Level Set and PDE Methods for Visualization
Ross Whitaker2005, IEEE Visualization 2005 - (VIS'05)
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Abstract
Level set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (isosurface) of a sampled, evolving nD function. This course is targeted for researchers interested in learning about level set and other PDE-based methods, and their application to visualization. The course material will be presented by several of the recognized experts in the field, and will include introductory concepts, practical considerations and extensive details on a variety of level set/PDE applications. The course will begin with preparatory material that introduces the concept of using partial differential equations to solve problems in visualization. This will include the structure and behavior of several different types of differential equations, e.g. the level set, heat and reaction-diffusion equations, as well as a general approach to developing PDE-based applications. The second stage of the course will describe the numerical methods and algorithms needed to implement the mathematics and methods presented in the first stage, including information on implementing the algorithms on GPUs. Throughout the course the technical material will be tied to applications, e.g. image processing, geometric modeling, dataset segmentation, model processing, surface reconstruction, anisotropic geometric diffusion, flow field post-processing and vector visualization. Prerequisites Knowledge of calculus, linear algebra, computer graphics, visualization, geometric modeling and computer vision. Some familiarity with differential geometry, differential equations, numerical computing and image processing is strongly recommended, but not required.
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Level set and PDE methods for computer graphics
ACM SIGGRAPH 2004 Course Notes, 2004
The great idea behind level set methods is to describe properties like density, velocity or color with a function over some domain (usually a region in 2D or 3D). Level sets allone are an inherently static concept. To allow modeling of dynamic processes, which means change of level set functions over time, partial differential equations (PDE) are introduced. We review basic principles and a few applications of level set and PDE methods for computer graphics that have recently been proposed.
A Diffusion-Augmented Level Set Method with Efficient Two Step Implementation
2013
The level set method was devised by Osher and Sethian [2] in as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field v [1]. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth function ϕ(x, t), i.e., Γ (t) = {x|ϕ(x, t) = 0}. ϕ is positive inside Ω, negative outside Ω and is zero on Γ (t) [1]. This paper presents a reaction-diffusion method used to describe a physicochemical phenomenon that comprises two elements, namely chemical reactions and diffusion for implicit active contours[21][37][39][40], which is completely free of the costly re-initialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in a diffusion-augmented level set method with efficien...
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Mathematics and Visualization, 2009
The visualization of stationary and time-dependent flow is an important and challenging topic in scientific visualization. Its aim is to represent transport phenomena governed by vector fields in an intuitively understandable way. In this paper, we review the use of methods based on partial differential equations (PDEs) to post-process flow datasets for the purpose of visualization. This connects flow visualization with image processing and mathematical multi-scale models. We introduce the concepts of flow operators and scale-space and explain their use in modeling post processing methods for flow data. Based on this framework, we present several classes of PDE-based visualization methods: anisotropic linear diffusion for stationary flow; transport and diffusion for non-stationary flow; continuous clustering based on phase-separation; and an algebraic clustering of a matrix-encoded flow operator. We illustrate the presented classes of methods with results obtained from concrete flow applications, using datasets in 2D, flows on curved surfaces, and volumetric 3D fields.
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Journal of Computational Physics, 1999
We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O(N ), where N is the number of points in space, not O(N log N ) as in works by , Proc. Nat. Acad. Sci. 93, 1591 and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method. 411 interface problem has been transformed into a two dimensional problem. In three space dimensions, considerable computational labor (O(n 3 )) is required per time step."
A PDE-Based Fast Local Level Set Method1
1991
We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O(N ), where N is the number of points in space, not O(N log N ) as in works by , Proc. Nat. Acad. Sci. 93, 1591 and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method. 411 interface problem has been transformed into a two dimensional problem. In three space dimensions, considerable computational labor (O(n 3 )) is required per time step."
A gradient-augmented level set method with an optimally local, coherent advection scheme
Journal of Computational Physics, 2010
The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the
On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion
Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.
A nonlinear PDE model for reconstructing a regular surface from sampled data using a level set formulation on triangular meshes
Journal of Computational Physics, 2011
In this paper, we propose a nonlinear PDE model for reconstructing a regular surface from sampled data. At first, we show the existence and the uniqueness of a viscosity solution to this problem. Then we propose a numerical scheme for solving the nonlinear level set equation on unstructured triangulations adapted to the data sample. We show the consistency of this scheme. In addition, we show how to compute nodewise first and second order derivatives. Some application examples of curve or surface construction are provided to illustrate the potential and to demonstrate the accuracy of this method.
Interactive deformation and visualization of level set surfaces using graphics hardware
IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control
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A level set technique applied to unsteady free surface flows
International Journal for Numerical Methods in Fluids, 2001
An unsteady Navier-Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions. The two-phase flow of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface. The free surface location is captured as the zero level of a distance function convected by the flow field. To validate the numerical procedure, two classical two-dimensional free surface problems in hydrodynamics, namely the oscillating flow in a tank and the waves generated by the flow over a bottom bump, are studied in non-breaking conditions, and the results are compared with those obtained with other numerical approaches. To check the capability of the method in dealing with complex free surface configurations, the breaking regime produced by the flow over a high bump is analyzed. The analysis covers the successive stages of the breaking phenomenon: the steep wave evolution, the falling jet, the splash-up and the air entrainment. In all phases, numerical results qualitatively agree with the experimental observations. Finally, to investigate a flow in which viscous effects are relevant, the numerical scheme is applied to study the wavy flow past a submerged hydrofoil.
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Figure 1: Dense flow fields are first converted into a scalar field, and then displayed and analyzed by means of level-sets in this field. An oracle that is based on the discrete curvature of level-sets allows for the automatic separation and extraction of homogeneous streams.
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2003
We introduce a new computational technique for evolving interfaces, the flux-based level set method. A nonlinear degenerate advection-diffusion level set equation is discretized by a finite volume method using a complementary volume strategy. It enables to solve the problem in an efficient and stable way. Using a flux-based method of characteristics for the advective part and a semi-implicit treatment of diffusive part, it removes the standard CFL condition on time step and it decreases CPU times significantly. The method is presented for 2D and 3D interface motions driven in normal direction by a constant and spatially varying driving force and (mean) curvature. Comparisons with known exact solutions and further numerical experiments, including topological changes of the interface, are presented.
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In this talk we present a brief introduction to level set methods. The basic mathematical formulation will be recalled, as well as the most common numerical methods used for the discretization of the equation. Some application to the computation of signed distance function, motion by mean curvature, and crystal gwowth will be illustrated.
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Level Set is a deformable contour model where the user specifies a starting contour that is evolved to the image contour. As opposed to other contour models, e.g. Snakes [1], where the contour is described in a parametric manner, the Level Set method is a geometric deformable model. The contour is described as a surface developed by partial differential equations, where the contour is the zero level of the surface. In this paper we give novel pure-particle algorithm for the simulation of reaction-diffusion systems on deforming surfaces[5], which represent an important class of biological models, Because they provide explanations to complex phenomena such as pattern formation or morphogenesis. The algorithm uses an implicit Lagrangian level-set representation to track the motion of the surface, the framework of discretizationcorrected PSE operators to discretize the spatial derivatives of the governing equations as well as pseudo-forces to adapt the particle distribution to local resolution requirements, which renders the use of Cartesian grids unnecessary. A diffusion term is introduced into LSE, resulting in a TSSM equation, to which a piecewise constant solution can be derived. We propose a two-step splitting method (TSSM) to iteratively solve the RD-LSE equation: first iterating the LSE equation, and then solving the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly re-initialization procedure is completely eliminated from LSE.
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This paper is devoted to further modification of the Level Set approach, which is well-known for simulation of gas-liquid flows with the interface. In our development, we addressed to the case of a strong velocity gradient at the free interface. This is a typical situation, for example, when this interface interacts with the turbulent flow. In this case, the gradients of the level set scalar, in the vicinity of the interface, increase with time very rapidly. In order to maintain the accuracy of the numerical solution, the Level Set methods are combined usually with the Eikonal equation for a signed distance function from the zero level set. In the standard procedure (Sussman et al., J. Comput. Phys. 114, 1994), in order to be consistent with evolutional type of the Level Set equation, the nonevolutional Eikonal equation is replaced by quasi-evolutional one, with the artificial time providing iterations at each time step. Our idea is to modify the Level Set equation, in such a way that the Eikonal equation is satisfied directly by the form of the modified equation. This was done in the proposed paper. The efficiency of the proposed method is demonstrated by using various tests problems with interface.
A level set approach to anisotropic flows with curvature regularization
Journal of Computational Physics, 2007
Modeling and simulation of faceting effects on surfaces are topics of growing importance in modern nanotechnology. Such effects pose various theoretical and computational challenges, since they are caused by non-convex surface energies, which lead to ill-posed evolution equations for the surfaces. In order to overcome the ill-posedness, regularization of the energy by a curvature-dependent term has become a standard approach, which seems to be related to the actual physics, too. The use of curvature-dependent energies yields higher order partial differential equations for surface variables, whose numerical solution is a very challenging task.
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