Rotation of 3D volumes by Fourier-interpolated shears

Algorithms for rotation of 2D images by multiple shearing transformations are well known; algorithms which use as few as four shears to perform an arbitrary 3D rotation on a 3D volume have also been described. By using Fourier transform methods to implement these shears, rotations can be performed completely reversibly and without loss of information. This is of great utility when the rotated data are used as input to statistical calculations, for example in human brain imaging. In general, six different patterns of four shears can be used to implement a given 3D rotation. We examine the mathematical and implementation details of these rotation algorithms. This paper provides a classification of these patterns, demonstrates that a consistent scheme must be used to select shear patterns for various rotations, and presents several such consistent schemes.