Level Set Formulation Research Articles

IMAGE SEGMENTATION WITH COMPLEMENTARY USE OF EDGE AND REGION INFORMATION

Two familiar approaches to image segmentation are the salient contour extraction approach and the closed-contour deformation approach. The former uses Gestalt laws to link individual edge elements and construct segmentation boundaries. However, it is often difficult to have both closure and precision of the boundary addressed at the same time. The latter starts with a closed contour and deforms the contour to localize the segmentation boundary more precisely whilst maintaining the closure. The approach does not have the closure problem, but how to assign a proper initial contour for it remains an open issue. In this work, we propose a scheme that puts together the two approaches to let them work complementarily. Specifically, we design a salient contour extraction process that extracts a proper initialization of the closed contours; the process looks into edge evidence and proximity to the desired segmentation boundaries. Then, a region-based active contour in a level set formulation is adopted to refine the contour position to locate the segmentation boundaries more precisely. The scheme requires neither manual input on contour initialization nor prior knowledge about the imaged scene. Experiments on extensive benchmarking image-sets are presented to illustrate the performance of the scheme.

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) Runge-Kutta scheme are employed. Polynomial spaces of quadratic order are used. A numerical example with a change of topology is provided, and the order of convergence is studied on the Frank sphere example.

Active contours driven by weighted region-scalable fitting energy based on local entropy

In this paper, we present a scheme of improvement on the region-scalable fitting (RSF) model proposed by Li et al. (Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing 17(10) (2008) 1940–1949) in terms of robustness to initialization and noise. First, the Gaussian kernel for the RSF energy is replaced with a “mollifying” kernel with compact support. Second, the RSF energy is redefined as a weighted energy integral, where the weight is local entropy deriving from a grey level distribution of image. The total energy functional is then incorporated into a variational level set formulation with two extra internal energy terms. The new RSF model not only handles better intensity inhomogeneity, but also allows for more flexible initialization and more robustness to noise compared to the original RSF model.

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k – ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.

Active contour model combining region and edge information

A novel active contour model is proposed by combining region and edge information. Its level set formulation consists of the edge-related term, the region-based term and the regularization term. The edge-related term is derived from the image gradient, and facilitates the contours evolving into object boundaries. The region-based term is constructed using both local and global statistical information, and related to the direction and velocity of the contour propagation. The last term ensures stable evolution of the contours. Finally, a Gaussian convolution is used to regularize the level set function. In addition, a new quantitative metric named modified root mean squared error is defined, which can be used to evaluate the final contour more accurately. Experimental results show that the proposed method is efficient and robust, and can segment homogenous images and inhomogenous images with the initial contour being set freely.

A coupling strategy based on anisotropic mesh adaptation for solving two‐fluid flows

AbstractWe propose a coupling strategy for solving efficiently bifluid flows based on the Stokes equations. Our approach relies on a level set formulation of the interface‐capturing problem, and involves a finite element discretization for the fluid resolution, the method of characteristics for solving the advection of the interface and the anisotropic mesh adaptation of the computational domain in the vicinity of the interface for better accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

Active contours with selective local or global segmentation property for multiobject image

In this paper, a novel region-based active contour model is proposed. Unlike the Chan-Vese model, the energy function of our model only uses information outside the evolving curve. A restriction matrix is introduced for restricting the curve to evolve in the desired region. By minimizing the proposed energy function, the corresponding level set function is obtained. The proposed method has four advantages over the traditional region-based level set methods. First, the level set function is no longer required to be initialized as a sign distance function. Second, due to the simplified level set formulation and its fast curve evolution speed, the level set function does not need to be re-initialized and the computational cost is low. Third, our model is applicable to multiobject images since it can segment objects by separating background from an image. Finally, the proposed model has the property of selective local or global segmentation; it can segment not only the desired object but also the other objects in an image. The advantages of our model are demonstrated using synthetic and real images. C 2011 Society of Photo-Optical

A least-square weighted residual method for level set formulation

In this paper, a least-square weighted residual method (LSWRM) for level set (LS) formulation is introduced to achieve interface capturing in two-dimensional (2D) and three-dimensional (3D) problems. An LSWRM was adopted for two semi-discretized advection and reinitialization equations of the LS formulation. The present LSWRM provided good mathematical properties such as natural numerical diffusion and the symmetry of the resulting algebraic systems for the advection and reinitialization equations. The proposed method was validated by solving some 2D and 3D benchmark problems such as those involving a rotating slotted disk, the rotation of a slotted sphere, and a time-reversed single-vortex flow and a deformation problem of a spherical fluid. The numerical results were compared with those obtained from essentially non-oscillatory type formulations and particle LS methods. Further, the proposed LSWRM for the LS formulation was coupled with a splitting finite element method code to solve the incompressible Navier–Stokes equations, and then, the collapse of a 3D broken dam flow was well simulated; in the simulation, the entrapping of air and the splashing of the surge front of water were reproduced. The mass conservation of the present method was found to be satisfactory during the entire simulation. Copyright © 2011 John Wiley & Sons, Ltd.

A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI

Intensity inhomogeneity often occurs in real-world images, which presents a considerable challenge in image segmentation. The most widely used image segmentation algorithms are region-based and typically rely on the homogeneity of the image intensities in the regions of interest, which often fail to provide accurate segmentation results due to the intensity inhomogeneity. This paper proposes a novel region-based method for image segmentation, which is able to deal with intensity inhomogeneities in the segmentation. First, based on the model of images with intensity inhomogeneities, we derive a local intensity clustering property of the image intensities, and define a local clustering criterion function for the image intensities in a neighborhood of each point. This local clustering criterion function is then integrated with respect to the neighborhood center to give a global criterion of image segmentation. In a level set formulation, this criterion defines an energy in terms of the level set functions that represent a partition of the image domain and a bias field that accounts for the intensity inhomogeneity of the image. Therefore, by minimizing this energy, our method is able to simultaneously segment the image and estimate the bias field, and the estimated bias field can be used for intensity inhomogeneity correction (or bias correction). Our method has been validated on synthetic images and real images of various modalities, with desirable performance in the presence of intensity inhomogeneities. Experiments show that our method is more robust to initialization, faster and more accurate than the well-known piecewise smooth model. As an application, our method has been used for segmentation and bias correction of magnetic resonance (MR) images with promising results.

Improved level-set framework-based algorithm for small infrared target detection

An improved algorithm integrating wavelet decomposition, multilevel filtering, and an additive operator splitting (AOS)-based level-set framework for infrared small target detection is proposed. This model has two components: a filtering operation, and level-set evolution. In the filtering step, the original image is first decomposed using a wavelet transform. After determining the location of sea-sky line, we construct a subimage based on the sea-sky-line position, and then execute multilevel filtering on this subimage. This filtering framework provides the input image for the level-set evolution. Using the level-set formulation, complex curves can be detected while naturally handling topological changes of the evolving curves. To reduce the computational cost required by an explicit implementation of the level-set formulation, a new solver named AOS is proposed. Additionally, the quantitative analyses for our algorithm are also given. Experiments on real infrared image sequences indicate that our method is efficient and robust.

Adaptive level-set evolution without initial contours for image segmentation

We propose a novel variational formulation (external energy) for implicit active contours that forces the level-set function to have the opposite sign along the edges at convergence. This external energy is then incorporated into a variational level-set formulation with two extra regularization terms (internal energy). The resulting evolution of the level-set function is the gradient flow that minimizes the overall energy functional. Because of the external energy, the level-set function can be initialized to any bounded function (e.g., a constant function), which completely eliminates the need of initial contours. This implies that the new formulation is robust to initialization or even free of manual initialization. The proposed model is been applied to both real and synthetic images with promising results. Besides, our model is especially appropriate for segmentation of real images with a complex and /or nonuniform background.

Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows

G-equations are well-known front propagation models in turbulent combustion and describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex ($L^1$ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $\bar H$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed $s_T$. An important problem in turbulent combustion theory is to study properties of $s_T$, in particular how $s_T$ depends on the flow amplitude $A$. In this paper, we will study the behavior of $\bar H=\bar H(A,d)$ as $A\to +\infty$ at any fixed diffusion constant $d > 0$. For the cellular flow, we show that $$ \bar H(A,d)\leq O(\sqrt {\mathrm {log}A}) \quad \text{for all $d>0$}. $$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit $\lim_{A\to +\infty}{\bar H(A,d)\over A} = \lambda (d) >0$ where $\lambda (d)$ is strictly decreasing in $d$, and has zero derivative at $d=0$. The linear growth law is also valid for $s_T$ of the curvature dependent G-equation in shear flows.

A nonlinear PDE model for reconstructing a regular surface from sampled data using a level set formulation on triangular meshes

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.

Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation

Optimal control (motion planning) of the free interface in classical two-phase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The first-order optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function. Suitable discretization schemes for the forward and adjoint systems are described. Numerical examples verify the correctness and flexibility of the proposed scheme.

Image segmentation by optimizing a homogeneity measure in a variational framework

This article describes a mechanism that segments an image by optimizing a homogeneity measure in a level-set formulation. The mechanism has uniform treatment toward texture, gray level, and color boundaries. The mechanism commences by quantizing the entire range of intensity values into a finite number of classes, thereby approximating the image as a class map, in which each image position carries a particular class label. For any region, each class label in it generally appears in a number of positions, which together define the spread-size of the class label in the region. The average spread-size of the class labels in a region, as a ratio to the region size, then constitutes a measure of how homogeneous the region is. The segmentation problem can be formulated as the optimization of such a measure of all the segmented regions. This work contributes chiefly by expressing the above optimization functional in such a way that the optimization can be conducted in a variational formulation, and that the solution can be reached by the deformation of an active contour. In addition, this work incorporates an additional geodesic term into the optimization functional, which maintains the contour's mobility at certain adverse conditions.

Subgrid analysis of liquid jet atomization

The objective of this paper is to study the feasibility of large eddy simulations of a liquid fuel injection in combustion chambers. To do so, a priori analyses of direct numerical simulations are carried out. A complete liquid jet atomization, from the injector down to the end of the liquid core, is simulated thanks to the coupling of both level-set and VOF formulations. To avoid the apparition of a subgrid term in the right hand side of the continuity equation, the choice was made to consider an incompressible formulation as far as the filtering operator is concerned. The corresponding LES transport equations and various subgrid contributions are thus presented. Results are first dedicated to the estimation of the various orders of magnitude of these subgrid terms. In a second part, classical eddy viscosity scale similarity models are tested against the prevalent ones. It appears that, contrary to a Smagorinsky formulation, the scale similarity assumption provides a better estimation of the subgrid terms. This result is found for all locations that have been considered in the jet: at the injection level or in the atomized area. The major drawback is the presence of a constant that needs to be estimated. Various values are found depending on the filter size.

Comparison of a least-square weighted residual method and the Taylor Galerkin method based on level set formulation for interface capturing

In this study, a least-square weighted residual method and the Taylor?Galerkin method have been adopted for the discretisation of the two hyperbolic-type equations of level set method: advection and reinitialisation equations. The accuracy and efficiency of the two methods were compared by solving a time-reversed vortex flow and a three-dimensional broken dam flows. A four-step splitting P1P1 finite element method was used for the solution of the incompressible Navier?Stokes equations. These numerical experiments showed that both the least-square weighted residual method and the Taylor?Galerkin method were first-order accurate in space and that the least-square method was more accurate and conservative than Taylor?Galerkin method in terms of mass error.

3D finite element simulation of the matter flow by surface diffusion using a level set method

AbstractWithin the context of the sintering process simulation, this paper proposes a numerical strategy for the direct simulation of the matter transport by surface diffusion, in two and three dimensions. The level set formulation of the surface diffusion problem is first established. The resulting equations are solved by using a finite element method. A stabilization technique is then introduced, in order to avoid the spurious oscillations of the grain boundary that are a consequence of the dependence of the surface velocity on the fourth‐order derivative of the level set function. The convergence and the accuracy of this approach are proved by investigating the change in an elliptic interface under surface diffusion. Cases in direct relation with the sintering process are analyzed besides: sintering between two grains of the same size or of two different sizes. Finally, 3D simulations involving a small number of particles show the ability of the proposed strategy to deal with strong deformations of the grain surface (formation of necks) and to access directly important parameters such as the closed porosity rate. Copyright © 2010 John Wiley & Sons, Ltd.

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

AbstractThis paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the Lp‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical Analysis of Bubble Growth and Departure from a Microcavity

A numerical approach is presented for analysis of bubble growth and departure from a microcavity during nucleate boiling. The level-set formulation for tracking the phase interfaces is modified to include the effect of phase change on the liquid–vapor interface and to treat the no-slip and contact angle conditions on the immersed (or irregularly shaped) solid surface of the microcavity. Also, the formulation is coupled with a simple and efficient model for predicting the evaporative heat flux from the liquid microlayer on an immersed solid surface. The effects of cavity size and geometry on the bubble growth and departure in nucleate boiling are investigated.