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This paper presents a nonlinear image registration algorithm based on the setting of Large Deformation Diffeomorphic Metric Mapping (LDDMM), but with a more efficient optimisation scheme — both in terms of memory required and the number of iterations required to reach convergence. Rather than perform a variational optimisation on a series of velocity fields, the algorithm is formulated to use a geodesic shooting procedure, so that only an initial velocity is estimated. A Gauss–Newton optimisation strategy is used to achieve faster convergence. The algorithm was evaluated using freely available manually labelled datasets, and found to compare favourably with other inter-subject registration algorithms evaluated using the same data.

This technology report describes the longitudinal registration approach that we intend to incorporate into SPM12. It essentially describes a group-wise intra-subject modeling framework, which combines diffeomorphic and rigid-body registration, incorporating a correction for the intensity inhomogeneity artifact usually seen in MRI data. Emphasis is placed on achieving internal consistency and accounting for many of the mathematical subtleties that most implementations overlook. The implementation was evaluated using examples from the OASIS Longitudinal MRI Data in Non-demented and Demented Older Adults.

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.

In the context of large deformations by diffeomorphisms , we propose a new diffeomorphic registration algorithm for 3D images that performs the optimization directly on the set of geodesic flows. The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics. Since the initial momentum has proven to be a key tool for statistics on shape spaces, our algorithm enables more reliable statistical comparisons for 3D images. Our proposed algorithm is a gradient descent on the initial momentum, where the gradient is calculated using standard methods from optimal control theory. To improve the numerical efficiency of the gradient computation, we have developed an integral formulation of the adjoint equations associated with the geodesic equations. We then apply it successfully to the registration of 2D phantom images and 3D cerebral images. By comparing our algorithm to the standard approach of Beg et al. (Int. J. Comput. Vis. 61:139–157, 2005 ), we show that it provides a more reliable estimation of the initial momentum for the optimal path. In addition to promising statistical applications, we finally discuss different perspectives opened by this work, in particular in the new field of longitudinal analysis of biomedical images.

Purpose Diffeomorphic image registration is essential in many medical imaging applications. Several registration algorithms of such type have been proposed, but primarily for intra-contrast alignment. Currently, efficient inter-modal/contrast diffeomorphic registration, which is vital in numerous applications, remains a challenging task. Methods We proposed a novel inter-modal/contrast registration algorithm that leverages Robust PaTch-based cOrrelation Ratio metric to allow inter-modal/contrast image alignment and bandlimited geodesic shooting demonstrated in Fourier-Approximated Lie Algebras (FLASH) algorithm for fast diffeomorphic registration. Results The proposed algorithm, named DiffeoRaptor, was validated with three public databases for the tasks of brain and abdominal image registration while comparing the results against three state-of-the-art techniques, including FLASH, NiftyReg, and Symmetric image Normalization (SyN). Conclusions Our results demonstrated that DiffeoRaptor offered comparable or better registration performance in terms of registration accuracy. Moreover, DiffeoRaptor produces smoother deformations than SyN in inter-modal and contrast registration. The code for DiffeoRaptor is publicly available at https://github.com/nimamasoumi/DiffeoRaptor .

We present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane, targeted at robust registration between curves that require large deformations. This condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the Hamiltonian equations of motion; this means that each iteration of the nonlinear optimisation algorithm produces a geodesic (up to numerical discretisation). We ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition (conjugate momentum is normal to the curve). Explicitly introducing and solving for a reparameterisation variable allows the use of a point-to-point matching condition. The equations are discretised using the variational particle-mesh method. We provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations, to show that it is a viable, efficient and accurate method for obtaining geodesics between curves.