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

Medical image analysis via Fréchet means of diffeomorphisms

The construction of average models of anatomy, as well as regression analysis of anatomical structures, are key issues in medical research, e.g., in the study of brain development and disease progression. When the underlying anatomical process can be modeled by parameters in a Euclidean space, classical statistical techniques are applicable. However, recent work suggests that attempts to describe anatomical differences using flat Euclidean spaces undermine our ability to represent natural biological variability. In response, this dissertation contributes to the development of a particular nonlinear shape analysis methodology. This dissertation uses a nonlinear deformable model to measure anatomical change and define geometry-based averaging and regression for anatomical structures represented within medical images. Geometric differences are modeled by coordinate transformations, i.e., deformations, of underlying image coordinates. In order to represent local geometric changes and accommodate large deformations, these transformations are taken to be the group of diffeomorphisms with an associated metric. A mean anatomical image is defined using this deformation-based metric via the Fréchet mean—the minimizer of the sum of squared distances. Similarly, a new method called manifold kernel regression is presented for estimating systematic changes—as a function of a predictor variable, such as age—from data in nonlinear spaces. It is defined by recasting kernel regression in terms of a kernel-weighted Fréchet mean. This method is applied to determine systematic geometric changes in the brain from a random design dataset of medical images. Finally, diffeomorphic image mapping is extended to accommodate extraneous structures—objects that are present in one image and absent in another and thus change image topology—by deflating them prior to the estimation of geometric change. The method is applied to quantify the motion of the prostate in the presence of transient bowel gas

Proceedings of the Second International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'08) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceThe goal of computational anatomy is to analyze and to statistically model the anatomy of organs in different subjects. Computational anatomic methods are generally based on the extraction of anatomical features or manifolds which are then statistically analyzed, often through a non-linear registration. There are nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behavior of intra-subject deformations. However, it is more difficult to relate the anatomies of different subjects. In the absence of any justified physical model, diffeomorphisms provide a general mathematical framework that enforce topological consistency. Working with such infinite dimensional space raises some deep computational and mathematical problems, in particular for doing statistics. Likewise, modeling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed (e.g. smooth left-invariant metrics, focus on well-behaved subspaces of diffeomorphisms, modeling surfaces using courants, etc.) The goal of the Mathematical Foundations of Computational Anatomy (MFCA) workshop is to foster the interactions between the mathematical community around shapes and the MICCAI community around computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop aims at being a forum for the exchange of the theoretical ideas and a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the very successful first edition of this workshop in 2006 (see http://www.inria.fr/sophia/asclepios/events/MFCA06/), the second edition was held in New-York on September 6, in conjunction with MICCAI 2008. Contributions were solicited in Riemannian and group theoretical methods, Geometric measurements of the anatomy, Advanced statistics on deformations and shapes, Metrics for computational anatomy, Statistics of surfaces. 34 submissions were received, among which 9 were accepted to MICCAI and had to be withdrawn from the workshop. Each of the remaining 25 paper was reviewed by three members of the program committee. To guaranty a high level program, 16 papers only were selected