# Helical symmetry

## Symmetrizing a particle

### Helical operator syntax

Symmetrizing a helical particle can be done with the usual tool dsym:

``` vh = dsym(v,<operator>);
```

where operator represents any symmetry operator. The syntax for helical operators is:

``` h [<phi>,<dz>,<repetitions[optional]>,<rotational symmetry[optional]>]
```

Here:

• phi is the angular increment in degrees
• dz is the axial rise in pixels
• the number of repetitions (defaults to the maximum definible for a given volume).
Can be passed as 0 to force Dynamo to create a default value.
• additional Cn rotation to account for n-start helices.

Example:

``` vh = dsym(v,'h[15.6,4.7]');
vh = dsym(v,'h[15.6,4.7,5]');       % uses 5 repetitions
vh = dsym(v,'h[15.6,4.7,5,8]');    % uses 5 repetitions and adds a C8 symmetrization
vh = dsym(v,'h[15.6,4.7,0,8]');    % computes internally the maximal number of repetitions
```

## Using symmetrization in alignment projects

The simplest way is to input one operator with the syntax described #Helical operator syntax above in the Numerical settings area of the dcm GUI.

This allows to impose a given symmetrization. In order to estimate the symmetry, you will need to create a plugin.

## Estimating the helical symmetry of a density map

You can use the command dynamo_symmetry_scan.

This command operates by creating a matrix of symmetrization opertors, and comparing the transformed volumes with the original one. The transform that produces the highest cross correlation is considered the best match.

## Number of repetitions

Symmetrization of a volume is computed by applying onto the original volume the transformation (phi,dz) repeatedly in both directions a number of times (called repetitions). The transformed volumes are averaging together. For instance, 17 repetitions imply 8 subsequent applications of (phi,dz) to the original volume and 8 applications of -(phi,dz).

Each time (phi,dz) is applied, the transformed volume will have a "bottom" of dz layers of pixels that are undefined: (they correspond to the theoretically part of the infinite helix below our volume). The averaging procedure will discard contributions of such pixels. As a result, the central area of the symmetrized volume will be representing contributions of more transformed volumes, attaining thus a higher SNR.

If the user does not specify a number of repetitions, Dynamo will take the maximal number of repetitions for the given volume. If the volume is of size N pixels and the operator carries an axial rise of dz pixels, the maximal number of translations along the z axis is floor(N/dz)