Parallel Transport with Pole Ladder: Application to Deformations of Time Series of Images

Group-wise analysis of time series of images requires to compare observed longitudinal evolutions. In medical imaging, longitudinal anatomical changes can be modeled by using deformations resulting from the non-rigid registration of follow-up images. The comparison of longitudinal trajectories is therefore the transport of longitudinal deformations in a common reference frame. We previously showed that the Schild’s Ladder is an efficient and simple method for the parallel transport of diffeomorphic deformations parameterized by tangent velocity fields. The Schild’s Ladder is based on the construction of a geodesic parallelogram. The base vertices of the parallelogram are the pairs of follow-up images and another vertex is the reference frame. By building the geodesic diagonals of the parallelogram, Schild’s Ladder computes the missing vertex which corresponds to the transported follow-up image. However, Schild’s Ladder may be inefficient in case of time series of multiple time points, in which the computation of the geodesic diagonals is required several times. In this paper we propose a new algorithm, the Pole Ladder, in which one diagonal of the parallelogram is the baseline-to-reference frame geodesic. This way we have to compute only one diagonal for each time point along the curve. In this work we show that the transport of the Pole ladder and the Schild’s Ladder are equivalent. Moreover, we show how the Pole ladder can be succesfully applied to the clinical problem of the measurement of the longitudinal atrophy progression in the brain for a group of patients affected by Alzheimer’s disease.