Conventions

Multi-dimensional arrays

• Volume data is stored in x-y-z order (from major to minor).
• Projection data is stored in row-column order (from major to minor).

Slice orientation

We need a convention for representing the orientation of a slice. The orientation is inside volume space and is completely independent from the number of pixels (i.e. the ‘size’ of an individual pixel is implied by the bounding square of a slice). We represent the orientation as 9 real numbers \((a, b, \ldots, i)\) so that:

\[\begin{split}\begin{equation} \begin{pmatrix} a & d & g \\ b & e & h \\ c & f & i \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_s \\ y_s \\ 1 \end{pmatrix} = \begin{pmatrix} x_w \\ y_w \\ z_w \\ 1 \end{pmatrix} \end{equation}\end{split}\]

Where the vector \(\vec{x}_s = (x_s, y_s)\) lives inside a slice (i.e. the normalized pixel coordinates of a slice, in the interval \([0, 1]\)), and where \(\vec{x}_w = (x_w, y_w, z_w)\) lives inside the volume geometry at the correct place. The pixel coordinates of a slice have the following convention:

\[\begin{split}\begin{equation} \begin{pmatrix} (0, m) & \cdots & (n, m) \\ \vdots & \ddots & \vdots \\ (0, 0) & \cdots & (n, 0) \end{pmatrix} \end{equation}\end{split}\]

i.e. we start counting from the bottom-left and use a standard cartesian \(xy\) convention.

Using this convention, the vector \(\vec{b} = (g, h, i)\) is the base point of the slice in world space (i.e. the world coordinates of the bottom left point of a slice). \(\vec{x} = (a,b,c)\) is the direction in world space corresponding to the \(x\) direction of the slice, and \(\vec{y} = (d, e, f)\) corresponds to the \(y\) direction.

Acquisition geometries

The fields of the acquisition geometry packets:

• tomop.parallel_beam_geometry_packet
• tomop.parallel_vec_geometry_packet
• tomop.cone_beam_geometry_packet
• tomop.cone_vec_geometry_packet

follow the conventions of the constructors of 3D Geometries of the ASTRA Toolbox. See http://www.astra-toolbox.com/docs/geom3d.html for an overview.