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43
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 217 (26 self)
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Euclidean metrics on diffusion tensors. Total word count: 6400.
VectorValued Image Regularization with PDEs: A Common Framework for Different Applications
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new reg ..."
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Cited by 181 (8 self)
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We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regularization PDE's and corresponding numerical schemes that respect the local geometry of vectorvalued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.
A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape
 International Journal of Computer Vision
, 2007
"... Abstract. Since their introduction as a means of front propagation and their first application to edgebased segmentation in the early 90’s, level set methods have become increasingly popular as a general framework for image segmentation. In this paper, we present a survey of a specific class of reg ..."
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Cited by 169 (4 self)
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Abstract. Since their introduction as a means of front propagation and their first application to edgebased segmentation in the early 90’s, level set methods have become increasingly popular as a general framework for image segmentation. In this paper, we present a survey of a specific class of regionbased level set segmentation methods and clarify how they can all be derived from a common statistical framework. Regionbased segmentation schemes aim at partitioning the image domain by progressively fitting statistical models to the intensity, color, texture or motion in each of a set of regions. In contrast to edgebased schemes such as the classical Snakes, regionbased methods tend to be less sensitive to noise. For typical images, the respective cost functionals tend to have less local minima which makes them particularly wellsuited for local optimization methods such as the level set method. We detail a general statistical formulation for level set segmentation. Subsequently, we clarify how the integration of various low level criteria leads to a set of cost functionals and point out relations between the different segmentation schemes. In experimental results, we demonstrate how the level set function is driven to partition the image plane into domains of coherent color, texture, dynamic texture or motion. Moreover, the Bayesian formulation allows to introduce prior shape knowledge into the level set method. We briefly review a number of advances in this domain.
Fast Anisotropic Smoothing of MultiValued Images using CurvaturePreserving PDE’s
 Research Report “Les Cahiers du GREYC”, No 05/01. Equipe IMAGE/GREYC (CNRS UMR 6072), Février
, 2005
"... We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the liter ..."
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Cited by 65 (3 self)
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We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the literature. Then, we introduce a new tensordriven PDE, regularizing images while taking the curvatures of specific integral curves into account. We show that this constraint is particularly well suited for the preservation of thin structures in an image restoration process. A direct link is made between our proposed equation and a continuous formulation of the LIC’s (Line Integral Convolutions by Cabral and Leedom [11]). It leads to the design of a very fast and stable algorithm that implements our regularization method, by successive integrations of pixel values along curved integral lines. Besides, the scheme numerically performs with a subpixel accuracy and preserves then thin image structures better than classical finitedifferences discretizations. Finally, we illustrate the efficiency of our generic curvaturepreserving approach in terms of speed and visual quality with different comparisons and various applications requiring image smoothing: color images denoising, inpainting and image resizing by nonlinear interpolation.
O.: Regularizing flows for constrained matrixvalued images
 J. Math. Imaging Vision
, 2004
"... Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differentialgeometric framework to define PDEs acting on some manifol ..."
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Cited by 47 (8 self)
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Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differentialgeometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structurepreserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging. Note: This is the draft
Orthonormal Vector Sets Regularization with PDE’s and Applications
, 2002
"... We are interested in regularizing fields of orthonormal vector sets, using constraintpreserving anisotropic diffusion PDE’s. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vectors ..."
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Cited by 44 (3 self)
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We are interested in regularizing fields of orthonormal vector sets, using constraintpreserving anisotropic diffusion PDE’s. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vectors or orthogonal matrices (among other examples). We first develop a general variational framework that solves this regularization problem, thanks to a constrained minimization of φfunctionals. This leads to a set of coupled vectorvalued PDE’s preserving the orthonormal constraints. Then, we focus on particular applications of this general framework, including the restoration of noisy direction fields, noisy chromaticity color images, estimated camera motions and DTMRI (Diffusion Tensor MRI) datasets.
Constrained flows of matrixvalued functions: Application to diffusion tensor regularization
 In European Conference on Computer Vision
, 2002
"... Abstract. Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows actingon constrained d ..."
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Cited by 34 (6 self)
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Abstract. Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows actingon constrained datasets. We focus our interest on flows of matrixvalued functions undergoing orthogonal and spectral constraints. The correspondingevolution PDE’s are found by minimization of cost functionals, and depend on the natural metrics of the underlyingconstrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable numerical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of diffusion tensor volumes (DTMRI).
A Segmentation Based Variational Model for Accurate Optical Flow Estimation
 ECCV
, 2008
"... Segmentation has gained in popularity in stereo matching. However, it is not trivial to incorporate it in optical flow estimation due to the possible nonrigid motion problem. In this paper, we describe a new optical flow scheme containing three phases. First, we partition the input images and integ ..."
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Cited by 22 (3 self)
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Segmentation has gained in popularity in stereo matching. However, it is not trivial to incorporate it in optical flow estimation due to the possible nonrigid motion problem. In this paper, we describe a new optical flow scheme containing three phases. First, we partition the input images and integrate the segmentation information into a variational model where each of the segments is constrained by an affine motion. Then the errors brought in by segmentation are measured and stored in a confidence map. The final flow estimation is achieved through a global optimization phase that minimizes an energy function incorporating the confidence map. Extensive experiments show that the proposed method not only produces quantitatively accurate optical flow estimates but also preserves sharp motion boundaries, which makes the optical flow result usable in a number of computer vision applications, such as image/video segmentation and editing.
A framework based on spin glass models for the inference of anatomical connectivity from diffusionweighted MR data
, 2002
"... A family of methods aiming at the reconstruction of a putative fascicle map from any diffusionweighted dataset is proposed. This fascicle map is defined as a tradeoff between local information on voxel microstructure provided by diffusion data and a priori information on the low curvature of plausi ..."
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Cited by 21 (8 self)
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A family of methods aiming at the reconstruction of a putative fascicle map from any diffusionweighted dataset is proposed. This fascicle map is defined as a tradeoff between local information on voxel microstructure provided by diffusion data and a priori information on the low curvature of plausible fascicles. The optimal fascicle map is the minimum energy configuration of a simulated spin glass in which each spin represents a fascicle piece. This spin glass is embedded into a simulated magnetic external field that tends to align the spins along the more probable fiber orientations according to diffusion models. A model of spin interactions related to the curvature of the underlying fascicles introduces a low bending potential constraint. Hence, the optimal configuration is a tradeoff between these two kind of forces acting on the spins. Experimental results are presented for the simplest spin glass model made up of compass needles located in the center of each voxel of a tensor based acquisition. Copyright 2002 John Wiley & Sons, Ltd.