. The information can then be used towards putting together a three-dimensional model. For example, in the random conical tilt reconstruction method, the actual reconstruction is done from the tilted particles, but the alignment and classification comes from the untilted images. By determining the angles between untilted and tilted particles, it is a straighforward transformation to go from one coordinate plane to the other in later steps.  Figure 3. ACFs of an unrotated (left) and rotated plus shifted (right) particle pair.

Alignment involves placing the images of particles into a similar orientation on the screen. In the simplest case, two particles are just shifted (by [x,y]) and rotated (by [φ]) relative to each other in the plane of the image (Figure 2). Correlation, a method for calculating similarities, is used to determine the [x,y,φ] parameters. The correlation can be done either between images in the data set (crosscorrelation: CC) or by comparing an image to itself (autocorrelation: AC). Usually a mixture of the two is used, as described below.

For this example, two similar objects will be aligned. The only difference between them is an [x,y] shift and a [φ] rotation (Figure 2). The autocorrelation function (ACF) is independent of translation, so in this case, the two autocorrelation functions (ACF) will just be rotated by the angle [φ]. The offset angle is computed by doing an angular correlation search of the two separate ACFs. When this angle is calculated, one of the two original particles is then rotated by the offset angle, in order to align it with the other particle (Figure 3).  Figure 4. An unshifted (left) and shifted (right) particle pair.

With the two particles now in the same orientation, they need to be shifted in-plane to overlap. Crosscorrelation (CC; like autocorrelation but between two images) coupled with a peak search of the CCF similarly provides the necessary [x,y] shift values (Figure 4 and Figure 5). Then, one of the two particles is shifted by [x,y] to get a correct overlap. If needed for future study, the aligned images can be summed and averaged (see Frank 1996 Fig 3.11e). This procedure is repeated for all particles that have a similar orientation.  Figure 5. ACF (left) and CCF (right) of a shifted, unrotated particle.

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Classifying Particles