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331
Anisotropic Geometric Diffusion in Surface Processing
, 2000
"... INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling o ..."
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Cited by 98 (0 self)
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INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth. On the other hand, curves and surfaces are essential objects in computer aided geometric design and computer graphics. Here, issues are fairing, modelling, deformation, and motion. Recently, geometric evolution problems and variational approaches have entered this research field as well and have turned out to be powerful tools. The aim of our work in the field of surface fairing and surface modelling is to modify "classical" curvature motion in a suitable way and apply it in computer graphics. 2 A GENERAL SCHEME Consider an image I :# R. A well known approach to image processing c
Analysis of the heterogeneous multiscale method for ordinary differential equations
 Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 56 (7 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.
A finite element method for domain decomposition with nonmatching grids
 ESAIM: Mathematical Modeling and Numerical Analysis
"... Abstract. In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to selfadjoint elliptic problems, using Poisson’s equation a ..."
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Cited by 53 (8 self)
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Abstract. In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to selfadjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
A Finite Element Method for Surface Restoration with Smooth Boundary Conditions
, 2004
"... In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundar ..."
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Cited by 49 (7 self)
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In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as system of two coupled second order equations. As it is wellknown, fourth order problems require an implicit time discretization.
Nonlinear Diffusion in Graphics Hardware
 In Proceedings of EG/IEEE TCVG Symposium on Visualization VisSym ’01
, 2001
"... Multiscale methods have proved to be successful tools in image denoising, edge enhancement and shape recovery. They are based on the numerical solution of a nonlinear diffusion problem where a noisy or damaged image which has to be smoothed or restorated is considered as initial data. Here a novel a ..."
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Cited by 38 (2 self)
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Multiscale methods have proved to be successful tools in image denoising, edge enhancement and shape recovery. They are based on the numerical solution of a nonlinear diffusion problem where a noisy or damaged image which has to be smoothed or restorated is considered as initial data. Here a novel approach is presented which will soon be capable to ensure real time performance of these methods. It is based on an implementation of a corresponding finite element scheme in texture hardware of modern graphics engines. The method regards vectors as textures and represents linear algebra operations as texture processing operations. Thus, the resulting performance can profit from the superior bandwidth and the build in parallelism of the graphics hardware. Here the concept of this approach is introduced and perspectives are outlined picking up the basic Perona Malik model on 2D images. 1
Towards Fast NonRigid Registration
 IN INVERSE PROBLEMS, IMAGE ANALYSIS AND MEDICAL IMAGING, AMS SPECIAL SESSION INTERACTION OF INVERSE PROBLEMS AND IMAGE ANALYSIS
, 2002
"... A fast multiscale and multigrid method for the matching of images in 2D and 3D is presented. Especially in medical imaging this problem  denoted as the registration problem  is of fundamental importance in the handling of images from multiple image modalities or of image time series. The paper res ..."
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Cited by 36 (16 self)
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A fast multiscale and multigrid method for the matching of images in 2D and 3D is presented. Especially in medical imaging this problem  denoted as the registration problem  is of fundamental importance in the handling of images from multiple image modalities or of image time series. The paper restricts to the simplest matching energy to be minimized, i.e., E[] = R jf 1 f2 j , where f1 , f2 are the intensity maps of the two images to be matched and is a deformation. The focus is on a robust and efficient solution strategy. Matching of
Using Graphics Cards for Quantized FEM Computations
 in IASTED Visualization, Imaging and Image Processing Conference
, 2001
"... Graphics cards exercise increasingly more computing power and are highly optimized for high data transfer volumes. In contrast typical workstations perform badly when data exceeds their processor caches. Performance of scientific computations very often is wrecked by this deficiency. Here we present ..."
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Cited by 30 (4 self)
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Graphics cards exercise increasingly more computing power and are highly optimized for high data transfer volumes. In contrast typical workstations perform badly when data exceeds their processor caches. Performance of scientific computations very often is wrecked by this deficiency. Here we present a novel approach by shifting the computational load from the CPU to the graphics card. We represent data in images and operations on vectors in graphics operations on images. Broad access to graphics memory and parallel processing of image operands thus turns the graphics card into an ultrafast vector coprocessor. The presented strategy opens up a wide area of numerical applications for hardware acceleration. The implementations of Finite Element solvers for the linear heat equation and the anisotropic diffusion method in image processing underline its practicability. We explain the vector processor usage of graphics cards in detail. An extensive correspondence of vector and graphics operations is given and the decomposition of complex operations into hardware supported is explicated. We also sketch the realization of arbitrary number formats in graphics hardware and the consequences of the restricted precision. Finally, we propose slight modifications and extensions which would further improve computational benefits and extend the range of applicability of the proposed approach. Computing in image processing at ms for an Jacobi iteration on images is exemplarily depicted as an ideal field, where Finite Element methods can be greatly accelerated and ultimate number precision is not required.
A level set formulation for Willmore flow
 INTERFACES FREE BOUNDARIES
, 2004
"... A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set f ..."
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Cited by 30 (8 self)
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A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. The approach in particular allows to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretization are discussed and some numerical simulations are presented.
Error Estimates With Smooth And Nonsmooth Data For A Finite Element Method For The CahnHilliard Equation
 Math. Comp
, 1992
"... . A finite element method for the CahnHilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained f ..."
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Cited by 29 (8 self)
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. A finite element method for the CahnHilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate. 1. Introduction. The CahnHilliard equation (1.1) u t + \Delta 2 u \Gamma \DeltaOE(u) = 0; x 2\Omega ae R 3 ; t ? 0; w...