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Preface to the Second Edition | |
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Introduction | |
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Equations of Motion for Moving Interfaces | |
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Formulation of Interface Propagation | |
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A boundary value formulation | |
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An initial value formulation | |
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Advantages of these perspectives | |
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A general framework | |
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A look ahead/A look back | |
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A larger perspective | |
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Theory and Algorithms | |
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Theory of Curve and Surface Evolution | |
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Fundamental formulation | |
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Total variation: stability and the growth of oscillations | |
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The role of entropy conditions and weak solutions | |
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Effects of curvature | |
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Viscosity Solutions and Hamilton-Jacobi Equations | |
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Viscosity solutions of Hamilton-Jacobi equations | |
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Some additional comments and references | |
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Traditional Techniques for Tracking Interfaces | |
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Marker/string methods | |
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Volume-of-fluid techniques | |
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Constructing an approximation to the gradient | |
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Hyperbolic Conservation Laws | |
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The linear wave equation | |
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The non-linear wave equation | |
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Basic Algorithms for Interface Evolution | |
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Convergence of schemes for Hamilton-Jacobi equations | |
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Hyperbolic schemes and Hamilton-Jacobi equations | |
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The example of a propagating one-dimensional graph | |
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The initial value problem: the Level Set Method | |
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The boundary value problem: the stationary method | |
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Schemes for non-convex speed functions | |
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Approximations to geometric variables | |
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Calculating additional quantities | |
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Initialization | |
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Computational domain boundary conditions | |
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Putting it all together | |
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Efficiency, Adaptivity, and Extensions | |
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Efficient Schemes: the Narrow Band Level Set Method | |
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Parallel algorithms | |
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Adaptive mesh refinement | |
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Narrow banding and fast methods | |
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Details of the Narrow Band implementation | |
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Efficient Schemes: Fast Marching Methods | |
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Iteration | |
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Causality | |
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The update procedure for the Fast Marching Method | |
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Heap sorts and computational efficiency | |
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Initial conditions | |
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Network path algorithms | |
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Optimal orderings | |
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Higher accuracy Fast Marching Methods | |
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Non-uniform orthogonal grids | |
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General static Hamilton-Jacobi equations | |
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Some clarifying comments | |
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Triangulated Versions of Level Set Methods | |
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Fundamentals and notation | |
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A monotone scheme for H ([down triangle, open]u) | |
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A positive scheme for homogeneous H ([down triangle, open]u) | |
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A Petrov-Galerkin formulation | |
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Time integration schemes | |
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Algorithms | |
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Schemes for curvature flow | |
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Mesh adaptivity | |
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Triangulated Fast Marching Methods | |
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The update procedure | |
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A scheme for a particular triangulated domain | |
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Fast Marching Methods on triangulated domains | |
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Constructing Extension Velocities | |
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The need for extension velocities | |
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Various approaches to extension velocities | |
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Equations for extension velocities | |
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Building extension velocities | |
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A quick demonstration | |
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Re-initialization | |
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Tests of Basic Methods | |
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The basic Cartesian Level Set Method | |
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Triangulated Level Set Methods for H-J equations | |
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Accuracy of Fast Marching Methods | |
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Tests of extension velocity methodology | |
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Building Level Set and Fast Marching Applications | |
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Applications | |
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Geometry | |
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Statement of problem | |
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Equations of motion | |
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Results | |
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Flows under more general metrics | |
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Volume-preserving flows | |
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Motion under the second derivative of curvature | |
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Triple points: variational and diffusion methods | |
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Grid Generation | |
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Statement of problem | |
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Equations of motion | |
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Results, complications, and future work | |
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Image Enhancement and Noise Removal | |
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Statement of problem | |
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Equations of motion | |
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Results | |
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Related work | |
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Computer Vision: Shape Detection and Recognition | |
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Shape-from-shading | |
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Shape detection/recovery | |
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Surface evolution and the stereo problem | |
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Reconstruction of obstacles in inverse problems | |
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Shape recognition | |
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Combustion, Solidification, Fluids, and Electromigration | |
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Combustion | |
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Crystal growth and dendritic solidification | |
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Fluid mechanics | |
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Additional applications | |
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Void evolution and electromigration | |
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Computational Geometry and Computer-aided Design | |
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Shape-Offsetting | |
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Voronoi diagrams | |
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Curve flows with constraints | |
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Minimal surfaces and surfaces of prescribed curvature | |
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Extensions to surfaces of prescribed curvature | |
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Boolean operations on shapes | |
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Extracting and combining two-dimensional shapes | |
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Shape smoothing | |
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Optimality and First Arrivals | |
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Optimal path planning | |
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Constructing shortest paths on weighted domains | |
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Constructing shortest paths on manifolds | |
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Seismic traveltimes | |
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Aircraft collision avoidance using Level Set Methods | |
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Visibility evaluations | |
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Etching and Deposition in Microchip Fabrication | |
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Physical effects and background | |
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Equations of motion for etching/deposition | |
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Additional numerical issues | |
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Two-dimensional results | |
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Three-dimensional simulations | |
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Timings | |
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Validation with experimental results | |
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Summary/New Areas/Future Work | |
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Bibliography | |
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Index | |