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LogEuclidean metrics for fast and simple calculus on diffusion tensors
 Magnetic Resonance in Medicine
, 2006
"... Euclidean metrics on diffusion tensors. Total word count: 6400. ..."
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Cited by 129 (23 self)
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Euclidean metrics on diffusion tensors. Total word count: 6400.
A Riemannian approach to diffusion tensor images segmentation
 In Proc. IPMI, 591–602
, 2005
"... Abstract. We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images. Our approach is grounded on the theoretically wellfounded differential geometrical properties of the space of multivariate normal distributions. We introduce a variational formulat ..."
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Cited by 20 (4 self)
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Abstract. We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images. Our approach is grounded on the theoretically wellfounded differential geometrical properties of the space of multivariate normal distributions. We introduce a variational formulation, in the level set framework, to estimate the optimal segmentation according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. Moreover, we must respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the diffusion tensor image. We validate our algorithm on synthetic data and report interesting results on real datasets. We focus on two structures of the white matter with different properties and respectively known as the corpus callosum and the corticospinal tract. 1
Diffusion Tensor Analysis with Invariant Gradients and Rotation Tangents
"... Abstract—Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees ..."
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Cited by 11 (2 self)
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Abstract—Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees of freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a thirdorder tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourthorder diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy. Index Terms—Diffusion tensor magnetic resonance imaging, fourthorder covariance tensor, tensor feature detection, tensor invariants, thirdorder gradient tensor. I.
PDEs for Tensor Image Processing
, 2005
"... Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey paper the most important PDEs for discontinuitypreserving denoising of tensor fields are reviewed such that the unde ..."
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Cited by 3 (1 self)
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Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey paper the most important PDEs for discontinuitypreserving denoising of tensor fields are reviewed such that the underlying design principles becomes evident. We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. These filters preserve positive semidefiniteness of any positive semidefinite initial tensor field. Finally we discuss geodesic active contours for segmenting tensor fields. Experiments are presented that illustrate the behaviour of all these methods.
Mathematical Morphology on Tensor Data Using the Loewner Ordering
"... The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensorvalued data. Based upon the Loewner ordering for symmetric matrices this paper extends the maximum and minimum operation to the tensorvalued ..."
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Cited by 2 (2 self)
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The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensorvalued data. Based upon the Loewner ordering for symmetric matrices this paper extends the maximum and minimum operation to the tensorvalued setting. This provides the ground to establish matrixvalued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DTMRI data. The morphological operations resulting from a componentwise maximum/minimum of the matrix channels
Morphology for Matrix Data: Ordering versus PDEBased Approach
, 2005
"... Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in th ..."
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Cited by 2 (1 self)
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Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrixvalued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the
A General Structure Tensor Concept and CoherenceEnhancing Diffusion Filtering For Matrix Fields
"... Coherenceenhancing diffusion filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flowlike features in images. The completion of linelike structures is also a major conce ..."
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Cited by 1 (1 self)
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Coherenceenhancing diffusion filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flowlike features in images. The completion of linelike structures is also a major concern in diffusion tensor magnetic resonance imaging (DTMRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3 × 3matrices, and it helps to visualise, for example, the nerve fibers in brain tissue. As any physical measurement DTMRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers. In this paper we address that problem by proposing a coherenceenhancing diffusion filtering methodology for matrix fields. The approach is based on a generic structure tensor concept for matrix fields that relies on the operatoralgebraic properties of symmetric matrices, rather than their channelwise treatment of earlier proposals. Numerical experiments with artificial and real DTMRI data confirm the gapclosing and flowenhancing qualities of the technique presented.
Magnetic Resonance in Medicine 56:411–421 (2006) LogEuclidean Metrics for Fast and Simple Calculus on Diffusion Tensors
"... Diffusion tensor imaging (DTMRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positivedefinite matrices), called “tensors ” here, has proved difficult in recent years. Usual Euclidean operations ..."
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Diffusion tensor imaging (DTMRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positivedefinite matrices), called “tensors ” here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affineinvariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called LogEuclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new 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. Theoretical aspects are presented and the Euclidean, affineinvariant, and LogEuclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data. Magn Reson Med
apport de rechercheDTI Segmentation by Statistical Surface Evolution
"... Abstract: We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). DTI produces, from a set of diffusionweighted MR images, tensorvalued images where each voxel is assigned with a 3 × 3 symmetric, positivedefinite matrix. This second order ..."
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Abstract: We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). DTI produces, from a set of diffusionweighted MR images, tensorvalued images where each voxel is assigned with a 3 × 3 symmetric, positivedefinite matrix. This second order tensor is simply the covariance matrix of a local Gaussian process, with zero mean, modeling the average motion of water molecules. As we will show in this article, the definition of a dissimilarity measure and statistics between such quantities is a non trivial task which must be tackled carefully. We claim and demonstrate that, by using the theoretically wellfounded differential geometrical properties of the manifold of multivariate normal distributions, it is possible to improve the quality of the segmentation results obtained with other dissimilarity measures such as the Euclidean distance or the KullbackLeibler divergence. The main goal of this work is to prove that the choice of the probability metric, i.e. the dissimilarity measure, has a deep impact on the tensor statistics and, hence, on the achieved results. We introduce a variational formulation, in the levelset framework, to estimate the optimal segmentation of a diffusion tensor image according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. We must also respect the geometric constraints imposed by the interfaces
Feature Extraction for DWMRI Visualization: The State of the Art and Beyond ∗
"... By measuring the anisotropic selfdiffusion rates of water, Diffusion Weighted Magnetic Resonance Imaging (DWMRI) provides a unique noninvasive probe of fibrous tissue. In particular, it has been explored widely for imaging nerve fiber tracts in the human brain. Geometric features provide a quick v ..."
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By measuring the anisotropic selfdiffusion rates of water, Diffusion Weighted Magnetic Resonance Imaging (DWMRI) provides a unique noninvasive probe of fibrous tissue. In particular, it has been explored widely for imaging nerve fiber tracts in the human brain. Geometric features provide a quick visual overview of the complex datasets that arise from DWMRI. At the same time, they build a bridge towards quantitative analysis, by extracting explicit representations of structures in the data that are relevant to specific research questions. Therefore, features in DWMRI data are an active research topic not only within scientific visualization, but have received considerable interest from the medical image analysis, neuroimaging, and computer vision communities. It is the goal of this paper to survey contributions from all these fields, concentrating on streamline clustering, edge detection and segmentation, topological methods, and extraction of anisotropy creases. We point out interrelations between these topics and make suggestions for future research.