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Delaunay Deformable Models: Topologyadaptive Meshes Based on the Restricted Delaunay triangulation
, 2006
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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.
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
A levelset method for interfacial flows with surfactant
, 2006
"... ... drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid i ..."
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Cited by 17 (1 self)
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... drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid interfaces with surfactant.
Eulerian finite element method for parabolic PDEs on implicit surfaces
 INTERFACES AND FREE BOUNDARIES 10 (2008), 119–138
, 2008
"... We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface Γ contained in a domain Ω ⊂ R n+1. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function Φ whose zero level set is Γ. Euleri ..."
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Cited by 16 (3 self)
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We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface Γ contained in a domain Ω ⊂ R n+1. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function Φ whose zero level set is Γ. Eulerian surface gradients are formulated by using a projection of the gradient in R n+1 onto the level surfaces of Φ. These Eulerian surface gradients are used to define weak forms of surface elliptic operators and so generate weak formulations of surface elliptic and parabolic equations. The resulting equation is then solved in one dimension higher but can be solved on a mesh which is unaligned to the level sets of Φ. We consider both second order and fourth order elliptic operators with natural second order splittings. The finite element method is applied to the weak form of the split system of second order equations using piecewise linear elements on a fixed grid. The computation of the mass and element stiffness matrices is simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.
How to Deal With Point Correspondences and Tangential Velocities in the Level Set Framework
, 2003
"... In this report, we overcome a major drawback of the level set framework: the lack of point correspondences. We maintain explicit backward correspondences from the evolving interface to the initial one by advecting the initial point coordinates with the same velocity as the level set function. Our me ..."
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Cited by 14 (6 self)
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In this report, we overcome a major drawback of the level set framework: the lack of point correspondences. We maintain explicit backward correspondences from the evolving interface to the initial one by advecting the initial point coordinates with the same velocity as the level set function. Our method leads to a system of coupled Eulerian partial differential equations. We show in a variety of numerical experiments that it can handle both normal and tangential velocities, large deformations, shocks, rarefactions and topological changes. Applications are many since our method can upgrade virtually any level set evolution. We complement our work with the design of non zero tangential velocities that preserve the relative area of interface patches; this feature may be crucial in such applications as computational geometry, grid generation or unfolding of the organs' surfaces, e.g. brain, in medical imaging. This report also tackles a diffeomorphic approach to level set evolution, a family of volumepreserving smoothing flows, and some numerical aspects of the intrinsic heat ow on implicit surfaces.
Maintaining the Point Correspondence in the Level Set Framework
, 2006
"... In this paper, we propose a completely Eulerian approach to maintain a point correspondence during a level set evolution. Our work is in the spirit of some recent methods (Adalsteinsson & Sethian, J. Comp. Phys. 2003; Xu & Zhao, J. Sci. Comp. 2003) for handling interfacial data on moving lev ..."
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Cited by 12 (4 self)
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In this paper, we propose a completely Eulerian approach to maintain a point correspondence during a level set evolution. Our work is in the spirit of some recent methods (Adalsteinsson & Sethian, J. Comp. Phys. 2003; Xu & Zhao, J. Sci. Comp. 2003) for handling interfacial data on moving level set interfaces. Our approach maintains an explicit backward correspondence from the evolving interface to the initial one, by advecting the initial point coordinates with the same velocity as the level set function. It leads to a system of coupled Eulerian partial differential equations. We describe in detail a robust numerical implementation of our approach, in accordance with the narrow band methodology. We show in a variety of numerical experiments that it can handle both normal and tangential velocities, large deformations, shocks, rarefactions and topological changes. The possible applications of our approach include scientific visualization, computer graphics and image processing.
Spacetimecoherent Geometry Reconstruction from Multiple Video Streams
"... By reconstructing timevarying geometry one frame at a time, one ignores the continuity of natural motion, wasting useful information about the underlying videoimage formation process and taking into account temporally discontinuous reconstruction results. In 4D spacetime, the surface of a dynamic ..."
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Cited by 10 (0 self)
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By reconstructing timevarying geometry one frame at a time, one ignores the continuity of natural motion, wasting useful information about the underlying videoimage formation process and taking into account temporally discontinuous reconstruction results. In 4D spacetime, the surface of a dynamic object describes a continuous 3D hypersurface. This hypersurface can be implicitly defined as the minimum of an energy functional designed to optimize photoconsistency. Based on an EulerLagrange reformulation of the problem, we find this hypersurface from a handful of synchronized video recordings. The resulting object geometry varies smoothly over time, and intermittently invisible object regions are correctly interpolated from previously and/or future frames.
OutofCore and Compressed Level Set Methods
"... This article presents a generic framework for the representation and deformation of level set surfaces at extreme resolutions. The framework is composed of two modules that each utilize optimized and application specific algorithms: 1) A fast outofcore data management scheme that allows for resolu ..."
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Cited by 9 (2 self)
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This article presents a generic framework for the representation and deformation of level set surfaces at extreme resolutions. The framework is composed of two modules that each utilize optimized and application specific algorithms: 1) A fast outofcore data management scheme that allows for resolutions of the deforming geometry limited only by the available disk space as opposed to memory, and 2) compact and fast compression strategies that reduce both offline storage requirements and online memory footprints during simulation. Outofcore and compression techniques have been applied to a wide range of computer graphics problems in recent years, but this article is the first to apply it in the context of level set and fluid simulations. Our framework is generic and flexible in the sense that the two modules can transparently be integrated, separately or in any combination, into existing level set and fluid simulation software based on recently proposed narrow band data structures like the DTGrid of Nielsen and Museth [2006] and the HRLE of Houston et al. [2006]. The framework can be applied to narrow band signed distances, fluid velocities, scalar fields, particle properties as well as standard graphics attributes like colors, texture coordinates, normals, displacements etc. In fact, our framework is applicable to a large body of computer graphics problems that involve sequential or random access to very large codimension one (level set) and zero (e.g. fluid) data sets. We demonstrate this with several applications, including fluid simulations interacting with large boundaries ( ≈ 15003), surface deformations ( ≈ 20483), the solution of partial differential equations on large surfaces ( ≈ 40963) and meshtolevel set scan conversions of resolutions up to ≈ 350003 (7 billion voxels in the narrow band). Our outofcore framework is shown to be several times faster than current stateoftheart level set data structures relying on OS paging. In particular we show sustained throughput (grid points/sec) for gigabyte sized level sets as high as 65 % of stateoftheart throughput for incore simulations. We also demonstrate that our compression techniques outperform stateoftheart
C.: An hnarrow band finiteelement method for elliptic equations on implicit surfaces
 IMA Journal of Numerical Analysis
, 2010
"... In this article we define a finiteelement method for elliptic partial differential equations (PDEs) on curves or surfaces, which are given implicitly by some level set function. The method is specially designed for complicated surfaces. The key idea is to solve the PDE on a narrow band around the ..."
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Cited by 9 (0 self)
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In this article we define a finiteelement method for elliptic partial differential equations (PDEs) on curves or surfaces, which are given implicitly by some level set function. The method is specially designed for complicated surfaces. The key idea is to solve the PDE on a narrow band around the surface. The width of the band is proportional to the grid size. We use finiteelement spaces that are unfitted to the narrow band, so that elements are cut off. The implementation nevertheless is easy. We prove error estimates of optimal order for a Poisson equation on a surface and provide numerical tests and examples.