# Radon

#

Salvatore Lanzavecchia, Francesca Cantele and Pier Luigi Bellon

(Dip. Chimica Strutturale e Stereochimica Inorganica, Via Venezian 21, 20133 Milano, Italy)

(email: francesca.cantele@unimi.it)

Bernard Heymann

(Rm 1515, 50 South Dr., NIH, Bethesda, MD, 20892, USA)

(email: bernard_heymann@nih.gov)

2018-01-09

The Radon package is now fully incorporated into the Bsoft package

The Radon package provides the following two programs:
1) radon: Forward and backward Radon transformation, and POCS filtering of a Radon transform.
2) radonrecon: Reconstruction of a Radon transform from 2D projection images.

## Usage:

Typing the program name only produces a list of options. There are defaults for most of the options.

All images must be square for 2D and cubic for 3D, with the edges powers of 2. If the image size is not a power of 2, it will be resized to the next power of 2. A Radon transform is an array of size radii * angles * angles, where the radial size (NCOL) is the same as the edge of the image, and the number of angles (NTHETA) covers the range 0 - 2*PI. Because half of this angular range is redundant, a smaller transform is calculated with the number of angles covering 0 - PI equal to the image edge size (NTHETA/2 = NCOL).The POCS filter imposes consistency on a Radon transform and can be used to fill in missing points.

### 1) radon

    	radon [options] input.img output.img

There are three actions defined to do the forward and backward transformations, and the POCS filtering. One or more of these actions must be used to get output:

1. To compute a Radon transform:
   radon -v 7 -forward 4 -kernel 11,2 ico.map ico.rad.map
2. To compute a backward Radon transform:
   radon -v 7 -backward 4 -kernel 15,4 ico.rad.map ico.img.map
3. To filter a Radon transform:
   radon -v 7 -pocs 5,7 -mask ico.mask.map ico.rad.map ico.radf.map
4. To do all three operations with one command line:
   radon -v 7 -for 4 -pocs 10,1 -back 4 -ker 11,2 ico.map ico.img.map

### 2) radonrecon

    	radonrecon [options] input.star [input.star]

All the required parameters are obtained from the STAR file, including the projection image files (with paths). The origins and orientations of the projection images must be specified correctly in the STAR file, following the conventions in Bsoft. The STAR file also incorporates a mechanism to indicate selected projection images, which can be manipulated using several Bsoft programs. Symmetry-related views are calculated for each projection image when it is incorporated into the reconstruction.

    	radonrecon -v 7 -origin 32,32,32 -mask ico.mask.map -out out.star -rec ico.rec.map -sym I90 ico.star

A mask image is calculated for the angular coverage of orientation space and used to weigh the reconstruction. This image can be used with the POCS filter to fill in missing points in the reconstruction.

The real space map is then calculated (shown below):

    	radon -v 7 -backward 4 -kernel 15,4 ico.rec.map ico.real.map

## Conventions:

The conventions in the Radon package follow those of the Bsoft package.

Euler angles are defined with respect to the view vector {x,y,z} and the rotation angle {a} around this vector:

phi = atan(y, x)
theta = acos(z)
psi = a - phi

The origin for 2D and 3D images is defined as its offset (in pixels) from the first pixel (voxel) in the image array and typically coincides with symmetry origin for symmetric objects. Shifts are calculated as the difference between the projection image origin and the reconstruction origin.

Symmetry notation:

Symmetry | Notation | Examples
Cyclic | C<n> | C2, C3, C4 ...
Dihedral | D<n> | D2, D3, D4 ...
Tetrahedral | T | T
Octahedral | O | O
Icosahedral | I | I
(alternative) | I90 |

The alternative I90 notation for icosahedral objects indicates a 90 degree rotation of the standard icosahedral view and is adopted by some electron microscopy packages.

## References:

Lanzavecchia S., Bellon P.L. and Radermacher M. "Fast and accurate three dimensional reconstruction from projections with random orientations via Radon transforms." J. Struct. Biol. 128, 152-164, 1999.

Lanzavecchia S., Cantele F., Radermacher M. and Bellon P.L. "Symmetry embedding in the reconstruction of macromolecular assemblies via the discrete Radon transform." J. Struct. Biol., 137, 259-272, 2002.