A Variational Level Set Approach to Multiphase Motion

Zhao, Hongkai; Chan, Tony F.; Merriman, Barry; Osher, Stanley

doi:10.1006/jcph.1996.0167

Cited by 945 publications

(740 citation statements)

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“…If V is the outward normal velocity of the interface, then we update the position of the interface implicitly via (7) where Ṽ is an extension of V off of the interface. The extension Ṽ is often obtained using a Hamilton-Jacobi PDE (e.g., see [55] and [2]). The fast marching method developed by Adalsteinsson and Sethian in [2] constructs an extension Ṽ while simultaneously reinitializing the level set function using an ordered sequence of discrete operations, but is only first-order accurate.…”

Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: Narrow Band/local Level Set Methodsmentioning

confidence: 99%

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…Later, it has been used for many different kind of physical problems, see [12][13][14]16]. The main idea behind this method is to represent the interface at each time t as the zero level set of a function ϕ.…”

Section: An Overview Of Level Set Methodsmentioning

confidence: 99%

Numerical Optimization of the Method of Auxiliary Sources by Using Level Set Technique

Bouzidi¹,

Aguili²

2011

PIER B

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“…So, C = {(x, y) ∈ Ω : φ(x, y) = 0}, and we choose φ to be positive inside C and negative outside C. For the level set formulation of our variational active contour model we essentially follow [20]. Therefore, we replace the unknown variable C by the unknown variable φ and the new energy, still denoted by F (φ, c 1 , c 2 ), becomes:…”

Section: The Level Set Formulation Of the Modelmentioning

confidence: 99%

“…A first possible regularization by C 2 and respectively C 1 functions, as proposed for instance in [20], is:…”

Section: Using the Heaviside Function H Defined Bymentioning

confidence: 99%

“…We want to formally explain here why we need to introduce the second approximations, instead of the first approximations, which have been used in previous papers (for instance in [20]). Because our energy is non-convex (allowing therefore many local minima), and because δ 1,ε has a very small compact support, the interval [−ε, ε], the iterative algorithm may depend on the initial curve, and will not necessarily compute a global minimizer.…”

Section: Using the Heaviside Function H Defined Bymentioning

confidence: 99%

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An Active Contour Model without Edges

Chan

Vese

1999

Lecture Notes in Computer Science

490

538

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Abstract. In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. The model is a combination between more classical active contour models using mean curvature motion techniques, and the Mumford-Shah model for segmentation. We minimize an energy which can be seen as a particular case of the so-called 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. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable.

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A Variational Level Set Approach to Multiphase Motion

Cited by 945 publications

References 10 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Numerical Optimization of the Method of Auxiliary Sources by Using Level Set Technique

An Active Contour Model without Edges

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