Spatial Transformation of Motion and Deformation Fields Using Nonrigid Registration

Abstract

In this paper, we present a technique that can be used to transform the motion or deformation fields defined in the coordinate system of one subject into the coordinate system of another subject. Such a transformation accounts for the differences in the coordinate systems of the two subjects due to misalignment and size/shape variation, enabling the motion or deformation of each of the subjects to be directly quantitatively and qualitatively compared. The field transformation is performed by using a nonrigid registration algorithm to determine the intersubject coordinate system mapping from the first subject to the second subject. This fixes the relationship between the coordinate systems of the two subjects, and allows us to recover the deformation/motion vectors of the second subject for each corresponding point in the first subject. Since these vectors are still aligned with the coordinate system of the second subject, the inverse of the intersubject coordinate mapping is required to transform these vectors into the coordinate system of the first subject, and we approximate this inverse using a numerical line integral method. The accuracy of our numerical inversion technique is demonstrated using a synthetic example, after which we present applications of our method to sequences of cardiac and brain images.

A bi-ventricular cardiac atlas built from 1000+ high resolution MR images of healthy subjects and an analysis of shape and motion

“…This can be computed using numerical integration, as in (Rao et al, 2004). There are also other approaches to transport a vector field from one space to another, such as the parallel transport techniques described in (Lorenzi and Pennec, 2013;Qiu et al, 2009;Younes, 2007;Younes et al, 2008).…”

Schild’s Ladder for the Parallel Transport of Deformations in Time Series of Images

“…A drawback of this method is that the longitudinal deformation is fully combined to the inter-subject one. The method proposed by [10] uses the transformation conjugation (change of coordinate system) from the group theory in order to compose the longitudinal inter-subject deformation with the subject-totemplate one. As pointed in [3], this practice could potentially introduce variations in the transported deformation and relies on the inverse consistency of the estimated deformations, which can raise problems for large deformations.…”

Similarity Metrics for Groupwise Non-rigid Registration

“…The inverses of these deformations (calculated using a numerical scheme [15]) were applied to a slice of the MNI Brainweb image to create a population of 100 subjects, whose average shape is the original, undeformed slice. In addition, the MNI Brainweb image has probabilistic and ground truth (hard) segmentations of white matter (WM), grey matter (GM), cerebrospinal fluid (CSF) and background (BG) classes.…”