| Medical Image Analysis | Biological Image Analysis | Computational Anatomy | Computational Physiology | Previous Themes |
In recent years, the need for rigorous frameworks to compute statistics in non-linear spaces has grown considerably in the bio-medical imaging community. First, a number imaging modalities, like diffusion MRI, provide researchers with data which do not live in a linear space, and nonetheless require post-processing (re-sampling, regularization, statistics, etc.). Second, the one-to-one registration of bio-medical images naturally deals with data living in non-linear spaces, since many types of 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 parameterized with time-varying speed vector fields.
Among statistics, the most fundamental is certainly the mean value, which extracts from the data a central point, minimizing in some sense the dispersion of the data around it. In this project, we have focused on the 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. But this Riemannian approach is completely satisfactory only if a bi-invariant metric exists, which is for example the case for compact groups such as rotations. The bi-invariant Frechet 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. Unfortunately, bi-invariant Riemannian metrics do not always exist. In particular, in this work, we have proved the novel result 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 a bi-invariant mean generalizing the Frechet mean induced by bi-invariant metrics, even in cases when such metrics do not exist. The intuition of the existence of such a mean was first given by R.P. Woods in 2003 (without precise definition).In this work, we have proposed a general framework to define rigorously bi-invariant means, this time in any finite dimensional real Lie group. To do this, we rely on a general barycentric equation, whose solution is by definition the bi-invariant mean. We have shown the existence and uniqueness of this novel type of mean, provided the dispersion of the data is small enough, and the convergence of the classical iterative algorithm of R.P. Woods is also shown. In the case of rigid transformations, we have been able to determine a simple criterion for the general existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations.
Contacts: Vincent Arsigny, and Xavier Pennec. Some publications can be found here.
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. Last but not least, SPD matrices are becoming a common tool in numerical analysis to generate adapted meshes to reduce the computational cost of solving partial differential equations (PDEs) in 3D.
As a consequence, there has been a growing need to carry out computations with these objects, for instance to interpolate, restore, enhance images of tensors. To this end, one needs to define a complete operational framework. This is necessary to fully generalize to the SPD case the usual statistical tools or PDEs on vector-valued images. In this work, we have 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 alogarithm particularly simple to recycle.
Contacts: Vincent Arsigny, Pierre Fillard and Xavier Pennec. Some publications can be found here.
The goal of this project is the development of model-based methods for automatic 3D segmentation and recognition of anatomical structures in medical images. A statistical shape model is generated using point cloud representation of the instances. In order to avoid typical problems of direct point-to-point correspondences, a correspondence probability approach is realized when registering the instances (see figures 1 and 2).
Contacts: Heike Hufnagel and Xavier Pennec
Some publications can be found here.
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 structure 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 function of the heart. The knowledge about fibre orientation has been recently eased with the use of Diffusion Tensor Imaging (DTI) since there is a correlation between the myocardium fibre structure and diffusion tensors. DTI also has the advantage to provide directly this information in 3D with a high resolution, unfortunately it is not available in vivo due to the cardiac motion. A statistical study of ex vivo cardiac DTI will help in understanding the cardiac fibre structure and in modeling the electromechanical behaviour of the heart.
Contact: Jean-Marc Peyrat
The emergence of diffusion tensor imaging (DTI) in the medical imaging community led to challenges in mathematics in order to manipulate 2nd order symmetric positive-defined matrices, so called tensors. In DT-MRI, each voxel of the brain contains a tensor, which is the 2nd order approximation of the brownian motion of water at this specific location. As water tends to move along oriented tissues (such as white matter neural fibers) diffusion tensors are likely to be aligned with the underlying structures. Due to the noise corrupting the data, the tensor field needs adequate post-processing before any further analysis. However, one cannot manipulate tensors like scalars (the tensor space is not a vector space). We used results in differential geometry to manipulate tensors while ensuring to remain on the tensor space, i.e. to keep the positive-defined constrain verified at all time. Applications are: tensor field regularization (PDE, etc.) and shape statistics (see the collaboration Epidaure-LONI).
Contacts: Pierre Fillard and Xavier Pennec.
Some publications can be found here.
This study is part of the partnership between the Epidaure team and the LONI (Laboratory of Neuro Imaging) at Los Angeles. While our team has been focusing on developping tools adapted to the analysis and registration of multidimensional and multimodal medical images, the LONI has been leading studies to build cerebral atlases of specific diseases (e.g. Alzheimer, schizophrenia) and thus has been collecting anatomical images for years. Today, this amount of images consitutes a unique databasis of more than 500 T1 MRI. Moreover, each images comes with an accurate delineation of 32 pairs of sulci, each sulcal line being segmented by experts who followed a meticulous protocol.
Basically, this work aims at learning the global brain variability from the variability of sulcal lines. The size of the databasis being statistically significant, this work could lead to new results in neuro-anatomy. In the one hand, a better knowledge of the brain variability among a population could help to early detect neurological pathologies and in the other hand, providing a map of the variability could help to guide non-rigid registration algorithms. Indeed, if we are able to predict in which direction each position of the brain statiscally moves, we will be able to better constrain registration algorithms. This work makes an intensive use of a large panel of scientific domains: computer vision, image analysis, statistics and differential geometry to quote just a few.
See the collaboration Asclepios-LONI for further information.
Contacts: Pierre Fillard, Vincent Arsigny and Xavier Pennec.
Some publications can be found here.
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