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.
[1]
Nicholas Ayache,et al.
A Log-Euclidean Framework for Statistics on Diffeomorphisms
,
2006,
MICCAI.
[2]
Elsayed Ahmed,et al.
Quantum Group Approach to q-Special Functions
,
2000
.
[3]
Charles W. Misner,et al.
重力理論 : gravitation--古典力学から相対性理論まで, 時空の幾何学から宇宙の構造へ
,
2011
.
[4]
Nicholas Ayache,et al.
LCC-Demons: a robust and accurate dieomorphic registration algorithm
,
2013
.
[5]
Marco Lorenzi,et al.
Deformation-based morphometry of the brain for the development of surrogate markers in Alzheimer's disease
,
2012
.
[6]
Alain Trouvé,et al.
Diffeomorphisms Groups and Pattern Matching in Image Analysis
,
1998,
International Journal of Computer Vision.
[7]
Nicholas Ayache,et al.
LCC-Demons: A robust and accurate symmetric diffeomorphic registration algorithm
,
2013,
NeuroImage.
[8]
李幼升,et al.
Ph
,
1989
.
[9]
I. Holopainen.
Riemannian Geometry
,
1927,
Nature.
[10]
Lasse Riis Østergaard,et al.
Active Surface Approach for Extraction of the Human Cerebral Cortex from MRI
,
2006,
MICCAI.
[11]
W. Miller,et al.
Schild's Ladder Parallel Transport Procedure for an Arbitrary Connection
,
2000
.
[12]
Nicholas Ayache,et al.
Schild's Ladder for the Parallel Transport of Deformations in Time Series of Images
,
2011,
IPMI.
[13]
Alain Trouvé,et al.
Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
,
2005,
International Journal of Computer Vision.
[14]
M. Postnikov,et al.
Geometry VI: Riemannian Geometry
,
2001
.