The pose of the acquired image slices, tools, and other objects are defined by specifying a 3D Cartesian coordinate system (a.k.a. reference frame) for each object and transformations between them. The transformation is assumed to be rigid and each transformation is represented by 4x4 homogeneous transformation matrix. Each coordinate system is defined by its name, origin position, axis directions, and unit. These definitions must be completed for all coordinate systems and archived along with the corresponding configuration files to avoid any chance of misunderstandings.
One of the most frequently needed operations in image-guided intervention systems is to compute transformation between two arbitrary reference frames. Although this operation is very simple in theory – just a multiplication and inversion of selected matrices in a certain order – the implementation can be quite complex and error-prone in practice, when there are a large number of transformations. Hard-coded implementation of the computations also greatly reduces the flexibility of the software. The number of potentially needed transformations is high even for a simple case: a tracker with 3 tracked objects typically uses 7 coordinate systems (3 markers, 3 objects that the markers are attached to, 1 tracker), which leads to 42 different transformations.
Therefore, in Plus a common repository is used for storing all transformations as edges of a graph, where each vertex of the graph corresponds to a coordinate system (identified by its unique name). Plus can then compute transformations between any coordinate systems automatically: first the path (list of transformations) between the two coordinate systems (vertices) is searched in the graph, then the corresponding transformations are chained in the correct order and inverted as needed.
Coordinate systems naming convention
Plus uses a standardized naming convention for constructing transformation names. The name is constructed from the name of the coordinate system that it transforms from (e.g., 'FrameA'), the 'To' word, and the reference frame name it transforms to (e.g., 'FrameB'): FrameAToFrameB. To ensure unambigous interpretation of the transformation name, the reference frame names must not contain the 'To' word.
Transformation matrix definition
If coordinate values of a point are known in the 'FrameA' coordinate system and coordinates of the same point are needed in the 'FrameB' coordinate system: multiply the coordinates by the FrameAToToFrameB matrix from the left.
The transformation that computes coordinates of point P in FrameB ('To' coordinate system) from its coordinates in FrameA ('From' coordinate system):
FrameAToFrameBTransform = [Rxx] [Rxy] [Rxz] [Tx] [Ryx] [Ryy] [Ryz] [Ty] [Rzx] [Rzy] [Rzz] [Tz] 0 0 0 1
where
[ Position_in_FrameB_x ] = [ Rxx Rxy Rxz Tx ] * [Position_in_FrameA_x ] [ Position_in_FrameB_y ] [ Ryx Ryy Ryz Ty ] [Position_in_FrameA_y ] [ Position_in_FrameB_z ] [ Rzx Rzy Rzz Tz ] [Position_in_FrameA_z ] [ 1 ] [ 0 0 0 1 ] [ 1 ]
Unit of T... values is the same as the unit of the 'FrameB' coordinate system (as the Tx, Ty, and Tz values are multiplied by 1 and the result is a position in the To coordinate system).
Unit of R... values are the ratio of unit of the To coordinate system divided by the unit of the From coordinate system (as a position value in the From coordinate system's unit multiplied by Rij results in a position value in the To coordinate system's unit).
Note that the standardized naming convention is better than the some commonly used generic transformation names such as "CalibrationMatrix" or "ReferenceTransform", because when using such ad-hoc names the associated reference frames and transformation direction must be documented separately. This separate documentation may be missing, incomplete, or incorrect. The transformation names generated with the above described scheme are inherently correct and fully specify the transformation. Using a simple string as transformation name is also beneficial when the transformations are saved into file or transmitted to another system.
