Results 11  20
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127
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).
Shedding light on stereoscopic segmentation
 In CVPR
, 2004
"... We propose a variational algorithm to jointly estimate the shape, albedo, and light configuration of a Lambertian scene from a collection of images taken from different vantage points. Our work can be thought of as extending classical multiview stereo to cases where point correspondence cannot be e ..."
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Cited by 33 (3 self)
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We propose a variational algorithm to jointly estimate the shape, albedo, and light configuration of a Lambertian scene from a collection of images taken from different vantage points. Our work can be thought of as extending classical multiview stereo to cases where point correspondence cannot be established, or extending classical shape from shading to the case of multiple views with unknown light sources. We show that a first naive formalization of this problem yields algorithms that are numerically unstable, no matter how close the initialization is to the true geometry. We then propose a computational scheme to overcome this problem, resulting in provably stable algorithms that converge to (local) minima of the cost functional. Although we restrict our attention to Lambertian objects with uniform albedo, extensions of our framework are conceivable. 1
Generalized gradients: priors on minimization flows
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2007
"... This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literatu ..."
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Cited by 31 (4 self)
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This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our applicationspecific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.
Estimation of 3D surface shape and smooth radiance from 2D images: A level set approach
 JOURNAL OF SCIENTIFIC COMPUTING
, 2003
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Fourth order partial differential equations on general geometries
 UNIVERSITY OF CALIFORNIA LOS ANGELES
, 2005
"... We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the ..."
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Cited by 26 (4 self)
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We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the level set of a smooth function, φ ¢ we compute the PDE using only finite differences on a standard Cartesian mesh in ¡ N. The higher order equations introduce a number of challenges that are of small concern when applying this method to first and second order PDEs. Many of these problems, such as timestepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the CahnHilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.
Brain surface conformal parameterization using riemann surface structure
 IEEE Trans. Med. Imaging
, 2007
"... Abstract—In medical imaging, parameterized 3D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surfacebased registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a s ..."
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Cited by 25 (17 self)
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Abstract—In medical imaging, parameterized 3D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surfacebased registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3D brain surface manifolds to modified PDEs on a twodimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified
Level Set Methods and Their Applications in Image Science
 Comm. Math Sci
"... this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applicatio ..."
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Cited by 23 (1 self)
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this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multidisciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field
Level set equations on surfaces via the Closest Point Method
, 2007
"... Level set methods have been used in a great number of applications in R 2 and R 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in R 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recen ..."
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Cited by 23 (6 self)
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Level set methods have been used in a great number of applications in R 2 and R 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in R 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recent technique for solving partial differential equations (PDEs) on surfaces, the Closest Point Method [RM06]. Our main modification is to introduce a Weighted Essentially NonOscillatory (WENO) interpolation step into the Closest Point Method. This, in combination with standard WENO for Hamilton–Jacobi equations, gives highorder results (up to fifthorder) on a variety of smooth test problems including passive transport, normal flow and redistancing. The algorithms we propose are straightforward modifications of standard codes, are carried out in the embedding space in a welldefined band around the surface and retain the robustness of the level set method with respect to the selfintersection of interfaces. Numerous examples are provided to illustrate the flexibility of the method with respect to geometry. 1
Inviscid and Incompressible Fluid Simulation on Triangle Meshes
 JOURNAL OF COMPUTER ANIMATION AND VIRTUAL WORLDS
, 2004
"... ... In this paper, we introduce a novel method for inviscid fluid simulation over meshes. It can enforce incompressibility on closed surfaces by utilizing a discrete vector field decomposition algorithm. It also includes effective implementations of semiLagrangian tracing and velocity interpolation ..."
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Cited by 22 (1 self)
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... In this paper, we introduce a novel method for inviscid fluid simulation over meshes. It can enforce incompressibility on closed surfaces by utilizing a discrete vector field decomposition algorithm. It also includes effective implementations of semiLagrangian tracing and velocity interpolation schemes. Different from previous work, our method performs simulations directly on triangle meshes and thus eliminates parametrization distortions. Our implementation can produce convincing fluid motion on surfaces and has interactive performance for meshes with tens of thousands of faces
The isophotic metric and its application to feature sensitive morphology on surfaces
 Computer Vision — ECCV 2004, Part IV, volume 3024 of Lecture Notes in Computer Science
, 2004
"... Abstract. We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface curve is not just dependent on the curve itself, but also on the variation of the surface normals along it. A weak variation of the normals brings the isophotic length of a curve close to its Eucl ..."
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Cited by 22 (5 self)
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Abstract. We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface curve is not just dependent on the curve itself, but also on the variation of the surface normals along it. A weak variation of the normals brings the isophotic length of a curve close to its Euclidean length, whereas a strong normal variation increases the isophotic length. We actually have a whole family of metrics, with a parameter that controls the amount by which the normals influence the metric. We are interested here in surfaces with features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. The isophotic metric is sensitive to those features: paths along features are close to geodesics in the isophotic metric, paths across features have high isophotic length. This shape effect makes the isophotic metric useful for a number of applications. We address feature sensitive image processing with mathematical morphology on surfaces, feature sensitive geometric design on surfaces, and feature sensitive local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects. 1