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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 216 (26 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 25 (5 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
 IEEE Transactions on Medical Imaging
, 2007
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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 8 (3 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
PDEdriven adaptive morphology for matrix fields
 In Proc. Second International Conference on Scale Space and Variational Methods in Computer Vision
, 2009
"... Abstract. Matrix fields are important in many applications since they are the adequate means to describe anisotropic behaviour in image processing models and physical measurements. A prominent example is diffusion tensor magnetic resonance imaging (DTMRI) which is a medical imaging technique usef ..."
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Cited by 6 (2 self)
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Abstract. Matrix fields are important in many applications since they are the adequate means to describe anisotropic behaviour in image processing models and physical measurements. A prominent example is diffusion tensor magnetic resonance imaging (DTMRI) which is a medical imaging technique useful for analysing the fibre structure in the brain. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to three dimensional fields of symmetric positive definite matrices. In this article we propose a novel method to incorporate adaptivity into the matrixvalued, PDEdriven dilation process. The approach uses a structure tensor concept for matrix data to steer anisotropic morphological evolution in a way that enhances and completes linelike structures in matrix fields. Numerical experiments performed on synthetic and realworld data confirm the gapclosing and linecompleting qualities of the proposed method. 1
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 5 (3 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.
Morphology for higherdimensional tensor data via Loewner ordering
 Mathematical Morphology: 40 Years On, volume 30 of Computational Imaging and Vision
, 2005
"... Keywords: The operators of greyscale morphology rely on the notions of maximum and minimum which regrettably are not directly available for tensorvalued data since the straightforward componentwise approach fails. This paper aims at the extension of the maximum and minimum operations to the tensor ..."
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Cited by 4 (2 self)
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Keywords: The operators of greyscale morphology rely on the notions of maximum and minimum which regrettably are not directly available for tensorvalued data since the straightforward componentwise approach fails. This paper aims at the extension of the maximum and minimum operations to the tensorvalued setting by employing the Loewner ordering for symmetric matrices. This prepares the ground for matrixvalued analogs of the basic morphological operations. The novel definitions of maximal/minimal matrices are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DTMRI data. Furthermore, they depend continuously on the input data which makes them viable for the design of morphological derivatives such as the Beucher gradient or a morphological Laplacian. Experiments on DTMRI images illustrate the properties and performance of our morphological operators. Mathematical morphology, dilation, erosion, matrixvalued images, diffusion tensor MRI, Loewner ordering
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 3 (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
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.
Flexible segmentation and smoothing of DTMRI fields through a customizable structure tensor
 Advances in Visual Computing (Proc. ISVC), volume 4291 of LNCS
, 2006
"... Abstract. We present a novel structure tensor for matrixvalued images. It allows for user defined parameters that add flexibility to a number of image processing algorithms for the segmentation and smoothing of tensor fields. We provide a thorough theoretical derivation of the new structure tensor, ..."
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Abstract. We present a novel structure tensor for matrixvalued images. It allows for user defined parameters that add flexibility to a number of image processing algorithms for the segmentation and smoothing of tensor fields. We provide a thorough theoretical derivation of the new structure tensor, including a proof of the equivalence of its unweighted version to the existing structure tensor from the literature. Finally, we demonstrate its advantages for segmentation and smoothing, both on synthetic tensor fields and on real DTMRI data. 1