Registration of Anatomical Images Using Paths of Diffeomorphisms Parameterized with Stationary Vector Field Flows

Abstract

Computational Anatomy aims for the study of variability in anatomical structures from images. Variability is encoded by the spatial transformations existing between anatomical images and a template selected as reference. In the absence of a more justified model for inter-subject variability, transformations are considered to belong to a convenient family of diffeomorphisms which provides a suitable mathematical setting for the analysis of anatomical variability. One of the proposed paradigms for diffeomorphic registration is the Large Deformation Diffeomorphic Metric Mapping (LDDMM). In this framework, transformations are characterized as end points of paths parameterized by time-varying flows of vector fields defined on the tangent space of a Riemannian manifold of diffeomorphisms and computed from the solution of the non-stationary transport equation associated to these flows. With this characterization, optimization in LDDMM is performed on the space of non-stationary vector field flows resulting into a time and memory consuming algorithm. Recently, an alternative characterization of paths of diffeomorphisms based on constant-time flows of vector fields has been proposed in the literature. With this parameterization, diffeomorphisms constitute solutions of stationary ODEs. In this article, the stationary parameterization is included for diffeomorphic registration in the LDDMM framework. We formulate the variational problem related to this registration scenario and derive the associated Euler-Lagrange equations. Moreover, the performance of the non-stationary vs the stationary parameterizations in real and simulated 3D-MRI brain datasets is evaluated. Compared to the non-stationary parameterization, our proposal provides similar results in terms of image matching and local differences between the diffeomorphic transformations while drastically reducing memory and time requirements.

Temporal diffeomorphic free-form deformation: Application to motion and strain estimation from 3D echocardiography

“…Beg et al (2005); Vercauteren et al (2007); Rueckert et al (2006); Hernandez et al (2009);Ashburner (2007); Avants et al (2008)) ensure a continuous and differentiable correspondence with continuous inverse between the features to register. They are therefore particularly well suited to handle medical image sequences as they preserve topology and orientation of the observed anatomical structures over time.…”

Discrete Ladders for Parallel Transport in Transformation Groups with an Affine Connection Structure

“…This is the role of the regularization term of the SVF registration algorithms [45,18] or of the spline parametrization of the SVF in [4,31]: this restricts the Lie algebra to the subalgebra of sufficiently regular velocity fields. The subgroup of diffeomorphisms considered is then generated by the flow of these stationary velocity fields and their finite composition.…”

A Multi-scale Kernel Bundle for LDDMM: Towards Sparse Deformation Description across Space and Scales

“…Besides LDDMM, many methods for non-rigid registration are in use today including elastic methods [11], parametrizations using static velocity fields [1,8] and the demons algorithm [13,16]. The deformable template model pioneered by Grenander in [7] and the flow approach by Christensen et al [4] was paramount in the development of LDDMM together with the theoretical contributions of Dupuis et al and Trouvé [6,14].…”