Joint T1 and Brain Fiber Log-Demons Registration Using Currents to Model Geometry

International audienceWe present an extension of the diffeomorphic Geometric Demons algorithm which combines the iconic registration with geometric constraints. Our algorithm works in the log-domain space, so that one can efficiently compute the deformation field of the geometry. We represent the shape of objects of interest in the space of currents which is sensitive to both location and geometric structure of objects. Currents provides a distance between geometric structures that can be defined without specifying explicit point-to-point correspondences. We demonstrate this framework by registering simultaneously T1 images and 65 fiber bundles consistently extracted in 12 subjects and compare it against non-linear T1, tensor, and multi-modal T1+ Fractional Anisotropy (FA) registration algorithms. Results show the superiority of the Log-domain Geometric Demons over their purely iconic counterparts

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome

Symmetric Log-Domain Diffeomorphic Registration: A Demons-based Approach

pmid 18979814International audienceModern morphometric studies use non-linear image registration to compare anatomies and perform group analysis. Recently, log-Euclidean approaches have contributed to promote the use of such computational anatomy tools by permitting simple computations of statistics on a rather large class of invertible spatial transformations. In this work, we propose a non-linear registration algorithm perfectly fit for log-Euclidean statistics on diffeomorphisms. Our algorithm works completely in the log-domain, i.e. it uses a stationary velocity field. This implies that we guarantee the invertibility of the deformation and have access to the true inverse transformation. This also means that our output can be directly used for log-Euclidean statistics without relying on the heavy computation of the log of the spatial transformation. As it is often desirable, our algorithm is symmetric with respect to the order of the input images. Furthermore, we use an alternate optimization approach related to Thirion's demons algorithm to provide a fast non-linear registration algorithm. First results show that our algorithm outperforms both the demons algorithm and the recently proposed diffeomorphic demons algorithm in terms of accuracy of the transformation while remaining computationally efficient

Parallel Transport with Pole Ladder: Application to Deformations of time Series of Images

Abstract. 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 e cient and simple method for the parallel transport of diffeomorphic deformations parameterized by tangent velocity elds. 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 ine cient 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 a ected by Alzheimer's disease.

Joint T1 and Brain Fiber Diffeomorphic Registration Using the Demons

International audienceNon-linear image registration is one of the most challenging task in medical image analysis. In this work, we propose an extension of the well-established diffeomorphic Demons registration algorithm to take into account geometric constraints. Combining the deformation field induced by the image and the geometry, we define a mathematically sound framework to jointly register images and geometric descriptors such as fibers or sulcal lines. We demonstrate this framework by registering simultaneously T 1 images and 50 fiber bundles consistently extracted in 12 subjects. Results show the improvement of fibers alignment while maintaining, and sometimes improving image registration. Further comparisons with non-linear T 1 and tensor registration demonstrate the superiority of the Geometric Demons over their purely iconic counterparts

Template Estimation for Large Database: A Diffeomorphic Iterative Centroid Method Using Currents

International audienceComputing a template in the Large Deformation Diffeomorphic Metric Mapping framework is a key step for the shape analysis of anatomical structures, but can lead to very computationally expensive algorithms in the case of large databases. We present an iterative method which quickly provides a centroid of the population in shape space. This centroid can be used as a rough template estimate or as initialization for template estimation methods

Spatially-Varying Metric Learning for Diffeomorphic Image Registration: A Variational Framework

International audienceThis paper introduces a variational strategy to learn spatially-varying metrics on large groups of images, in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. Spatially-varying metrics we learn not only favor local deformations but also correlated deformations in different image regions and in different directions. In addition, metric parameters can be efficiently estimated using a gradient descent method. We first describe the general strategy and then show how to use it on 3D medical images with reasonable computational ressources. Our method is assessed on the 3D brain images of the LPBA40 dataset. Results are compared with ANTS-SyN and LDDMM with spatially-homogeneous metrics

Left-invariant metrics for diffeomorphic image registration with spatially-varying regularisation

International audienceWe present a new framework for diffeomorphic image registration which supports natural interpretations of spatially-varying metrics. This framework is based on left-invariant diffeomorphic metrics (LIDM) and is closely related to the now standard large deformation diffeomorphic metric mapping (LDDMM). We discuss the relationship between LIDM and LDDMM and introduce a computationally convenient class of spatially-varying metrics appropriate for both frameworks. Finally, we demonstrate the effectiveness of our method on a 2D toy example and on the 40 3D brain images of the LPBA40 dataset

A Log-Euclidean Framework for Statistics on Diffeomorphisms

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