A Variational Level Set Approach to Multiphase Motion

doi:10.1006/JCPH.1996.0167

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A Variational Level Set Approach to Multiphase Motion

Journal Article•DOI•

A Variational Level Set Approach to Multiphase Motion

Hongkai Zhao¹, Tony F. Chan¹, Barry Merriman¹, Stanley Osher¹•Institutions (1)

University of California, Los Angeles¹

01 Aug 1996-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 127, Iss: 1, pp 179-195

TL;DR: In this paper, a coupled level set method for the motion of multiple junctions (of, e.g., solid, liquid, and grain boundaries), which follows the gradient flow for an energy functional consisting of surface tension and bulk energies, is developed.

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About: This article is published in Journal of Computational Physics.The article was published on 1996-08-01. It has received 1158 citations till now. The article focuses on the topics: Level set method & Level set (data structures).

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Journal Article•DOI•

Active contours without edges

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Tony F. Chan¹, Luminita A. Vese¹•Institutions (1)

University of California, Los Angeles¹

01 Feb 2001-IEEE Transactions on Image Processing

TL;DR: A new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets is proposed, which can detect objects whose boundaries are not necessarily defined by the gradient.

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Abstract: We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

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10,404 citations

Cites background or methods from "A Variational Level Set Approach to..."

...When working with level sets and Dirac delta functions, a standard procedure is to reinitialize to the signed distance function to its zero-level curve, as in [25] and [27]....
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...We mention that, in order to extend the evolution to all level sets of , another possibility is to replace by (see [27])....
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...First possible regularization of by functions, as proposed in [27], is...
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...For the level set formulation of our variational active contour model, we replace the unknown variable by the unknown variable , and we follow [27]....
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Proceedings Article•DOI•

Image inpainting

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Marcelo Bertalmío¹, Guillermo Sapiro¹, V. Caselles², Coloma Ballester²•Institutions (2)

University of Minnesota¹, Pompeu Fabra University²

01 Jul 2000

TL;DR: A novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators, and does not require the user to specify where the novel information comes from.

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Abstract: Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators. After the user selects the regions to be restored, the algorithm automatically fills-in these regions with information surrounding them. The fill-in is done in such a way that isophote lines arriving at the regions' boundaries are completed inside. In contrast with previous approaches, the technique here introduced does not require the user to specify where the novel information comes from. This is automatically done (and in a fast way), thereby allowing to simultaneously fill-in numerous regions containing completely different structures and surrounding backgrounds. In addition, no limitations are imposed on the topology of the region to be inpainted. Applications of this technique include the restoration of old photographs and damaged film; removal of superimposed text like dates, subtitles, or publicity; and the removal of entire objects from the image like microphones or wires in special effects.

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3,830 citations

Journal Article•DOI•

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

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Alex M. Andrew

01 Mar 2000-Kybernetes

3,822 citations

Journal Article•DOI•

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

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Luminita A. Vese¹, Tony F. Chan¹•Institutions (1)

University of California, Los Angeles¹

01 Dec 2002-International Journal of Computer Vision

TL;DR: A new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations, and validated by numerical results for signal and image denoising and segmentation.

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Abstract: We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

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2,649 citations

Cites background or methods from "A Variational Level Set Approach to..."

...In order to keep the phases disjoint (no overlap) and their union the domain (no vacuum), the authors in Zhao et al. (1996) have added an additional term to the energy which is minimized, in the form λ ∫ ( ∑ i H (φi ) − 1)2dxdy....
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...Other possibilities for the extension step can be found in Zhao et al. (1996), Chen et al. (1997), Fedkiw et al. (1999), Fedkiw (1999), Jensen (1993), and Caselles et al. (1997)....
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...Most of the models need three level set functions, as in Zhao et al. (1996) and Samson et al. (1999, 2000)....
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...A standard rescaling can be made, as in Zhao et al. (1996), by replacing δε(φ) by |∇φ|, giving the following equations, already introduced in Osher and Sethian (1988) in the context of the level set theory: ∂φ ∂t = |∇φ|div ( ∇φ |∇φ| ) , or ∂φ ∂t = |∇φ| (4) (motion by mean curvature minimizing the…...
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...As mentioned above, a first idea was proposed in Zhao et al. (1996), and then applied in Samson et al. (1999, 2000): a level set function is associated to each phase or each connected component i (this is also used in Paragios and Deriche (2000))....
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Journal Article•DOI•

Level set methods: an overview and some recent results

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Stanley Osher¹, Ronald Fedkiw²•Institutions (2)

University of California, Los Angeles¹, Stanford University²

30 May 2001-Journal of Computational Physics

TL;DR: The level set method is couple to a wide variety of problems involving external physics, such as compressible and incompressible flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films, vortex-dominated flows, and extensions to multiphase motion.

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2,174 citations

Cites background from "A Variational Level Set Approach to..."

...The algorithm in [2] claims O(N) complexity, but this is not borne out by the numerical evidence presented there....
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...High order accurate, essentially non-oscillatory discretizations to general Hamilton-Jacobi equations including (3) were obtained in [64], see also [65] and [43]....
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References

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Journal Article•DOI•

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

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Stanley Osher¹, James A. Sethian²•Institutions (2)

University of California, Los Angeles¹, University of California, Berkeley²

01 Nov 1988-Journal of Computational Physics

TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.

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13,020 citations

Journal Article•DOI•

Efficient implementation of essentially non-oscillatory shock-capturing schemes,II

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Chi-Wang Shu¹, Stanley Osher²•Institutions (2)

Brown University¹, University of California, Los Angeles²

01 Jul 1989-Journal of Computational Physics

TL;DR: Two methods of sharpening contact discontinuities-the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework-are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations.

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5,292 citations

Journal Article•DOI•

A level set approach for computing solutions to incompressible two-phase flow

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Mark Sussman¹, Peter Smereka¹, Stanley Osher¹•Institutions (1)

University of California, Los Angeles¹

01 Sep 1994-Journal of Computational Physics

TL;DR: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number.

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4,148 citations

A level set approach for computing solutions to incompressible two- phase flow II

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Mark Sussman, Emad Fatemi, Stanley Osher, Peter Smereka

01 Jun 1995

TL;DR: In this article, a level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number.

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Abstract: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables one to compute flows with large density ratios (1000/1) and flows that are surface tension driven; with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, to name a few. We validate our code against experiments and theory.

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3,556 citations

Journal Article•DOI•

Motion of multiple junctions: a level set approach

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Barry Merriman¹, James K. Bence¹, Stanley Osher¹•Institutions (1)

University of California, Los Angeles¹

01 Jun 1994-Journal of Computational Physics

TL;DR: In this paper, a coupled level set method for the motion of multiple junctions is proposed, which retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front.

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521 citations