MULTI-PHASE ACTIVE CONTOUR MODEL FOR IMAGE SEGMENTATION BASED ON LEVEL SETS

A new multi-phase active contour model is proposed for the image segmentation. It is a generalization of the C-V model with the following characteristics： （1） A key technique, called the technique of painting background （TPBG）, is developed to remove the information of the background, which blocks the detection of weak boundaries in the object; （2） The two-phase level set is applied multiple times for getting the multi-phase segmentation model （n-1 times for the n-phase model, n〉1）; （3） A scaling-based method is introduced to improve the basic model. Experimental results show that the proposed model is effective for detecting weak boundaries.

A NEW DEFORMABLE MODEL USING LEVEL SETS FOR SHAPE SEGMENTATION

In this paper, we present a new deformable model for shape segmentation, which makes two modifications to the original level set implementation of deformable models.The modifications are motivated by difficulties that we have encountered in applying deformable models to segmentation of medical images.The level set algorithm has some advantages over the classical snake deformable models.However, it could develop large gaps in the boundary and holes within the objects.Such boundary gaps and holes of objects can cause inaccurate segmentation that requires manual correction.The proposed method in this paper possesses an inherent property to detect gaps and holes within the object with a single initial contour and also does not require specific initialization.The first modification is to replace the edge detector by some area constraint, and the second modification utilizes weighted length constraint to regularize the curve under evolution.The proposed method has been applied to both synthetic and real images with promising results.

LEVEL SETS AND EQUIVALENCES OF MORAN-TYPE SETS

In the paper, we consider Moran-type sets E;given by sequences {a;};and{n;};. we prove that E;may be decompose into the disjoint union of level sets. Moreover,we define three type of equivalence between two dimension functions associated to two Morantype sets, respectively, and we classify Moran-type sets by these equivalent relations.

MOTION OF LEVEL SETS BY A GENERALIZED MEAN CURVATURE

Short time existence and uniqueness for the classical motion are studied by the function of the principal curvatures of a smooth surface and the Evans and Spruck's results are generalized.

Active Contours and Mumford-Shah Segmentation Based on Level Sets

This paper is to detect regions (objects) boundaries, also to isolate and extract individual components from a medical image. This can be done using an active contours to detect regions in a given image, based on techniques of curve evolution, Mumford Shah functional for segmentation and level sets. The paper classified the images into different intensity regions based on Markov random field, then detected regions whose boundaries are not necessarily defined by gradient by minimizing an energy of Mumford Shah functional for segmentation 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. The stopping term does not depend on the gradient of the image, as in the classical active contour and the initial curve of level set can be anywhere in the image, and interior contours are automatically detected. The final image segmentation is one closed boundary per actual region in the image.

A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

Buckling Optimization of Curved Grid Stiffeners through the Level Set Based Density Method

Stiffened structures have great potential for improvingmechanical performance,and the study of their stability is of great interest.In this paper,the optimization of the critical buckling load factor for curved grid stiffeners is solved by using the level set based density method,where the shape and cross section(including thickness and width)of the stiffeners can be optimized simultaneously.The grid stiffeners are a combination ofmany single stiffenerswhich are projected by the corresponding level set functions.The thickness and width of each stiffener are designed to be independent variables in the projection applied to each level set function.Besides,the path of each single stiffener is described by the zero iso-contour of the level set function.All the single stiffeners are combined together by using the p-norm method to obtain the stiffener grid.The proposed method is validated by several numerical examples to optimize the critical buckling load factor.

Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations

We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F(▽2u,▽u,u,t) under a structural condition,and give a geometric lower bound of the principal curvature of the spatial level surfaces.

Curvature Estimates for the Level Sets of Solutions to the Monge-Ampère Equation det D^2u = 1

For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.

Level Sets of Certain Subclasses of α-Analytic Functions

For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.

The Concavity of the Gaussian Curvature of the Convex Level Sets of p-Harmonic Functions with Respect to the Height

For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level sets of p-harmonic function.We prove that this curvature function is concave with respect to the height of the p-harmonic function.This auxiliary function is an affine function of the height when the p-harmonic function is the p-Green function on ball.

A Remark on the Level Sets of the Graph of Harmonic Functions Bounded by Two Circles in Parallel Planes

In this paper, we find two auxiliary functions and make use of the maximum principle to study the level sets of harmonic function defined on a convex ring with homogeneous Dirichlet boundary conditions in R2. In higher dimensions, we also have a similar result to Jagy＇s.

基于Level Set方法的40CrMnMoA合金钢的奥氏体晶粒长大模型

为准确描述40CrMnMoA合金钢的奥氏体晶粒长大行为,借助Gleeble-1500D型热模拟机对40CrMnMoA合金钢进行了保温试验,温度为900~1150℃,时间为10~2000 s。基于试验结果,通过非线性回归建立了40CrMnMoA合金钢Burke-Turnbull奥氏体晶粒长大模型,构建了基于Level Set方法的40CrMnMoA合金钢奥氏体晶粒长大的介观尺度有限元模型并进行计算。结果表明:保温温度与保温时间对40CrMnMoA合金钢的晶粒长大行为均具有较大的影响。40CrMnMoA合金钢的奥氏体平均晶粒直径随保温时间的延长呈抛物线增长。所建立的Burke-Turnbull晶粒长大动力学模型具有较高的精度,试验值与计算值相关系数(R)为0.9932。40CrMnMoA合金钢奥氏体再结晶晶粒长大激活能(Q_(m))为142845.37 J/mol。通过对比仿真计算与试验所得的晶粒组织,验证了本文所建有限元模型可有效地预测40CrMnMoA合金钢的奥氏体晶粒长大演变过程。

Topology Optimization for Steady-State Navier-Stokes Flow Based on Parameterized Level Set Based Method

In this paper,we consider solving the topology optimization for steady-state incompressibleNavier-Stokes problems via a new topology optimization method called parameterized level set method,which can maintain a relatively smooth level set function with a local optimality condition.The objective of topology optimization is to􀀀nd an optimal con􀀀guration of theuid and solid materials that minimizes power dissipation under a prescribeduid volume fraction constraint.An arti􀀀cial friction force is added to the Navier-Stokes equations to apply the no-slip boundary condition.Although a great deal of work has been carried out for topology optimization ofuidow in recent years,there are few researches on the topology optimization ofuidow with physical body forces.To simulate theuidow in reality,the constant body force(e.g.,gravity)is considered in this paper.Several 2D numerical examples are presented to discuss the relationships between the proposed method with Reynolds number and initial design,and demonstrate the feasibility and superiority of the proposed method in dealing with unstructuredmesh problems.Three 3D numerical examples demonstrate the proposedmethod is feasible in three-dimensional.

CONSTRUCTION OF GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS FOR LEVEL SETS

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction.In this work,a novel scheme based on a specific level set ghost fluid method and an implicit-explicit(IMEX)flux splitting is proposed to overcome this timestep restriction.A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface.In this part of the domain,the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases.It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method.Applica-tions to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.

Level-set法在液体晃荡研究中的应用

Level-set法是一种简便、通用、高效的计算分析空间运动界面的数值方法,本文尝试性地将Level-set方法应用于液体晃荡的研究,详细论述了应用该方法模拟液体晃荡的数学模型及数值解法。为了验证该方法处理晃荡问题的正确合理性,首先在势流假定下模拟了释放初始液面引起的自由晃荡,并将液面波动的数值结果与理论解析解进行了比较;然后模拟了二维矩形液箱做简谐横摇运动引起的液体受迫晃荡,计算结果以箱壁不同高度处的压强时间历经给出,并与实验结果进行了定量比较。计算结果初步表明采用Level-set方法研究液体晃荡现象是正确可行的。

基于Level-set法的液舱液体晃荡数值模拟

船舶液舱中液体晃荡现象已引起人们的深刻关注,液体晃荡载荷与效应成为航行中载液船舶安全性评估的重要内容之一。该文基于Level-set法,对一矩形液舱的两种工况下舱内的液体晃荡进行了数值模拟。通过计算,得到了液面起伏和压强的时间历经,结果显示Level-set方法可有效地模拟液体晃荡问题。