A probabilistic framework is presented that enables image registration, tissue classification, and bias correction to be combined within the same generative model. A derivation of a log-likelihood objective function for the unified model is provided. The model is based on a mixture of Gaussians and is extended to incorporate a smooth intensity variation and nonlinear registration with tissue probability maps. A strategy for optimising the model parameters is described, along with the requisite partial derivatives of the objective function.

Abstract. In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain. 1

Abstract. We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data. 1

Modern 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.

We propose an efficient non-parametric diffeomorphic image registration algorithm based on Thirion’s demons algorithm. In the first part of this paper, we show that Thirion’s demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. We provide strong theoretical roots to the different variants of Thirion’s demons algorithm. This analysis predicts a theoretical advantage for the symmetric forces variant of the demons algorithm. We show on controlled experiments that this advantage is confirmed in practice and yields a faster convergence. In the second part of this paper, we adapt the optimization procedure underlying the demons algorithm to a space of diffeomorphic transformations. In contrast to many diffeomorphic registration algorithms, our solution is computationally efficient since in practice it only replaces an addition of displacement fields by a few compositions. Our experiments show that in addition to being diffeomorphic, our algorithm provides results that are similar to the ones from the demons algorithm but with transformations that are much smoother and closer to the gold standard, available in controlled experiments, in terms of Jacobians.

We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.

Abstract—This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling’s 2 test to them, in a Log-Euclidean domain. In 2-D and 3-D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensor-derived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulative-value plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and large-scale studies of disease that use tensor-based morphometry. Index Terms—Brain, image analysis, Lie groups, magnetic resonance imaging (MRI), statistics. I.

Abstract. We propose a new methodology to analyze the anatomical variability of a set of longitudinal data (population scanned at several ages). This method accounts not only for the usual 3D anatomical variability (geometry of structures), but also for possible changes in the dynamics of evolution of the structures. It does not require that subjects are scanned the same number of times or at the same ages. First a regression model infers a continuous evolution of shapes from a set of observations of the same subject. Second, spatiotemporal registrations deform jointly (1) the geometry of the evolving structure via 3D deformations and (2) the dynamics of evolution via time change functions. Third, we infer from a population a prototype scenario of evolution and its 4D variability. Our method is used to analyze the morphological evolution of 2D profiles of hominids skulls and to analyze brain growth from amygdala of autistics, developmental delay and control children. 1 Methodology for Statistics on Longitudinal Data

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In large-deformation diffeomorphic metric mapping (LDDMM), the diffeomorphic matching of images are modeled as evolution in time, or a flow, of an associated smooth velocity vector field controlling the evolution. The initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image. Thus, methods such as principal component analysis (PCA) of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to smalldeformation assumption in the analysis of linear displacements. We apply this approach to a study of dementia of the Alzheimer type (DAT). The left hippocampus in the DAT group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality. Further, PCA of the initial momentum leads to correct classification of 12 out of 18 DAT subjects and 22 out of 26 control subjects.