# 3D Geometries

In all 3D geometries, the coordinate system is defined around the reconstruction volume. The center of the reconstruction volume is the origin, and the sides of the voxels in the volume have length 1.

All dimensions in the projection geometries are relative to this unit length.

## Volume geometries

Create a 3D volume geometry:

```vol_geom = astra_create_vol_geom(n_rows, n_cols, n_slices);
```

Specify the extent of the 3D volume (note that rows are oriented along the Y axis, columns along the X axis and slices along the Z axis):

```vol_geom = astra_create_vol_geom(n_rows, n_cols, n_slices, min_x, max_x, min_y, max_y, min_z, max_z);
```

This can be used to control the voxel size, including specifying anisotropic voxels (note that the FDK algorithm does not currently support anisotropic voxels and will raise an exception).

## Projection geometries

parallel3d

```proj_geom = astra_create_proj_geom('parallel3d', det_spacing_x, det_spacing_y, det_row_count, det_col_count, angles);
```

Create a 3D parallel beam geometry.

• det_spacing_x: distance between the centers of two horizontally adjacent detector pixels

• det_spacing_y: distance between the centers of two vertically adjacent detector pixels

• det_row_count: number of detector rows in a single projection

• det_col_count: number of detector columns in a single projection

• angles: projection angles in radians

cone

```proj_geom = astra_create_proj_geom('cone',  det_spacing_x, det_spacing_y, det_row_count, det_col_count, angles, source_origin, origin_det);
```

Create a 3D cone beam geometry.

• det_spacing_x: distance between the centers of two horizontally adjacent detector pixels

• det_spacing_y: distance between the centers of two vertically adjacent detector pixels

• det_row_count: number of detector rows in a single projection

• det_col_count: number of detector columns in a single projection

• angles: projection angles in radians

• source_origin: distance between the source and the center of rotation

• origin_det: distance between the center of rotation and the detector array

parallel3d_vec

```proj_geom = astra_create_proj_geom('parallel3d_vec',  det_row_count, det_col_count, vectors);
```

Create a 3D parallel beam geometry specified by 3D vectors.

• det_row_count: number of detector rows in a single projection

• det_col_count: number of detector columns in a single projection

• vectors: a matrix containing the actual geometry.

Each row of vectors corresponds to a single projection, and consists of:

```( rayX, rayY, rayZ, dX, dY, dZ, uX, uY, uZ, vX, vY, vZ )
```
• ray : the ray direction

• d : the center of the detector

• u : the vector from detector pixel (0,0) to (0,1)

• v : the vector from detector pixel (0,0) to (1,0)

To illustrate, this is a matlab script to convert a single projection in a projection geometry of type “parallel3d” into such a 12-element row:

```% ray direction
vectors(i,1) = sin(proj_geom.ProjectionAngles(i));
vectors(i,2) = -cos(proj_geom.ProjectionAngles(i));
vectors(i,3) = 0;

% center of detector
vectors(i,4) = 0;
vectors(i,5) = 0;
vectors(i,6) = 0;

% vector from detector pixel (0,0) to (0,1)
vectors(i,7) = cos(proj_geom.ProjectionAngles(i)) * proj_geom.DetectorSpacingX;
vectors(i,8) = sin(proj_geom.ProjectionAngles(i)) * proj_geom.DetectorSpacingX;
vectors(i,9) = 0;

% vector from detector pixel (0,0) to (1,0)
vectors(i,10) = 0;
vectors(i,11) = 0;
vectors(i,12) = proj_geom.DetectorSpacingY;
```

This conversion is also available as a function in the toolbox:

```proj_geom_vec = astra_geom_2vec(proj_geom);
```

cone_vec

```proj_geom = astra_create_proj_geom('cone_vec',  det_row_count, det_col_count, vectors);
```

Create a 3D cone beam geometry specified by 3D vectors.

• det_row_count: number of detector rows in a single projection

• det_col_count: number of detector columns in a single projection

• vectors: a matrix containing the actual geometry.

Each row of vectors corresponds to a single projection, and consists of:

```( srcX, srcY, srcZ, dX, dY, dZ, uX, uY, uZ, vX, vY, vZ )
```
• src : the ray source

• d : the center of the detector

• u : the vector from detector pixel (0,0) to (0,1)

• v : the vector from detector pixel (0,0) to (1,0)

To illustrate, this is a matlab script to convert a single projection in a projection geometry of type “cone” into such a 12-element row:

```% source
vectors(i,1) = sin(proj_geom.ProjectionAngles(i)) * proj_geom.DistanceOriginSource;
vectors(i,2) = -cos(proj_geom.ProjectionAngles(i)) * proj_geom.DistanceOriginSource;
vectors(i,3) = 0;

% center of detector
vectors(i,4) = -sin(proj_geom.ProjectionAngles(i)) * proj_geom.DistanceOriginDetector;
vectors(i,5) = cos(proj_geom.ProjectionAngles(i)) * proj_geom.DistanceOriginDetector;
vectors(i,6) = 0;

% vector from detector pixel (0,0) to (0,1)
vectors(i,7) = cos(proj_geom.ProjectionAngles(i)) * proj_geom.DetectorSpacingX;
vectors(i,8) = sin(proj_geom.ProjectionAngles(i)) * proj_geom.DetectorSpacingX;
vectors(i,9) = 0;

% vector from detector pixel (0,0) to (1,0)
vectors(i,10) = 0;
vectors(i,11) = 0;
vectors(i,12) = proj_geom.DetectorSpacingY;
```