Representing diffusion MRI in 5D for segmentation of white matter tracts with a level set method.

We present a method for segmenting white matter tracts from high angular resolution diffusion MR. images by representing the data in a 5 dimensional space of position and orientation. Whereas crossing fiber tracts cannot be separated in 3D position space, they clearly disentangle in 5D position-orientation space. The segmentation is done using a 5D level set method applied to hyper-surfaces evolving in 5D position-orientation space. In this paper we present a methodology for constructing the position-orientation space. We then show how to implement the standard level set method in such a non-Euclidean high dimensional space. The level set theory is basically defined for N-dimensions but there are several practical implementation details to consider, such as mean curvature. Finally, we will show results from a synthetic model and a few preliminary results on real data of a human brain acquired by high angular resolution diffusion MRI

Representing diffusion mri in 5D for segmentation of white matter tracts with a level set method

We present a method for segmenting white matter tracts from high angular resolution diffusion MR images by representing the data in a 5 dimensional space of position and orientation. Whereas crossing fiber tracts cannot be separated in 3D position space, they clearly disentangle in 5D position-orientation space. The segmentation is done using a 5D level set method applied to hyper-surfaces evolving in 5D position-orientation space. In this paper we present a methodology for constructing the positionorientation space. We then show how to implement the standard level set method in such a non-Euclidean high dimensional space. The level set theory is basically defined for N-dimensions but there are several practical implementation details to consider, such as mean curvature. Finally, we will show results from a synthetic model and a few preliminary results on real data of a human brain acquired by high angular resolution diffusion MRI

Segmentation of diffusion weighted MRI using the level set framework

Medical imaging is a rapidly growing field in which diffusion imaging is a recently developed modality. This novel imaging contrast permits in-vivo measurement of the diffusion of water molecules. This is particularly interesting in brain imaging where the diffusion reveals an amazing insight into the neuronal organization and cerebral cytoarchitecture. Diffusion images contain from six up to hundreds of values in each voxel and are represented as tensor fields (Diffusion Tensor Imaging (DTI)) or as fields of functions (High Angular Resolution Diffusion (HARD) imaging). To fully extract the large amount of data contained within these images new processing and analysis tools are needed. The aim of this thesis is the development of such tools. The method we have been mainly focusing on for this purpose is the level set method. The level set method is a numerical and theoretical tool for propagating interfaces. In image processing it is used for propagating curves in 2D or surfaces in 3D for delineation of objects or for regularization purposes. In this thesis we have explored some of the numerous aspects of the level set frame work to see how the diffusion properties can be used for segmentation. For segmentation of tensor fields we have considered similarity measures for comparison of tensors. From these similarity measures several applications of the level set method have been developed for the segmentation of different structures. Different measures of similarity have been used dependent on the application. When segmenting white matter regions in DTI, the similarity measure emphasizes anisotropic regions. The segmentation algorithm itself has a very local dependence since white matter, in general fiber tracts, experiences different diffusion in different parts of the structure. The presented results show segmentations of the major fiber tracts in the brain. Other structures, such as the deep cerebral nuclei, that are mainly composed of gray matter, have more homogenous diffusion properties than white matter structures. Therefore, in these structures we maximize the internal coherence within the entire structure by using a region based approach to the segmentation problem. Segmentations of the thalamus and its nuclei as well as on tensor fields from fluid mechanics are presented. For High Angular Resolution Diffusion (HARD) images, two methods for fiber tract segmentation are presented based on different types of coherence. The coherence is either measured as the similarity between fibers obtained from a tractography algorithm, or the similarity of scalar values in a five-dimensional non-Euclidean space. The similarity between two fibers is determined by a counting strategy and is equal to the number of voxels they have in common. A spectral clustering algorithm is then used for grouping fibers with a high inter-resemblance. When segmenting white matter with the level set method, we propose to expand the space we are working in from a three-dimensional space of Orientation Distribution Functions (ODF) to a five-dimensional space of position and orientation. By a careful definition of this space and an adaptation of the level set to five dimensions the fibers tracts can be segmented as separated structures. We show some preliminary results from segmentations in this 5D space. The approaches proposed in this thesis permit a consideration of the fiber tracts and gray matter structures as an entity, allowing quantitative measures of the diffusion without losing information by simplifying the images to scalar representations

Image segmentation with variational active contours

An important branch of computer vision is image segmentation. Image segmentation aims at extracting meaningful objects lying in images either by dividing images into contiguous semantic regions, or by extracting one or more specific objects in images such as medical structures. The image segmentation task is in general very difficult to achieve since natural images are diverse, complex and the way we perceive them vary according to individuals. For more than a decade, a promising mathematical framework, based on variational models and partial differential equations, have been investigated to solve the image segmentation problem. This new approach benefits from well-established mathematical theories that allow people to analyze, understand and extend segmentation methods. Moreover, this framework is defined in a continuous setting which makes the proposed models independent with respect to the grid of digital images. This thesis proposes four new image segmentation models based on variational models and the active contours method. The active contours or snakes model is more and more used in image segmentation because it relies on solid mathematical properties and its numerical implementation uses the efficient level set method to track evolving contours. The first model defined in this dissertation proposes to determine global minimizers of the active contour/snake model. Despite of great theoretic properties, the active contours model suffers from the existence of local minima which makes the initial guess critical to get satisfactory results. We propose to couple the geodesic/geometric active contours model with the total variation functional and the Mumford-Shah functional to determine global minimizers of the snake model. It is interesting to notice that the merging of two well-known and "opposite" models of geodesic/geometric active contours, based on the detection of edges, and active contours without edges provides a global minimum to the image segmentation algorithm. The second model introduces a method that combines at the same time deterministic and statistical concepts. We define a non-parametric and non-supervised image classification model based on information theory and the shape gradient method. We show that this new segmentation model generalizes, in a conceptual way, many existing models based on active contours, statistical and information theoretic concepts such as mutual information. The third model defined in this thesis is a variational model that extracts in images objects of interest which geometric shape is given by the principal components analysis. The main interest of the proposed model is to combine the three families of active contours, based on the detection of edges, the segmentation of homogeneous regions and the integration of geometric shape prior, in order to use simultaneously the advantages of each family. Finally, the last model presents a generalization of the active contours model in scale spaces in order to extract structures at different scales of observation. The mathematical framework which allows us to define an evolution equation for active contours in scale spaces comes from string theory. This theory introduces a mathematical setting to process a manifold such as an active contour embedded in higher dimensional Riemannian spaces such as scale spaces. We thus define the energy functional and the evolution equation of the multiscale active contours model which can evolve in the most well-known scale spaces such as the linear or the curvature scale space

Molecular Imaging

The present book gives an exceptional overview of molecular imaging. Practical approach represents the red thread through the whole book, covering at the same time detailed background information that goes very deep into molecular as well as cellular level. Ideas how molecular imaging will develop in the near future present a special delicacy. This should be of special interest as the contributors are members of leading research groups from all over the world