A novel family of geometrical transformations: Polyrigid transformations. Application to the registration of histological slices

We present in this report a novel kind of geometrical transformations, which we have named polyrigid. Within their framework, it is possible to define local rigid deformations in a given number of simple regions, while simultanously guaranteeing the smoothness and invertibility of the global transformation. Entirely parametric, this new type of tool is highly suitable for inference, and it is successfully applied to the non-rigid registration of histological slices. These general transformations are a nice alternative to classical B-Spline transformations (which do not guaranty invertibility). In future work, other applications will be considered, for instance in 3D registration

Bi-invariant Means in Lie Groups. Application to Left-invariant Polyaffine Transformations

In this work, we present a general framework to define rigorously a novel type of mean in Lie groups, called the bi-invariant mean. This mean enjoys many desirable invariance properties, which generalize to the non-linear case the properties of the arithmetic mean: it is invariant with respect to left- and right-multiplication, as well as inversion. Previously, this type of mean was only defined in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups such as the group of rotations. But Riemannian bi-invariant metrics do not always exist. In particular, we prove in this work that such metrics do not exist in any dimension for rigid transformations, which form but the most simple Lie group involved in bio-medical image registration. To overcome the lack of existence of bi-invariant Riemannian metrics for many Lie groups, we propose in this article to define bi-invariant means in any finite-dimensional real Lie group via a general barycentric equation, whose solution is by definition the bi-invariant mean. We show the existence and uniqueness of this novel type of mean, provided the dispersion of the data is small enough, and the convergence of an efficient iterative algorithm for computing this mean has also been shown. The intuition of the existence of such a mean was first given by R.P.Woods (without any precise definition), along with an efficient algorithm for computing it (without proof of convergence), in the case of matrix groups. In the case of rigid transformations, we give a simple criterion for the general existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. Interestingly, for general linear transformations, we show that similarly to the Log-Euclidean mean, which we proposed in recent work, the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is exactly equal to the geometric mean of the determinants of the data. Last but not least, we use this new type of mean to define a novel class of polyaffine transformations, called left-invariant polyaffine, which allows to fuse local rigid or affine components arbitrarily far away from the identity, contrary to Log-Euclidean polyaffine fusion, which we have recently introduced

Joint Estimation and Smoothing of Clinical DT-MRI with a Log-Euclidean Metric

Diffusion tensor MRI is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data has to be acquired rapidly at the detriment of the image quality. We propose a new variational framework that specifically targets low quality DT-MRI. The hypothesis of an additive Gaussian noise on the images leads us to estimate the tensor field directly on the image intensities. To further reduce the influence of the noise, we optimally exploit the spatial correlation by adding to the estimation an anisotropic regularization term. This criterion is easily optimized thanks to the use of the recently introduced Log-Euclidean metrics. Results on real clinical data show promising improvements of fiber tracking in the brain and we present the first successful attempt, up to our knowledge, to reconstruct the spinal cord

Statistics on Diffeomorphisms in a Log-Euclidean Framework

International audienceIn 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 can be used 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

A Fast and Log-Euclidean Polyaffine Framework for Locally Affine Registration

Projet ASCLEPIOSIn this article, we focus on the parameterization of non-rigid geometrical deformations with a small number of flexible degrees of freedom . In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called Log-Euclidean polyaffine, which overcomes these defects. We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently Log-Euclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated. Last but not least, we show in this article that the Log-Euclidean polyaffine framework is implicitely based on a Log-Euclidean framework for rigid and affine transformations, which generalizes to linear transformations the Log-Euclidean framework recently proposed for tensors. We detail in the Appendix of this article the properties of this novel framework, which allows a straightforward and efficient generalization to linear transformations of classical vectorial tools, with excellent theoretical properties. In particular, we propose here a simple generalization to locally rigid or affine deformations of a visco-elastic regularization energy used for dense transformations

Fast and Simple Computations on Tensors with Log-Euclidean Metrics.

Computations on tensors, i.e. symmetric positive definite real matrices in medical imaging, appear in many contexts. In medical imaging, these computations have become common with the use of DT-MRI. The classical Euclidean framework for tensor computing has many defects, which has recently led to the use of Riemannian metrics as an alternative. So far, only affine-invariant metrics had been proposed, which have excellent theoretical properites but lead to complex algorithms with a high computational cost. In this article, we present a new familly of metrics, called Log-Euclidean. These metrics have the same excellent theoretical properties as affine-invariant metrics and yield very similar results in practice. But they lead to much more simple computations, with a much lighter computational cost, very close to the cost of the classical Euclidean framework. Indeed, Riemannian computations become Euclidean computations in the logarithmic domain with Log-Euclidean metrics. We present in this article the complete theory for these metrics, and show experimental results for multilinear interpolation, dense extrapolation of tensors and anisotropic diffusion of tensor fields

Statistical Computing on Non-Linear Spaces for Computational Anatomy

International audienceComputational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings

Traitement de données dans les groupes de Lie : une approche algébrique. Application au recalage non-linéaire et à l'imagerie du tenseur de diffusion

Recently, the need for rigorous frameworks for the processing of non-linear data has grown considerably in medical imaging. In this thesis, we propose several general frameworks to process various types of non-linear data, which all belong to Lie groups. To this end, we rely on the algebraic properties of these spaces. Thus, we propose a general processing framework for symmetric and positive-definite matrices, named Log-Euclidean, very simple to use and which has excellent theoretical properties. It is particularly well-adapted to the processing of diffusion tensor MRI. We also propose several frameworks, called polyaffine, to parameterize locally rigid or affine transformations, in a way that guarantees their invertibility. Their use is illustrated in the case of the locally rigid registration of histological slices and of the locally affine 3D registration of MRIs of the human brain. This led us to propose two general frameworks for computing statistics in finite-dimensional Lie groups: first the Log-Euclidean one, which generalizes our work on tensors, and second a framework based on the novel notion of bi-invariant mean, whose properties generalize to Lie groups those of the arithmetic mean. Finally, we generalize our Log-Euclidean framework to diffeomorphic geometrical transformations, which opens the way to a general and consistent framework for statistics in computational anatomy.Ces dernières années, le besoin de cadres rigoureux pour traiter des données non-linéaires s'est développé considérablement en imagerie médicale. Ici, nous avons proposé plusieurs cadres généraux pour traiter certains de ces types de données, qui appartiennent à des groupes de Lie. Pour ce faire, nous nous sommes appuyés sur les propriétés algébriques de ces espaces. Ainsi, nous avons présenté un cadre de traitement général pour les matrices symétriques définies positives, appelé log-euclidien, très simple à utiliser et avec d'excellentes propriétés théoriques ; il est particulièrement adapté au traitement des images de tenseurs de diffusion. Nous avons également proposé des cadres, dits polyaffines, pour paramétrer les transformations localement rigides ou affines, en garantissant leur inversibilité avec d'excellentes propriétés théoriques. Leur utilisation est illustrée avec succès dans le cas du recalage localement rigide de coupes histologiques et du recalage 3D localement affine d'IRMs du cerveau humain. Ce travail nous a menés à proposer deux cadres généraux nouveaux pour le calcul de statistiques dans les groupes de Lie en dimension finie : d'abord le cadre log-euclidien, qui généralise notre travail sur les tenseurs, et un cadre basé sur la notion nouvelle de moyenne bi-invariante, dont les propriétés généralisent celles de la moyenne arithmétique des espaces euclidiens. Enfin, nous avons généralisé notre cadre log-euclidien aux déformations géométriques difféomorphes afin de permettre un calclul simple des statistiques sur ces transformations, ce qui ouvre la voie à un cadre général et cohérent pour les statistiques en anatomie computationnelle

Polyrigid and Polyaffine Transformations: a Novel Geometrical Tool to Deal with Non-Rigid Deformations - Application to the Registration of Histological Slices

We describe in this paper a novel kind of geometrical transformations, named polyrigid and polyaffine. These transformations efficiently code for locally rigid or affine deformations with a small number of intuitive parameters. They can describe compactly large rigid or affine movements, unlike most free-form deformation classes. Very flexible, this tool can be readily adapted to a large variety of situations, simply by tuning the number of rigid or affine components and the number of parameters describing their regions of influence. The displacement of each spatial position is defined by a continuous trajectory that follows a differential equation which averages the influence of each rigid or affine component. We show that the resulting transformations are diffeomorphisms, smooth with respect to their parameters. We devise a new and flexible numerical scheme to allow a trade-off between computational efficiency and closeness to the ideal diffeomorphism. Our algorithms are implemented within the Insight Toolkit, whose generic programming style offers rich facilities for prototyping. In this context, we derive an effective optimization strategy of the transformations which demonstrates that this new tool is highly suitable for inference. The whole framework is exemplified successfully with the registration of histological slices. This choice is challenging, because these data often present locally rigid deformations added during their acquisition, and can also present a loss of matter, which makes their registration even more difficult. Powerful and flexible, this new tool opens up large perspectives, in non-rigid 3D rigid registration as well as in shape statistics