Section: New Results
Log-Euclidean Processing of Tensors
Symmetric positive-definite matrices (or SPD matrices) of real numbers, also called here tensors by abuse of language, appear in many contexts. In medical imaging, their use has become common during the last ten years with the growing interest in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI or simply DTI). SPD matrices also provide a powerful framework to model the anatomical variability of the brain. More generally, they are widely used in image analysis, especially for segmentation, grouping, motion analysis and texture segmentation. They are also used intensively in mechanics, for example with strain or stress tensors. SPD matrices are also becoming a common tool in numerical analysis to generate adapted meshes to reduce the computational cost of solving partial differential equations (PDEs) in 3D.
Defining a complete operational framework to interpolate, restore, enhance images of tensors is necessary to fully generalize to the SPD case the usual statistical tools or PDEs on vector-valued images. Last year, we proposed a novel and general processing framework for tensors, called Log-Euclidean. It is based on Log-Euclidean Riemannian metrics, which have excellent theoretical properties, very close to those of the recently-introduced affine-invariant metrics and yield similar results in practice, but with much simpler and faster computations. This innovative approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms, which makes classical Euclidean processing algorithm particularly simple to recycle. Two journal articles have been published this year on this topic:  on the theoretical aspects of the Log-Euclidean framework for tensors and  for its application to diffusion tensor processing.
We also proposed a new methodology to analyze DT-MRI data sets of a rather low quality, typical of clinical acquisitions. We handle the Rician nature of the noise thanks to a Maximum Likelihood (ML) approach, combined with an anisotropic regularization term. We showed that this can correct for sides effect caused by the Rician noise: tensors tend to be smaller than normal. This work has been accepted for publication at ISBI this year  . An extended version, with a detailed quantitative analysis of 7 algorithms to estimate and regularize DT-MRI is currently submitted.
A statistical atlas of the cardiac fiber architecture
This work was funded in part by Siemens Corporate Research (NJ, USA).
While the main geometrical arrangement of myofibers has been known for decades, its variability between subjects and species still remains largely unknown. Understanding this variability is not only important for a better description of physiological principles but also for the planning of patient-specific cardiac therapies. Furthermore, the knowledge of the relation between the myocardium shape and its myofiber architecture is an important and required stage towards the construction of computational models of the heart since the fiber orientation plays a key role when simulating the electrical and mechanical functions of the heart.
The knowledge about fiber orientation has been recently eased with the use of diffusion tensor imaging (DT-MRI) since there is a correlation between the myocardium fiber structure and diffusion tensors. DT-MRI also has the advantage to provide directly this information in 3D with a high resolution but it is unfortunately not available in vivo due to the cardiac motion. There has been several works in the past decade that have studied the variability of fiber orientation from DT-MRI. Those studies estimated the fiber direction as the first eigenvector of each tensor and for instance compared its transmural variation with that observed from dissection experiments. We extended those studies by building a statistical model of the whole diffusion tensors. This tensor analysis allows us to study the variability of laminar sheets which are associated with the second and third eigenvectors.
To achieve it, we proposed a framework  to register and reorient DT-MRIs in a same reference frame based on anatomical MRIs in order to compute first and second order statistics on diffusion tensors in a Log-Euclidean framework at each voxel. We also developed statistical tools  to translate the covariance matrix of the whole tensors into variabilities of eigenvalues and eigenvector's frame orientation.
We applied this framework to a dataset of 9 ex vivo canine hearts. From the first order statistics results a smooth average DT-MRI suited to fiber tracking (see Fig 12 ). The second order statistics reveals high inter-subject similarities of the cardiac fiber orientation. This is the first step towards a validation of using an average model of the cardiac fiber architecture for medical image analysis and electromechanical simulations.
Modeling brain variability
This is joint work with the LONI (Laboratory of Neuroimaging) at UCLA (University of California at Los-Angeles), partly funded by the INRIA associated team program (Brain Atlas): http://www-sop.inria.fr/epidaure/Collaborations/UCLA .
This study builds on the work performed over the last two years to model the brain variability from sulcal lines. The whole process relies on manual delineations of sulcal lines done on MR images(http://www.loni.ucla.edu/ khayashi/Public/medial_surface/ ) affinely registered onto a template (the ICBM 305 space) coming from the LONI team at UCLA through the associated team (see Section 8.3 ). The mean sulcal lines were previously determined by iteratively optimizing over the position of the mean line and the correspondances with all the instances. We proposed this year to refine the affine transforms based on the sulcal mappings obtained between each subject's instance and the mean lines. This is realized by adding a third step to the previous iterative estimation scheme. The global effect is to concentrate the sulcal curves around the mean lines and to reduce the amplitude of the variability by about 10%, without modifying much the other results.
To compare two populations based on their sulcal tracings, we also proposed to compute global mean curves (including instances of both population), and to put into correspondence theses means and the population means. This allows to provide a common reference for comparing variability tensors of the two populations, without creating a bias towards one population. This work appeared in Neuroimage  , and in a research report  .
We are now applying this methodology to the mapping and understanding of gender differences (males vs. females) (see Fig. 13 ), pathologies (Williams syndrome), and hemispheric differences.
Registration Algorithms and Statistics on Deformation : Comparison of methods
There is nowadays a large interest of the scientific community for understanding the brain shape and its variability. The method is usually to extract some anatomical features like sulci or gyri in a population of subjects and to perform a statistical analysis of these features after a group-wise matching. To analyse the influence of the matching method used, we propose in this work to compare the results of the extrapolation of variability on sulcal lines of  (see also Section 6.4.3 ) with the diffeomorphic mapping framework developed in Miller's and Trouvé's teams.
In the diffeomorphic mapping framework, we look for a global coherent deformation of the whole space that best matches the data. This has major advantages: we not only find a correspondence between data points but also the complete trajectories of these points along the deformation process. Moreover, the deformation is not only defined on each data points individually but also on every point in the space. Eventually, the whole deformation could be seen as a geodesic in some space and therefore could be stored and recovered efficiently thanks to a finite number of parameters  ,  . Indeed, the whole deformation process is completely defined by the initial vector speed of each data points. Therefore, population-wise statistical analyses of the brain variability can be performed by measuring for instance the covariance matrix of these initial vector speeds.
We expect to see a few differences betwen the two methods because the mapping is defined on the whole space before computing the covariance structure on each point in the diffeomorphic mapping case while the covariance matrix of the displacement is computed on each lines independantly and then diffused in the whole space thanks to the log-Euclidean framework (see  ) with Fillard's method. We are currently performing experiments on the sulcal lines used in Section 6.4.3 . Figure 14 shows the registration of the mean to one subject. Quantitative measurements of the similarity and difference between the two methods are under way.
Bi-Invariant Means in Lie Groups
In recent years, the need for rigorous frameworks to compute statistics in non-linear spaces has grown considerably in the bio-medical imaging community. The registration of bio-medical images naturally deals with data living in non-linear spaces, since invertible geometrical deformations belong to groups of transformations which are not vector spaces. These groups can be finite-dimensional, as in the case of rigid or affine transformations, or infinite-dimensional as in the case of groups of diffeomorphisms.
In this work, we focused on the consistent generalization of the Euclidean mean to Lie Groups, which are a large class of non-linear spaces with relatively nice properties. Classically, in a Lie group endowed with a Riemannian metric, the natural choice of mean is called the Frechet mean. This Riemannian approach is completely satisfactory if a bi-invariant metric exists, for example in the case of compact groups such as rotations. The bi-invariant Frechet mean generalizes in this case the properties of the arithmetic mean: it is invariant with respect to left- and right-multiplication, as well as inversion. Unfortunately, bi-invariant Riemannian metrics do not always exist. In particular, we have showed 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 defined a bi-invariant mean generalizing the Frechet mean based on an implicit barycentric equation (see Fig. 15 ). Alternatively, we also proposed a simpler Log-Euclidean framework approach to statistics of linear invertible transformations, For matrices, the bi-invariant and Log-Euclidean means are both generalizations of the geometric mean of positive numbers, since their determinants are both exactly equal to the geometric mean of the determinant of the data. The Log-Euclidean mean is much simpler to compute, but has fewer invariance properties and is limited to transformations not too far away from the identity, contrary to the bi-invariant mean. A research report detailing this novel framework is available  .
Log-Euclidean Statistics of Diffeomorphisms
Currently, a large variety of non-linear registration algorithms have been proposed to deal with the non-rigid registration of medical images. However, the quantitative comparison of these algorithms remains to be done. To this end, having a consistent framework to compute statistics on general invertible transformations would be very useful. In this work, we introduced a novel parameterization of diffeomorphisms, based on the generalization of the principal logarithm to non-linear geometrical deformations. This corresponds to parameterizing diffeomorphisms with stationnary speed vectors fields. As for matrices, this logarithm can be used only for transformations close enough to the identity. However, our preliminary numerical experiments on 3D non-rigid registration suggest that this limitation affects only very large deformations, and may not be problematic for image registration results. This novel setting is the infinite-dimensional analogous of our Log-Euclidean framework for tensors and linear transformations. In this framework, usual Euclidean statistics can be performed on diffeomorphisms via their logarithms, with excellent mathematical properties like inversion-invariance.
In MICCAI'06 and MFCA'06  ,  , we presented two efficient algorithms to compute numerically logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Moreover we successfully applied 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 (see Fig. 16 ).
Characterization of Spine Deformations and Orthopedic Treatments Effects
This project is part of a partnership between the Asclepios team, the Montreal's Sainte-Justine hospital and the Polytechnic School of Montreal.
Spine surgeries are very delicate interventions that need to be carefully planned. In the case of scoliosis, a crucial part of the surgical planning process is to classify the patient's spine deformation in accordance to an accepted surgical classification scheme. Those classification schemes then offer guidance to the selection of fusion levels and to the selection of the associated orthopedic instrumentation. However, current classification schemes are two-dimensional while spine deformations are three-dimensional, mainly because analyzing large databases of 3D spine models is difficult and time consuming.
To facilitate this task, we proposed a method  that automatically extracts the most important deformation modes from a set of 3D spine models. The spine was expressed as a set of rigid transforms that superimpose local coordinates systems of neighboring vertebrae. To take into account the fact that rigid transforms belong to a Riemannian manifold, the Fréchet mean and a generalized covariance computed in the exponential chart at that point were used to construct a statistical shape model. The principal deformation modes were then extracted by performing a principal component analysis (PCA). The proposed method was applied to a group of 307 scoliotic patients and meaningful deformation modes were successfully extracted (see Figure 17 ). For example, patients' growth, double curves, simple thoracic curves and lumbar lordosis were extracted in the first four deformation modes.
Moreover, we were able to show with a logistic regression that there is a statistically significant relationship between conventional 2D surgical classifications such as King's classification and the first four principal deformation modes  . Thus, our method can be used to extract clinically significant deformation modes from a set of 3D spine models. This can help surgeons to refine arbitrary classes in 3D (King's or Lenke's classes, for instance), thus helping the design of new clinically relevant 3D classifications.
Although corrective surgeries are commonly performed on acute cases of scoliosis, there exist less invasive treatments that are prescribed for more common cases, such as Bracing. However, there is still no consensus about its actual effect. Previous studies were based on global descriptors of the spine shape (Cobb angle, plane of maximal deformity, etc.) and local shape was never directly assessed. We analyzed in  the braces effects at a finer scale to find which vertebral levels were significantly affected by this treatment. The 3D spine geometry of a group of 41 patients treated with a Boston brace and a control group of 28 untreated scoliotic patients was digitized with and without brace (first group) or twice without brace (control group). The modifications of the relative poses of successive vertebrae were extracted from 3D reconstructions. As before, the Fréchet mean and a generalized covariance were used to measure the centrality and dispersion of our manifold-valued primitives (illustrated in Figure 18 ). Multivariate hypothesis tests were used to compare the two groups: Significant differences (p<0.01) between the centrality and dispersion measures of the relative poses modifications were respectively found from T1 to T6 and from T8 to L1. This is in accordance with the back flattening effect and the spatially limited correction found in other studies. However our results offer a more specific evaluation of the localization of those effects.
Our current work now aims at exploiting our statistical model of the spine shape to develop 2D-3D registration method suitable to register an articulated model of the spine to per-operative radiographs, in view of an image guided surgery system for spine surgery. This problem is particularly challenging due to the flexible nature of the spine and the presence of sensitive anatomical structures near the surgical path, like the spinal chord or major blood vessels, which makes an image guided surgery system for spine surgery very useful.
Point-Based Statistical Shape Models Using Correspondence Probabilities
This work takes place in a cooperation with the medical imaging group of the university Hamburg-Eppendorf.
A fundamental problem when computing surface based statistical shape models is the determination of correspondences between the instances. Often, homologies between the point cloud representations are assumed which might be erroneous and might distort the results. In order to find a solution to that, we developed a novel algorithm based on the affine Expectation Maximization - Iterative Closest Point (EM-ICP) registration method. Here, exact correspondences are replaced by iteratively evolving correspondence probabilities which provide the basis for the computation of mean shape and variability model. The performance of this approach is investigated on different kind of organs (kidney, brain, prostate, ganglions...)  (see Fig. 19 ). The method is currently updated to perform automatic classification.