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352
Gradient flows and geometric active contour models
 in Proc. of the 5th International Conference on Computer Vision
, 1995
"... In this paper, we analyze the geometric active contour models discussed in [6, 181 from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel snake paradigm in which the feature of interes ..."
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Cited by 199 (16 self)
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In this paper, we analyze the geometric active contour models discussed in [6, 181 from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus the snake is attracted very naturally and eficiently to the desired feature. Moreover, we consider some 30 active surface models based on these ideas. 1
The Relaxation Schemes for Systems of Conservation Laws in Arbitrary Space Dimensions
 Comm. Pure Appl. Math
, 1995
"... We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with ..."
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Cited by 160 (19 self)
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We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solver temporally. Numerical results for 1D and 2D problems are presented. The second order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. 1. Introduction In this paper we present a class of nonoscillatory numerical schemes for systems of conservation laws in several space dimensions. The basic idea is to use a local relaxation approximation. For any given system of conservation laws, we will construct a corresponding linear hyp...
Fast Marching Methods
 SIAM Review
, 1998
"... Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static HamiltonJacobi equations. Based on entropysatisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are opti ..."
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Cited by 145 (4 self)
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Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static HamiltonJacobi equations. Based on entropysatisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N ), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shapefromshading, photolithographic development, computing rst arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings, provide details of the computational schemes including higher order versions,...
A geometric snake model for segmentation of medical imagery
 IEEE Transactions on Medical Imaging
, 1997
"... Abstract — In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining featurebased metrics on a given i ..."
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Cited by 120 (18 self)
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Abstract — In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining featurebased metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus, the snake is attracted very quickly and efficiently to the desired feature. Index Terms — Active contours, active vision, edge detection, gradient flows, segmentation, snakes. I.
Conformal Curvature Flows: From Phase Transitions to Active Vision
, 1995
"... In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be consid ..."
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Cited by 117 (30 self)
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In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edgeseeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the AllenCahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3D active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a levelset approach. Key words: Active vision, antiphase boundary, visual tracking, edge detection, segmentation, gradient flows, Riemannian metrics, viscosity solutions, geometric heat equ...
Color TV: Total Variation Methods for Restoration of Vector Valued Images
 IEEE Trans. Image Processing
, 1996
"... We propose a new definition of the total variation norm for vector valued functions which can be applied to restore color and other vector valued images. The new TV norm has the desirable properties of (i) not penalizing discontinuities (edges) in the image, (ii) rotationally invariant in the image ..."
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Cited by 111 (13 self)
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We propose a new definition of the total variation norm for vector valued functions which can be applied to restore color and other vector valued images. The new TV norm has the desirable properties of (i) not penalizing discontinuities (edges) in the image, (ii) rotationally invariant in the image space, and (iii) reduces to the usual TV norm in the scalar case. Some numerical experiments on denoising simple color images in RGB color space are presented. 1 Introduction During gathering and transfer of image data some noise and blur is usually introduced into the image. Several reconstruction methods based on the total variation (TV) norm have been proposed and studied for intensity (gray scale) images, see [9, 14, 21, 26, 29]. Since these methods have been successful in reducing noise and blur without smearing sharp edges for intensity images, it is natural to extend the TV norm to handle color and other vector valued images. Why do we need color restoration? It can be argued that si...
Uniformly accurate schemes for hyperbolic systems with relaxations
 SIAM J. Numer. Anal
, 1997
"... Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underreso ..."
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Cited by 61 (21 self)
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Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one cannot decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a secondorder scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a firstorder scheme and numerical convergence proof for the secondorder scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.
A Fully Global Approach to Image Segmentation via Coupled Curve Evolution Equations
 Journal of Visual Communication and Image Representation
, 2002
"... In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost func ..."
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Cited by 61 (12 self)
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In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost functional. The resulting active contour model, in contrast to many other edge and region based models, is fully global in that the evolution of each curve depends at all times upon every pixel in the image and is directly coupled to the evolution of every other curve regardless of their mutual proximity. As such evolving contours enjoy a very wide “field of view, ” endowing the algorithm with a robustness to initial contour placement above and beyond the significant improvement exhibited by other region based snakes over earlier edge based snakes. C ○ 2002 Elsevier Science (USA) Key Words: active contours; curve evolution; snakes; segmentation; gradient flows.
Tree Methods for Moving Interfaces
, 1999
"... Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effor ..."
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Cited by 56 (7 self)
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Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effort on the interface, so the methods moves an interface with N degrees of freedom in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. The methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging, anisotropy, faceting and curvature.
Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 54 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...