Diffeomorphic image registration is an essential procedure in many MRI analysis tasks, such as atlas building and segmentation, anatomical variability analysis, functional magnetic resonance image (fMRI) analysis, and so on. Diffeomorphism (one-one and smooth mapping) under large deformation can be achieved by temporally integrating smooth vector fields which are usually built as stationary or non-stationary ones.

Under non-stationary parameterization, smooth vector fields vary continuously in time, and the solution lies within full diffeomorphism space, which may benefit the mapping of highly complicated cortical structures. The representative method of this kind is the large deformation diffeomorphic metric mapping (LDDMM) [1] that has been shown as a powerful approach to characterize anatomical variabilities [2, 3], but its high computational complexity limits its application in large scale analysis.

On the other hand, stationary velocity parameterization has the advantage of implementation efficiency brought by the scaling and squaring procedure in deformation generation and has the ability to achieve satisfactory brain mapping accuracy. Though there have been some stationary approaches proposed, how to achieve a good trade-off between accuracy and efficiency is still a research issue worthy of further study. For instance, Hernandez et al. has introduced stationary velocity in context of LDDMM [4] and proposed a 1st-order approximation model for optimal velocity that only depends on the end-point deformation information, which showed much faster speed than LDDMM with comparable accuracy in terms of residual. But this model is just valid under small deformation setting, hence its ability to handle large deformation mapping is in question.

Some other stationary approaches [5] including DARTEL in SPM package adopted an integral model of optimal velocity to achieve validity under large deformation setting. Computation of this integral model relies on deformations uniformly sampled in time interval [0,1], which is similar to the operations in time-varying approach, so efficiency brought by the scaling and squaring procedure is compromised, and heavy computational load is incurred in velocity gradient computation, and this problem becomes more protrude for large size volumes.

Aiming to find a better solution to improve the efficiency of stationary parameterization while achieving its validity for large deformation, we have proposed a new simplified model for optimal stationary velocity. This model is derived by representing temporal integration form of transformation variation via a single point model, e.g. the middle time point, according to mean value theorem and smoothness of transformation. By this substitution, gradient computation complexity is downgraded to logarithmic level O(log_{2}N), as opposed to linear complexity O(N) for an integral model and non-stationary approach, where N is the number of time sampling points.

This method (termed DMMS) was evaluated on both synthesized and real brain image data, and it exhibited almost the same registration accuracy as a stationary integral model and non-stationary approach, while reduced time cost a lot. For instance, on synthesized MR brain images, DMMS achieved dice around 0.82 on white/gray matter segmentation, and 0.7 on CSF, which was best among the methods in comparison, including well-known brain registration methods such as diffeomorphic demons and DARTEL. For a typical brain registration task (resolution:1x1x1mm^{3}), the running time of DMMS was about 1.5 hours on a 3.4GHz dual-core CPU, which saved more than 30 minutes compared to the stationary integral model, and was several times faster than non-stationary approach. This model also showed a clear advantage over the first order approximation model [4] in handling large deformation and achieved comparable speed. It showed that dices by 1st-order model were 2~3% lower than DMMS.

In this study, we also comprehensively compared the performance of DMMS and non-stationary approach and found they resulted in similar deformation fields and registration residuals, despite that their implementations are quite different. It proves that DMMS can maintain the advantage of non-stationary approach while greatly improving efficiency. Based on these observations, we believe DMMS will be an efficient tool for MRI analysis and has promising application in large scale analysis.

These findings are described in the article entitled Efficient diffeomorphic metric image registration via stationary velocity, recently published in the *Journal of Computational Science*.

**References:**

- Beg, M.F., et al., Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 2005. 61(2): p. 139-157.
- Yang, X., et al., Evolution of hippocampal shapes across the human lifespan. Hum Brain Mapp, 2013. 34(11): p. 3075-85.
- Qiu, A. and M.I. Miller, Multi-structure network shape analysis via normal surface momentum maps. Neuroimage, 2008. 42(4): p. 1430-8.
- Hernandez, M., M.N. Bossa, and S. Olmos, Registration of Anatomical Images Using Paths of Diffeomorphisms Parameterized with Stationary Vector Field Flows. International Journal of Computer Vision, 2009. 85(3): p. 291-306.
- Yang, X.F., et al., Diffeomorphic Metric Landmark Mapping Using Stationary Velocity Field Parameterization. International Journal of Computer Vision, 2015. 115(2): p. 69-86.