Band-Limited Stokes Large Deformation Diffeomorphic Metric Mapping

Band-Limited Stokes Large Deformation Diffeomorphic Metric Mapping

Abstract: The class of registration methods proposed in the framework of Stokes Large Deformation Diffeomorphic Metric mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. Stokes-LDDMM methods are formulated as constrained variation problems, where the different physical models are imposed using the associated partial differential equations as hard constraints. The most significant limitation of Stokes-LDDMM framework is its huge computational complexity. The objective of this work is to promote the use of Stokes-LDDMM in Computational Anatomy applications with an efficient approximation of the original variation problem. Thus, we propose a novel method for efficient Stokes-LDDMM diffeomorphic registration. Our method poses the constrained variation problem in the space of band-limited vector fields and it is implemented in the GPU. The performance of Band-Limited Stokes-LDDMM has been compared and evaluated with original Stokes-LDDMM, EPDiff- LDDMM, and Band-Limited EPDiff-LDDMM. The evaluation has been conducted in 3D with the Non-Rigid Image Registration Evaluation Project database. Since the update equation in Stokes- LDDMM involves the action of low-pass filters, the computational complexity has been greatly alleviated with a modest accuracy lose.

We have obtained a competitive performance for some method configurations.

Overall, our proposed method may make feasible the extensive use of novel physically meaningful Stokes-LDDMM methods in different Computational Anatomy applications. In addition, our results reinforce the usefulness of bandlimited vector fields in diffeomorphic registration methods involving the action of low-pass filters in the optimization, even in algorithmically challenging environments such as Stokes- LDDMM. Existing system: On the other hand, there are clinical domains for which finding the biophysical model that leads to the most meaningful anatomical deformation is still an open question. Two relevant examples of such domains are intra-subject registration of longitudinal brain images in normal and diseased individuals and inter-subject registration of brain images from different individuals. In these cases, the diffeomorphic registration methods should impose a plausible physical model to the computed transformations. This is mostly approached regularizing the problem with the L2-norm of some differential expression associated with the physical model. Some other approaches augment the problem by introducing additional constraints from the partial differential equations involved in the definition of the physical model.

Proposed system: The class of diffeomorphic registration methods proposed in is especially interesting, where the physical partial differential equations are imposed using hard constraints. Extended from optical Stokes flow to the diffeomorphic setting, Stokes-LDDMM methods are formulated using a constrained variation problem.

Numerical optimization has been approached using gradient-descent and secondorder optimization in the form of inexact Newton-Krylov methods. The method of Lagrange multipliers is used to transform the constrained problem into an unconstrained one. The expressions of the gradient and the Hessian are typically obtained by formally computing the first and second variations of the Lagrangian with respect to the control, state, and ad joint variables. The constrained optimization approach provides the versatility to impose different physical models to the computed transformations by simply adding the partial differential equations

associated to the problem as hard constraints. However, Stokes-LDDMM methods exhibit a huge computational complexity. In fact, few 2D experiments were conducted in these works as proof of concepts. The first published approach to solving 3D image registration problems at a large scale consisted on a distributedmemory implementation running on the CPUs of a supercomputer cluster.

Advantages: Although the differentiability and invertibility of the transformations constitute crucial features for Computational Anatomy applications, the diffeomorphic constraint does not necessarily guarantee that a transformation computed with a given method is physically meaningful for the clinical domain of interest.

On the one hand, there are clinical domains where the underlying biophysical model of the transformation is known. In these cases, the diffeomorphic registration methods should be combined with priors of the particular model.

Among these clinical domains, important families of diffeomorphic registration methods have been applied to the estimation of healthy heart deformation in cardiac imaging series.

Disadvantages: As happens with deformable registration, diffeomorphic registration is a highly illposed problem. Ill-posedness means that a number of qualitatively different diffeomorphisms can provide plausible correspondences of the anatomical structures in the images.

This is mostly approached regularizing the problem with the L2-norm of some differential expression associated with the physical model .Some other approaches augment the problem by introducing additional constraints from the partial differential equations involved in the definition of the physical model , This physical constraint allows formulating the problem in the space of initial velocity fields.

This guarantees that the obtained transformations belong to geodesic paths of diffeomorphisms, which is desirable in important Computational Anatomy applications.

Modules: Physically meaningful registration methods: Although the differentiability and invertibility of the transformations constitute crucial features for Computational Anatomy applications, the diffeomorphic constraint does not necessarily guarantee that a transformation computed with a given method is physically meaningful for the clinical domain of interest. Current research in diffeomorphic registration methods is focused on the quest for the most sensible choice for each clinical domain. On the one hand, there are clinical domains where the underlying biophysical model of the transformation is known.

In these cases, the diffeomorphic registration methods should be combined with priors of the particular model . Among these clinical domains, important families of diffeomorphic registration methods have been applied to the estimation of healthy heart deformation in cardiac imaging series. The physical model requires that the local volume of the heart does not vary during the cardiac cycle (incompressible physical model). The incompressible physical model has also been relevant in optical flow estimation.

EPDiff-LDMM and Stokes-LDDMM: The first class of diffeomorphic image registration methods constrained to physical partial differential equations arose with the work of Younes et al. , followed by the proposals . In these works, the transformations are parameterized by time-varying velocity fields that satisfy the EPDiff equation, an evolution partial differential equation deriving from an application of the Euler-Poincare principle. The EPDiff equation can be found in the physical model of soliton dynamics driven by the Camassa-Holm equation. This physical constraint allows formulating the problem in the space of initial velocity fields. This guarantees that the obtained transformations belong to geodesic paths of diffeomorphisms, which is desirable in important Computational Anatomy applications. However, these methods require the computation of numerical solutions of partial differential equations that are both memory- and imeconsuming. The class of diffeomorphic registration methods proposed in is especially interesting, where the physical partial differential equations are imposed using hard constraints.

Diffeomorphic image registration:

In the last two decades, diffeomorphic image registration has arisen as a powerful paradigm for deformable image registration. Diffeomorphic registration methods compute differentiable and invertible transformations between the source and the target images. These transformations have become fundamental inputs in Computational Anatomy applications since they belong to spaces endowed with rigorous mathematical foundations for the computation of statistical atlases, population models, and growth models . As happens with deformable registration, diffeomorphic registration is a highly ill-posed problem. Ill-posedness means that a number of qualitatively different diffeomorphisms can provide plausible correspondences of the anatomical structures in the images. This justifies the vast literature on diffeomorphic registration methods with differences on diffeomorphism parameterization, regularizers, image similarity metrics, optimization methods, and additional constraints for representative examples).

Differentiation: The computation of differentials is approached using Fourier spectral methods as an alternative to commonly used finite difference approximations. The use of spectral differentiation and integration was introduced in diffeomorphic registration literature. Spectral methods allow solving ODEs and PDEs to high accuracy in simple domains for problems involving smooth data. During the development phase of the algorithms, we noticed that the use of spectral differentiation allowed resolving several numerical difficulties. Indeed, the use of spectral differentiation supposes working with the variables in Fourier domain. In consequence, the efficiency of methods formulated in the space of band-limited vector fields such as BL-Stk-LDDMM is further increased.