A Convex Semi-Definite Positive Framework for DTI Estimation and Regularization

International audienceIn this paper we introduce a novel variational method for joint estimation and regularization of diffusion tensor fields from noisy raw data. To this end, we use the classic quadratic data fidelity term derived from the Stejskal-Tanner equation with a new smoothness term leading to a convex objective function. The regularization term is based on the assumption that the signal can be reconstructed using a weighted average of observations on a local neighborhood. The weights measure the similarity between tensors and are computed directly from the diffusion images. We preserve the positive semi-definiteness constraint using a projected gradient descent. Experimental validation and comparisons with a similar method using synthetic data with known noise model, as well as classification of tensors towards understanding the anatomy of human skeletal muscle demonstrate the potential of our method

Inferring Preoperative Reconstructed Spine Models to Volumetric CT Data through High-Order MRFs

In this paper, we introduce a novel approach based on higher order energy functions which have the ability to encode global structural dependencies to infer articulated 3D spine models to CT volume data. A personalized geometrical model is reconstructed from biplanar X-rays before spinal surgery in order to create a spinal column representation which is modeled by a series of intervertebral transformations based on rotation and translation parameters. The shape transformation between the standing and lying poses is then achieved through a Markov Random Field optimization graph, where the unknown variables are the deformations applied to the intervertebral transformations. Singleton and pairwise potentials measure the support from the data and geometrical dependencies between neighboring vertebrae respectively, while higher order cliques are introduced to integrate consistency in regional curves. Optimization of model parameters in a multi-modal context is achieved using efficient linear programming and duality. A qualitative evaluation of the vertebra model alignment obtained from the proposed method gave promising results while the quantitative comparison to expert identification yields an accuracy of 1.8 +/- 0.7mm based on the localization of surgical landmarks

Geodesic Active Regions for Texture Segmentation

This paper proposes a framework for segmenting different textured areas over synthetic or real textured frames by curves propagation. We assume that the system has the ability to be taught over different texture prototypes. For each prototype a global statistical model is generated, as a set of probability density functions attributes from a multi-valued frame analysis, where different filter responses are used to create this multi-valued frame. Then, each prototype is represented by a reliable statistical model. Given an input frame composed of different texture types, the same bank of filters is applied. Over the generated multi-valued frame, we define an energy as a special form of a geodesic active contour model, a Geodesic Active Regi on Model, where we integrate boundary finding and region based segmentation approaches. This energy is minimized using a steepest gradient descend method, where smoothing, edge-based, and region statistics forces, move the curve toward the minimum of the designed objective function. Using the level set formulation scheme, complex curves can be detected, while topological changes for the evolving curves are naturally managed. In order to deal with the problem of noise influence, as well as to reduce the required computational cost, a multi-grid approach has also been considered. Finally, two different methods are used for the level set implementation, the Narrow Band and the Hermes Algorithm. Very promising experimental results are provided using synthetic and real textured frames

Classification of Tensors and Fiber Tracts Using Mercer-Kernels Encoding Soft Probabilistic Spatial and Diffusion Information

In this paper, we present a kernel-based approach to the clustering of diffusion tensors and fiber tracts. We propose to use a Mercer kernel over the tensor space where both spatial and diffusion information are taken into account. This kernel highlights implicitly the connectivity along fiber tracts. Tensor segmentation is performed using kernel-PCA compounded with a landmark-Isomap embedding and k-means clustering. Based on a soft fiber representation, we extend the tensor kernel to deal with fiber tracts using the multi-instance kernel that reflects not only interactions between points along fiber tracts, but also the interactions between diffusion tensors. This unsupervised method is further extended by way of an atlas-based registration of diffusion-free images, followed by a classification of fibers based on nonlinear kernel Support Vector Machines (SVMs). Promising experimental results of tensor and fiber classification of the human skeletal muscle over a significant set of healthy and diseased subjects demonstrate the potential of our approach

MRF-based Diffeomorphic Population Deformable Registration & Segmentation

In this report, we present a novel framework to deform mutually a population of n-examples based on an optimality criterion. The optimality criterion comprises three terms, one that aims to impose local smoothness, a second that aims to minimize the individual distances between all possible pairs of images, while the last one is a global statistical measurement based on "compactness" criteria. The problem is reformulated using a discrete MRF, where the above constraints are encoded in singleton (global) and pair-wise potentials (smoothness (intra-layer costs) and pair-alignments (inter-layer costs)). Furthermore, we propose a novel grid-based deformation scheme, that guarantees the diffeomorphism of the deformation while being computationally favorable compared to standard deformation methods. Towards addressing important deformations we propose a compositional approach where the deformations are recovered through the sub-optimal solutions of successive discrete MRFs. The resulting paradigm is optimized using efficient linear programming. The proposed framework for the mutual deformation of the images is applied to the group-wise registration problem as well as to an atlas-based population segmentation problem. Both articially generated data with known deformations and real data of medical studies were used for the validation of the method. Promising results demonstrate the potential of our method

A Kernel-based Approach to Diffusion Tensor and Fiber Clustering in the Human Skeletal Muscle

In this report, we present a kernel-based approach to the clustering of diffusion tensors in images of the human skeletal muscle. Based on the physical intuition of tensors as a means to represent the uncertainty of the position of water protons in the tissues, we propose a Mercer (i.e. positive definite) kernel over the tensor space where both spatial and diffusion information are taken into account. This kernel highlights implicitly the connectivity along fiber tracts. We show that using this kernel in a kernel-PCA setting compounded with a landmark-Isomap embedding and k-means clustering provides a tractable framework for tensor clustering. We extend this kernel to deal with fiber tracts as input using the multi-instance kernel by considering the fiber as set of tensors centered in the sampled points of the tract. The obtained kernel reflects not only interactions between points along fiber tracts, but also the interactions between diffusion tensors. We give an interpretation of the obtained kernel as a comparison of soft fiber representations and show that it amounts to a generalization of the Gaussian kernel Correlation. As in the tensor case, we use the kernel-PCA setting and k-means for grouping of fiber tracts. This unsupervised method is further extended by way of an atlas-based registration of diffusion-free images, followed by a classification of fibers based on non-linear kernel Support Vector Machines (SVMs) and kernel diffusion. The experimental results on a dataset of diffusion tensor images of the calf muscle of 25 patients (of which 5 affected by myopathies, i.e. neuromuscular diseases) show the potential of our method in segmenting the calf in anatomically relevant regions both at the tensor and fiber level

Manifold-driven Grouping of Skeletal Muscle Fibers

In this report, we present a manifold clustering method for the classification of fibers obtained from diffusion tensor images (DTI) of the human skeletal muscle. To this end, we propose the use of angular Hilbertian metrics between multivariate normal distributions to define a family of distances between tensors that we generalize to fibers. The obtained metrics between fiber tracts encompasses both diffusion and localization information. As far as clustering is concerned, we use two methods. The first approach is based on diffusion maps and k-means clustering in the spectral embedding space. The second approach uses a linear programming formulation of prototype-based clustering. This formulation allows for classification over manifolds without the necessity to embed the data in low dimensional spaces and determines automatically the number of clusters. The experimental validation of the proposed framework is done using a manually annotated significant dataset of DTI of the calf muscle for healthy and diseased subjects

Local Appearance Knowledge and Shape Variation Models for Muscle Segmentation

In this report, we present a novel prior knowledge representation of shape variation using diffusion wavelets and applied for medical image segmentation. One of the major advantage of our approach is that it can reflect arbitrary and continuous interdependencies in the training data. In contrast to state-of-the-art methods our framework during the learning stage optimizes the coefficients as well as the number and the position of landmarks using geometric (reconstructed surface) constraints. Saliency is encoded in the model and segmentation is expressed through the extraction of the corresponding features in a new data-set. The resulting paradigm supports hierarchies both in the model and the search space, can encode complex geometric and photometric dependencies of the structure of interest, and can deal with arbitrary topologies. In another hand, our report deals with a different model search methodology where we apply an approach related to active feature models; the location of landmarks is updated iteratively, using local features, and the canonical correlation analysis. We report promising results on two challenging medical data sets, that illustrate the potential of our method

Manifold-driven Grouping of Skeletal Muscle Fibers

In this report, we present a manifold clustering method for the classification of fibers obtained from diffusion tensor images (DTI) of the human skeletal muscle. To this end, we propose the use of angular Hilbertian metrics between multivariate normal distributions to define a family of distances between tensors that we generalize to fibers. The obtained metrics between fiber tracts encompasses both diffusion and localization information. As far as clustering is concerned, we use two methods. The first approach is based on diffusion maps and k-means clustering in the spectral embedding space. The second approach uses a linear programming formulation of prototype-based clustering. This formulation allows for classification over manifolds without the necessity to embed the data in low dimensional spaces and determines automatically the number of clusters. The experimental validation of the proposed framework is done using a manually annotated significant dataset of DTI of the calf muscle for healthy and diseased subjects