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A Variant of the Level Set Method and Applications to Image Segmentation
 Math. Comp
, 2003
"... Abstract. In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2 n phases. The novelty in our approach is to introduce a pie ..."
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Cited by 38 (10 self)
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Abstract. In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2 n phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If 2 n phases should be identified, the level set function must approach 2 n predetermined constants. We just need one level set function to represent 2 n unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches. 1. Introduction to related level set methods A function u:Ω↦ → R defined on an open and bounded domain Ω ∈ Rm may
A Binary Level Set Model and some Applications to MumfordShah Image Segmentation
"... In this work we propose a variant of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can ..."
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Cited by 34 (5 self)
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In this work we propose a variant of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e. it can only be 1 or1. Some of the properties of the standard level set function are preserved in the proposed method, while others are not. Using this new level set method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth and locally convex. We show numerical results using the method for segmentation of digital images.
Electrical Impedance Tomography Using Level Set Representation and Total Variational Regularization
, 2003
"... In this paper, we propose a numerical scheme for the identification of piecewise constant conductivity coe#cient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with di#er ..."
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Cited by 31 (2 self)
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In this paper, we propose a numerical scheme for the identification of piecewise constant conductivity coe#cient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with di#erent values of coe#cients. Numerical tests show that our method can be able to recover a sharp interface and can tolerate higher level of noise in the observation data. Results concerning the e#ects of number of measurements, noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given.
A SURVEY ON MULTIPLE LEVEL SET METHODS WITH APPLICATIONS FOR IDENTIFYING PIECEWISE CONSTANT FUNCTIONS
, 2004
"... We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the g ..."
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Cited by 30 (9 self)
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We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the general approach for applications to: image segmentation, optimal shape design, elliptic inverse coefficient identification, electricall impedance tomography and positron emission tomography. Numerical experiments are also presented for some of the problems.
On level set regularization for highly illposed distributed parameter estimation problems
, 2005
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A Method of biological tissues elasticity reconstruction using magnetic resonnance elastography measurements
, 2007
"... Magnetic resonance elastography (MRE) is an approach to measuring material properties using external vibration in which the internal displacement measurements are made with magnetic resonance. A variety of simple methods have been designed to recover mechanical properties by inverting the displaceme ..."
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Cited by 25 (12 self)
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Magnetic resonance elastography (MRE) is an approach to measuring material properties using external vibration in which the internal displacement measurements are made with magnetic resonance. A variety of simple methods have been designed to recover mechanical properties by inverting the displacement data. Currently, the remaining problems with all of these methods are that in general the homogeneous Helmholtz equation is used and therefore it fails at interfaces between tissues of different properties. The purpose of this work is to propose a new method for reconstructing both the location, the shape and the shear modulus of a small anomaly with Lame ́ parameters different from the background ones using internal displacement measurements.
Identification of Discontinuous Coefficients in Elliptic Problems Using Total Variation Regularization
 SIAM J. Sci. Comput
, 2003
"... . We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its wellestablished ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for s ..."
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Cited by 23 (9 self)
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. We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its wellestablished ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the outputleastsquares inverse problem. In addition to the basic outputleastsquares formulation, we introduce two new techniques to handle large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the outputleastsquares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors. 1. Introduction. Consider the partial differential equation ae \Gammar \Delta (q(x)ru) =...
Piecewise constant level set method for interface problems
, 2006
"... We apply the Piecewise Constant Level Set Method (PCLSM) to interface problems, especially for elliptic inverse and multiphase motion problems. PCLSM allows using one level set function to represent multiple phases, and the interfaces are represented implicitly by the discontinuity of a piecewise ..."
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Cited by 15 (4 self)
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We apply the Piecewise Constant Level Set Method (PCLSM) to interface problems, especially for elliptic inverse and multiphase motion problems. PCLSM allows using one level set function to represent multiple phases, and the interfaces are represented implicitly by the discontinuity of a piecewise constant level set function. The inverse problem is solved using a variational penalization method with total variation regularization of the coefficient, while the multiphase motion problem is solved by an Additive OperatorSplitting (AOS) scheme.
Image Segmentation Using Some Piecewise Constant Level Set Methods with MBO Type of Projection ∗
, 2005
"... Abstract. In this work, we are trying to propose fast algorithms for MumfordShah image segmentation using some recently proposed piecewise constant level set methods (PCLSM). Two variants of the PCLSM will be considered in this work. The first variant, which we call the binary level set method, nee ..."
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Cited by 10 (4 self)
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Abstract. In this work, we are trying to propose fast algorithms for MumfordShah image segmentation using some recently proposed piecewise constant level set methods (PCLSM). Two variants of the PCLSM will be considered in this work. The first variant, which we call the binary level set method, needs a level set function which only takes values ±1 to identify the regions. The second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions. For the MumfordShah image segmentation model with these new level set methods, one needs to minimize some smooth energy functionals under some constrains. A penalty method will be used to deal with the constraint. AOS (additive operator splitting) and MOS (multiplicative operator splitting) schemes will be used to solve the EulerLagrange equations for the minimization problems. By doing this, we obtain some algorithms which are essentially applying the MBO scheme for our segmentation models. Advantages and disadvantages are discussed for the proposed schemes.