Spatially adaptive techniques for level set methods and incompressible flow

Since the seminal work of [Sussman, M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] on coupling the level set method of [Osher S, Sethian J. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 1988;79:12–49] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both of its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton–Jacobi WENO [Jiang G-S, Peng D. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J Sci Comput 2000;21:2126–43], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle level set method for improved interface capturing. J Comput Phys 2002;183:83–116] and the coupled level set volume of fluid method [Sussman M, Puckett EG. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J Comput Phys 2000;162:301–37], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [Losasso F, Gibou F, Fedkiw R. Simulating water and smoke with an octree data structure, ACM Trans Graph (SIGGRAPH Proc) 2004;23:457–62].     

A fast level set based algorithm for topology-independent shape modeling

Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods, and overcomes some of their limitations. Our technique can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solid/liquid interfaces with curvature-dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a Hamilton-Jacobi type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. We also introduce a new algorithm for rapid advancement of the front using what we call a narrow-band update scheme. The efficacy of the scheme is demonstrated with numerical experiments on low contrast medical images.Supported in part by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, U.S. Dept. of Energy under Contract DE-AC03-76SD00098 and by the NSF ARPA under grant DMS-8919074.Supported in part by NSF grant ECS-9210648.     

Dynamic cluster formation using level set methods

Density-based clustering has the advantages for: 1) allowing arbitrary shape of cluster and 2) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But, when clusters become closed to each other, CIFs still clearly reveal cluster centers, cluster boundaries, and degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation, which are often oscillatory or oversmoothed. These problems of kernel density estimation are resolved using level set methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented which illustrate the advantages of our approach.     

Hybrid control synthesis for eventuality specifications using level set methods

This paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications. Given a hybrid system modelled by a hybrid automaton and a target set of states, the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time. Due to the existence of set-valued disturbance inputs, the problem is studied in a game-theoretic framework. Having shown that a least restrictive solution does not exist, we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property. To implement the algorithm, reachable sets of pursuit-evasion differential games need to be computed. For that reason level set methods are employed, where the boundary of the reachable set is characterized as the zero level set of a Hamilton–Jacobi equation. The procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology. Finally, to demonstrate the practical character of our results, a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology.     

脑MR图像分割和偏移场矫正的耦合水平集模型

脑核磁共振(MR)图像因需要偏移场矫正,传统分割方法很难获得准确的分割结果。针对这一问题,首先构造一组基函数拟合偏移场以保证偏移场的光滑特性,再将其融入到高斯概率密度函数中,结合统计分类准则建立脑MR图像的分割和偏移场矫正的能量方程,最后将该能量方程引入到三相位水平集的变分框架中得到脑MR图像的分割和偏移场矫正的耦合模型。实验表明该方法在得到准确的分割结果同时还可以得到较好的恢复结果。     

Shape modeling with front propagation: a level set approach

Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. The authors' techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of the authors' model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian (1988) to model propagating solid/liquid interfaces with curvature-dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a “Hamilton-Jacobi” type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. The authors present a variety of ways of computing the evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images     

Morphological region-based initial contour algorithm for level set methods in image segmentation

Multimedia Tools and Applications - Initial Contour (IC) is the essential step in level set image segmentation methods due to start the efficient process. However, the main issue with IC is how to...     

Numerical simulations for the motion of soap bubbles using level set methods

In this paper, we develop appropriate equations for bubble motions depending on curvature using the level set methodology. Our method consists of an appropriate finite difference scheme for solving our model equations and a level set approach for capturing the complicated motion between the bubbles. Results indicate that our models and the level set methods can handle complicated interfacial motions and topology changes, and that they can numerically simulate many of the physical features of bubble motions.     

基于光流场与水平集的运动目标检测

利用光流场信息及运动内极限约束确定运动目标区域的初始分割,提取光流大小与光流方向两个特征构成特征向量,使用K-means聚类算法获得运动目标区域,利用水平集方法对初始运动区域进行进一步分割,通过最小化定义的能量函数驱动代表运动目标的闭合曲线进行演化,曲线演化将在空间梯度较大的位置停止,从而得到运动目标的封闭边缘曲线。实验表明,该方法可有效地从图像序列中检测出完整的运动目标。     

Comparison of Yager's level set method for fuzzy logic control withMamdani's and Larsen's methods

A new fuzzy reasoning method for fuzzy control recently proposed by R. Yager is investigated. A comparison with the most useful fuzzy control schemes, for a first-order with time delay process, is carried out. The results obtained show that Yager's method is superior from the point of view of both computational burden and control system behavior     

Binary level set methods for topology and shape optimization of a two-density inhomogeneous drum

We optimize eigenvalues in optimal shape design using binary level set methods. The interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or ?1 at convergence. A binary constraint is added to the original model problems. We propose two variational algorithms to solve the constrained optimization problems. One is a hybrid type by coupling the Lagrange multiplier approach for the geometry constraint with the augmented Lagrangian method for the binary constraint. The other is devised using the Lagrange multiplier method for both constraints. The two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations. Intensive numerical results are presented and compared with those obtained by level set methods, which demonstrate the effectiveness and robustness of our algorithms.     

整体曲率变分水平集方法的曲面去噪模型

针对曲面去噪问题,提出了一种基于整体曲率变分水平集方法的曲面去噪模型。该模型是应用于图像去噪的ROF模型在几何形状处理中的自然拓展。它基于隐式水平集变分方法,能够自动处理曲面拓扑结构的变化。对该能量模型采用变分水平集方法求其梯度最速下降方程,通过演化该方程,最终得到模型最优解。为使计算结果更加准确,采用了半点差分格式离散。实验结果表明,该模型具有良好的去噪性能,同时能有效地保持曲面中的特征信息。     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

Compliant mechanism design based on the level set and arbitrary Lagrangian Eulerian methods

This paper deals with continuum-based compliant mechanism design. In topology optimization, filtering techniques are used for the regularization of the design space. In the density design representation, convolution-type filters inherently produce gray transition regions between solids and voids. In order to reduce the gray transition regions, projection schemes have recently been proposed. Binarization, however, still leads to one-node connected hinges in compliant mechanism design. In this paper, we propose a method that incorporates the level set and arbitrary Lagrangian Eulerian methods so that the gray transition regions are completely excluded and one-node connected hinges are not formed.     

Sparse field level set method for non-convex Hamiltonians in 3D plasma etching profile simulations

Level set method [S. Osher, J. Sethian, J. Comput. Phys. 79 (1988) 12] is a highly robust and accurate computational technique for tracking moving interfaces in various application domains. It originates from the idea to view the moving front as a particular level set of a higher dimensional function, so the topological merging and breaking, sharp gradients and cusps can form naturally, and the effects of curvature can be easily incorporated. The resulting equations, describing interface surface evolution, are of Hamilton-Jacobi type and they are solved using techniques developed for hyperbolic equations. In this paper we describe an extension of the sparse field method for solving level set equations in the case of non-convex Hamiltonians, which are common in the simulations of the profile surface evolution during plasma etching and deposition processes. Sparse field method itself, developed by Whitaker [R. Whitaker, Internat. J. Comput. Vision 29 (3) (1998) 203] and broadly used in image processing community, is an alternative to the usual combination of narrow band and fast marching procedures for the computationally effective solving of level set equations. The developed procedure is applied to the simulations of 3D feature profile surface evolution during plasma etching process, that include the effects of ion enhanced chemical etching and physical sputtering, which are the primary causes of the Hamiltonian non-convexity.     

基于变分水平集的三维模型复杂孔洞修复*

针对三维模型重建后存在大量复杂孔洞的问题,提出一种孔洞修补算法。首先构造符号距离函数,孔洞所在曲面用静态符号距离函数的零水平集表达,另一动态符号距离函数表示初始曲面;借助隐式曲面上的变分水平集,引入全局凸优化能量模型,通过对其极小化诱导,从而将提取孔洞边缘的问题转化为维体上隐式曲面的演化过程;最后以提取到的孔洞边缘曲面作为初始观察面,通过卷积和合成两个交替的步骤进行体素扩散完成孔洞修补。实验表明该算法能够有效恢复复杂孔洞区域的显著几何特征,且适用于含有网格较多的模型的孔洞修复。     

OR/AND neuron in modeling fuzzy set connectives

The paper introduces a neural network-based model of logical connectives. The basic processing unit consists of two types of generic OR and AND neurons structured into a three layer topology. Due to the functional integrity we will be referring to it as an OR/AND neuron. The specificity of the logical connectives is captured by the OR/AND neuron within its supervised learning. Further analysis of the connections of the neuron obtained in this way provides a better insight into the nature of the connectives applied in fuzzy sets by emphasizing their features of “locality” and interactivity. Afterward, we will study several architectures of neural networks comprising these neurons treated as their basic functional components. The numerical studies embrace both the structures formed by single OR/AND neurons and aimed at modeling logical connectives (including the Zimmermann-Zysno data set, 1980) and the networks representing various decision-making architectures. We will also propose a realization of a pseudo median filter in which the OR/AND neurons play an ultimate role     

Geometrical methods for level set based abdominal aortic aneurysm thrombus and outer wall 2D image segmentation

Abdominal aortic aneurysm (AAA) is a localized dilatation of the aortic wall. Accurate measurements of its geometric characteristics are critical for a reliable estimate of AAA rupture risk. However, current imaging modalities do not provide sufficient contrast to distinguish thrombus from surrounding tissue thus making the task of segmentation quite challenging. The main objective of this paper is to address this problem and accurately extract the thrombus and outer wall boundaries from cross sections of a 3D AAA image data set (CTA). This is achieved by new geometrical methods applied to the boundary curves obtained by a Level Set Method (LSM). Such methods address the problem of leakage of a moving front into sectors of similar intensity and that of the presence of calcifications. The versatility of the methods is tested by creating artificial images which simulate the real cases. Segmentation quality is quantified by comparing the results with a manual segmentation of the slices of ten patient data sets. Sensitivity to the parameter settings and reproducibility are analyzed. This is the first work to our knowledge that utilizes the level set framework to extract both the thrombus and external AAA wall boundaries.     

Resampling methods for ranked set samples

When measuring units are expensive or time consuming, while ranking them can be done easily, it is known that ranked set sampling (RSS) is preferred to simple random sampling (SRS). Available results for RSS are developed under specific parametric assumptions or are asymptotic in nature, with few results available for finite size samples when the underlying distribution of the observed data is unknown. We investigate the use of resampling techniques to draw inferences on population characteristics. To obtain standard error and confidence interval estimates we discuss and compare three methods of resampling a given ranked set sample. Chen et al. (2004. Ranked Set Sampling: Theory and Applications. Springer, New York) suggest a natural method to obtain bootstrap samples from each row of a RSS. We prove that this method is consistent for a location estimator. We propose two other methods that are designed to obtain more stratified resamples from the given sample. Algorithms are provided for these methods. We recommend a method that obtains a bootstrap RSS from the observations. We prove several properties of this method, including consistency for a location parameter. We define two types of L-estimators for RSS and obtain expressions for their exact moments. We discuss an application to obtain confidence intervals for the Winsorized mean of a RSS.