- Gary E. Christensen ,
University of Iowa, USA

*Inverse Consistent Image Registration and Evaluation*This talk will discuss Transitive Inverse-Consistent Manifold Registration (TICMR), Boundary-Constrained Inverse Consistent Image Registration (BICIR), and the Non-Rigid Image Registration Evaluation Project (NIREP).

The TICMR method jointly estimates correspondence maps between groups of three manifolds embedded in a higher dimensional image space while minimizing inverse consistency and transitivity errors. Registering three manifolds at once provides a means for minimizing the transitivity error which is not possible when registering only two manifolds. TICMR is an iterative method that uses the closest point projection operator to define correspondences between manifolds as they are non-rigidly registered.

The BICIR method performs boundary constrained intensity based image registration by combining surface correspondence with intensity based registration. The method registers region inside an object of interest and ignores everything outside the object. This eliminates the interference caused by surrounding regions due to the regularization constraints and the boundary conditions of the image. The boundaries of the two objects are first registered using a consistent boundary registration technique. This provides the boundary conditions, which are used to compute the displacement over the object using the Element Free Galerkin Method (EFGM). The EFGM solution is used as an initialization and is fine-tuned using the intensity information inside the object.

Many non-rigid image registration methods have been developed, but are especially difficult to evaluate since point-wise inter-image correspondence is usually unknown, i.e., there is no ``Gold Standard'' to evaluate performance. The Non-rigid Image Registration Evaluation Project (NIREP) has been started to develop, establish, maintain, and endorse a standardized set of relevant benchmarks and metrics for performance evaluation of non-rigid image registration algorithms.

- James Gee ,
University of Pennsylvania, USA

*Symmetric Image Normalization in the Diffeomorphic Space*Medical image analysis based on diffeomorphisms (differentiable one to one and onto maps with differentiable inverse) has placed computational analysis of anatomy and physiology on firm theoretical ground. We detail our approach to diffeomorphic computational anatomy while highlighting both theoretical and practical benefits. We first introduce the metric used to locate geodesics in the diffeomorphic space. Second, we give a variational energy that parameterizes the image normalization problem in terms of a geodesic diffeomorphism, enabling a fundamentally symmetric solution. This approach to normalization is extended for optimal template population studies using general imaging data. Finally, we show how the temporal parameterization and large deformation capabilities of diffeomorphisms make them appropriate for longitudinal analysis, particularly of neurodegenerative data.

- Xavier Pennec ,
INRIA, Sophia Antipolis, France

*Statistical Computing on Manifolds for Computational Anatomy*Computational anatomy is an emerging discipline that aim at analysing and modeling the biological variability of the human anatomy. The goal in not only to model the representative normal shape (the atls) and its normal variations among a poulation, but also discover morphological differences between normal and pathological populations, and possibly to detect, model and classify the pathologies from structural anomalities. To reach this goal, the method is to identify anatomically representative geometric features (points, tensors, curves, surfaces, volume transformations), and to describe their statistical distribution. This can be done for instance via a mean shape and covariance structure after a group-wise matching. Then, in order to compare populations, we need to compare feature distributions and to test for statistical differences.

Unfortunately, geometric features often belong to manifolds that are not vector spaces. Based on a Riemannian manifold structure, we previously develop a consistent framework for statistical computing on manifolds which proves to be also usefull for a number of more classical image analysis problems (e.g. DTI processing). For computational anatomy, we used this framework to model the brain variability from a dataset of lines on the cerebral cortex. As a result, we obtained a dense 3D variability map which can be seen as the diagonal elements of the Green's function of the Brain accross subjects. We will first present new results which extend this modeling with non-diagonal element by computing significantly correlated regions in the brain. Finally, we will discuss some recent methods for computing statistics on diffeomorphisms and show how the computational advances they bring practically improves non-linear registration algorithm.

- Brad Davis , Kitware / UNC, USA

Sarang Joshi, University of Utah, USA

*Simple statistics on Interesting Spaces: Regression Analysis on Manifolds for Computational Anatomy*Regression analysis is a powerful tool for the study of changes in a dependant variable as a function of an independent regressor variable. When the underlying process can be modeled by parameters in a Euclidean space, classical regression techniques are applicable and have been studied extensively. However, recent work suggests that attempts to describe anatomical shapes using

*flat Euclidean spaces*undermines our ability to represent natural biological variability. In this talk I will develop a method for regression analysis of general, manifold-valued data. Specifically, we extend Nadaraya-Watson kernel regression by recasting the regression problem in terms of Frechet expectation. Although this method is quite general, our driving problem is the study anatomical shape-change as a function of age from random-design image data.