Homogeneous coordinates in motion correction

Purpose Prospective motion correction for MRI and other imaging modalities are commonly based on the assumption of affine motion, i.e., rotations, shearing, scaling and translations. In addition it often involves transformations between different reference frames, especially for applications with an external tracking device. The goal of this work is to develop a computational framework for motion correction based on homogeneous transforms. Theory and Methods The homogeneous representation of affine transformations uses 4 × 4 transformation matrices applied to four-dimensional augmented vectors. It is demonstrated how homogenous transforms can be used to describe the motion of slice objects during an MRI scan. Furthermore, we extend the concept of homogeneous transforms to gradient and k-space vectors, and show that the fourth dimension of an augmented k-space vector encodes the complex phase of the corresponding signal sample due to translations. Results The validity of describing motion tracking in real space and k-space using homogeneous transformations only is demonstrated on phantom experiments. Conclusion Homogeneous transformations allows for a conceptually simple, consistent and computationally efficient theoretical framework for motion correction applications. Magn Reson Med 75:274–279, 2016. © 2015 Wiley Periodicals, Inc.