The last column of the matrix represents a translation (blue rectangle on right side). The upper-left 3 × 3 sub-matrix of the matrix shown above (blue rectangle on left side) represents a rotation transform, byt may also include scales and shears. ![]() Such a 4 by 4 matrix M corresponds to a affine transformation T() that transforms point (or vector) x to point (or vector) y. 3D Affine Transformation MatricesĪny combination of translation, rotations, scalings/reflections and shears can be combined in a single 4 by 4 affine transformation matrix: " M"), whereas scalars are written in italics (e.g. In the equations used in this chapter, variables representing vectors and matrices are written in bold font (e.g. The presented information is aimed towards advanced users who want to understand how position and orientation information is stored in matrices and how to convert transformation results from and to third party (neuroimaging) software. ![]() It will be described how sub-transformations such as scale, rotation and translation are properly combined in a single transformation matrix as well as how such a matrix is properly decomposed into elementary transformations that are useful e.g. This topic aims to provide knowledge about spatial transformations in general and how they are implemented in BrainVoyager, which is important to understand subsequent topics about coordinate systems used in BrainVoyager and relevant neuroimaging file formats. The topic describes how affine spatial transformation matrices are used to represent the orientation and position of a coordinate system within a "world" coordinate system and how spatial transformation matrices can be used to map from one coordinate system to another one. BrainVoyager v23.0 Spatial Transformation Matrices
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