Topology optimization of shell structures using adaptive inner-front (AIF) level set method

A new topology optimization using adaptive inner-front level set method is presented. In the conventional level set-based topology optimization, the optimum topology strongly depends on the initial level set due to the incapability of inner-front creation during the optimization process. In the present work, in this regard, an algorithm for inner-front creation is proposed in which the sizes, the positions, and the number of new inner-fronts during the optimization process can be globally and consistently identified. In the algorithm, the criterion of inner-front creation for compliance minimization problems of linear elastic structures is chosen as the strain energy density along with volumetric constraint. To facilitate the inner-front creation process, the inner-front creation map is constructed and used to define new level set function. In the implementation of inner-front creation algorithm, to suppress the numerical oscillation of solutions due to the sharp edges in the level set function, domain regularization is carried out by solving the edge smoothing partial differential equation (smoothing PDE). To update the level set function during the optimization process, the least-squares finite element method (LSFEM) is adopted. Through the LSFEM, a symmetric positive definite system matrix is constructed, and non-diffused and non-oscillatory solution for the hyperbolic PDE such as level set equation can be obtained. As applications, three-dimensional topology optimization of shell structures is treated. From the numerical examples, it is shown that the present method brings in much needed flexibility in topologies during the level set-based topology optimization process.     

Image segmentation and bias correction via an improved level set method

Intensity inhomogeneity causes considerable difficulty in the quantitative analysis of magnetic resonance (MR) images. Thus bias field estimation is a necessary pre-processing step before quantitative analysis of MR data. This paper presents a variational level set approach for bias correction and segmentation for images with intensity inhomogeneities. Our method is based on the observation that local intensity variations in relatively smaller regions are separable, despite the inseparability of the whole image. In the beginning we define a function for clustering the image pixels in a smaller neighborhood. The cluster centers in this objective function have a multiplicative factor that estimates the bias within the neighborhood. Generally the local intensity variations are described by the Gaussian distributions with different means and variances. In this work the objective functions are integrated over the entire domain with local Gaussian distribution of fitting energy, ultimately analyzing the data with a level set framework. Our method is able to capture bias of quite general profiles. Moreover, our model can also distinguish regions with similar intensity distribution with different variances. The proposed method has been rigorously validated with images acquired on variety of imaging modalities with promising results.     

A level set method for oil slick segmentation in SAR images

Despite much effort and significant progress in recent years, image segmentation remains a challenging problem in image processing, especially for the low contrast, noisy synthetic aperture radar (SAR) images. This paper explores the segmentation of oil slicks using a partial differential equation (PDE)‐based level set method, which represents the slick surface as an implicit propagation interface. Starting from an initial estimation with priori information, the level set method creates a set of speed functions to detect the position of the propagation interface. Specifically, the image intensity gradient and the curvature flow are utilized together to determine the speed and direction of the propagation. This allows the front interface to propagate naturally with topological changes, significant protrusions and narrow regions, giving rise to stable and smooth boundaries that discriminate oil slicks from the surrounding water. As the speckles are removed concurrently while the front interface propagates, the pre‐filtering of noise is saved. The proposed method has been illustrated by experiments on oil slick segmentation using the ERS‐2 SAR images. Its advantages over the traditional image segmentation approaches have also been demonstrated.     

A new level set method for inhomogeneous image segmentation

Intensity inhomogeneity often appears in medical images, such as X-ray tomography and magnetic resonance (MR) images, due to technical limitations or artifacts introduced by the object being imaged. It is difficult to segment such images by traditional level set based segmentation models. In this paper, we propose a new level set method integrating local and global intensity information adaptively to segment inhomogeneous images. The local image information is associated with the intensity difference between the average of local intensity distribution and the original image, which can significantly increase the contrast between foreground and background. Thus, the images with intensity inhomogeneity can be efficiently segmented. What is more, to avoid the re-initialization of the level set function and shorten the computational time, a simple and fast level set evolution formulation is used in the numerical implementation. Experimental results on synthetic images as well as real medical images are shown in the paper to demonstrate the efficiency and robustness of the proposed method.     

一种新颖的医学图像锐化增强算法

针对医学图像组织间不明显现象,提出了一种基于模糊规则和小波变换的医学图像锐化增强算法。对不同尺度的小波系数进行锐化增强时,首先根据该尺度低频系数中心像素与其邻域像素的相容性利用模糊规则自适应计算非线性细节增益系数。然后把增益系数与细节小波系数相乘,小波重建后得到增强图像。实验结果表明,提出的算法对图像细节进行增强的同时能够有效地抑制噪声。     

Towards nonideal iris recognition based on level set method, genetic algorithms and adaptive asymmetrical SVMs

We present algorithms for iris segmentation, feature extraction and selection, and iris pattern matching. To segment the inner boundary from a nonideal iris image, we apply a level set based curve evolution approach using the edge stopping function, and to detect the outer boundary, we employ the curve evolution approach using the regularized Mumford-Shah segmentation model with an energy minimization algorithm. Daubechies wavelet transform (DBWT) is used to extract the textural features, and genetic algorithms (GAs) are deployed to select the subset of informative features by combining the valuable outcomes from the multiple feature selection criteria without compromising the recognition accuracy. To speed up the matching process and to control the misclassification error, we apply a combined approach called the adaptive asymmetrical support vector machines (AASVMs). The parameter values of SVMs are also optimized in order to improve the overall generalization performance. The verification and identification performance of the proposed scheme is validated using the UBIRIS Version 2, the ICE 2005, and the WVU datasets.     

A new level set method for systematic design of hinge-free compliant mechanisms

This paper presents a new level set-based method to realize shape and topology optimization of hinge-free compliant mechanisms. A quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is employed to solve the Hamilton-Jacobi partial differential equation (PDE) in the level set method. The design of compliant mechanisms is mathematically represented as a general non-linear programming with a new objective function augmented by the high-order energy term. The structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. In doing so, it is expected that numerical difficulties such as the Courant-Friedrichs-Lewy (CFL) condition and periodically applied re-initialization procedures in most conventional level set methods can be eliminated. In addition, new holes can be created inside the design domain. The final mechanism configurations consist of strip-like members suitable for generating distributed compliance, and solving the de-facto hinge problem in the design of compliant mechanisms. Two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms.     

基于水平集的医学图像快速分割方法

针对高噪声、低对比度的医学图像难以快速准确分割的问题,结合基于像素的传统方法和基于水平集的活动轮廓模型,提出了一种混合的医学图像分割新技术.首先依据待分割对象的先验知识交互选取感兴趣区域.然后由传统的方法和基于水平集的C-V模型结合实现感兴趣区域图像的预分割.预分割的结果直接作为窄带变分水平集模型的初始轮廓,演化曲线在很短的时间内准确收敛到待分割物体的边缘.     

Application of spectral level set methodology in topology optimization

In this paper, two benchmark problems in structural boundary design are solved using the spectral level set methodology, which is a new approach to topology optimization of interfaces. This methodology is an extension of the level set methods, in which the interface is represented as the zero level set of a function. According to the proposed formulation, the Fourier coefficients of that function are the design variables describing the interface during the topology optimization. An advantage of the spectral level set methodology, in the case of a sufficiently regular interface, is to admit an upper bound error which is asymptotically smaller than the one for nonadaptive spacial discretizations of the level set function. Other advantages include the nucleation of holes in the interior of the interface and the avoidance of checkerboard-like designs. The theoretical framework of the methodology is presented and estimates on its convergence rate are discussed. The numerical applications consist in the design of short and long cantilevers subject to a vertical concentrated load opposite to the fixed end. The goal is to maximize the structural stiffness subject to a solid volume constraint. The zero level set of the level set function defines the structural boundary.     

Topological shape optimization of geometrically nonlinear structures using level set method

Using the level set method, a topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian formulation. The structural boundaries are implicitly represented by the level set function, obtainable from “Hamilton-Jacobi type” equation with “up-wind scheme,” embedded into a fixed initial domain. The method minimizes the compliance through the variations of implicit boundary, satisfying an allowable volume requirement. The required velocity field to solve the Hamilton-Jacobi equation is determined by the descent direction of Lagrangian derived from an optimality condition. Since the homogeneous material property and implicit boundary are utilized, the convergence difficulty is significantly relieved.     

A novel level set method for image segmentation by incorporating local statistical analysis and global similarity measurement

This paper presents a novel level set method for complex image segmentation, where the local statistical analysis and global similarity measurement are both incorporated into the construction of energy functional. The intensity statistical analysis is performed on local circular regions centered in each pixel so that the local energy term is constructed in a piecewise constant way. Meanwhile, the Bhattacharyya coefficient is utilized to measure the similarity between probability distribution functions for intensities inside and outside the evolving contour. The global energy term can be formulated by minimizing the Bhattacharyya coefficient. To avoid the time-consuming re-initialization step, the penalty energy term associated with a new double-well potential is constructed to maintain the signed distance property of level set function. The experiments and comparisons with four popular models on synthetic and real images have demonstrated that our method is efficient and robust for segmenting noisy images, images with intensity inhomogeneity, texture images and multiphase images.     

Individual tooth segmentation from CT images using level set method with shape and intensity prior

3D visualization of teeth from CT images provides important assistance for dentists performing orthodontic surgery and treatment. However, dental CT images present several major challenges for the segmentation of tooth, which touches with adjacent teeth as well as surrounding periodontium and jaw bones. Moreover, tooth contour suffers from topological changes and splits into several branches. In this work, we focus on the segmentation of individual teeth with complete crown and root parts. To this end, we propose adaptive active contour tracking algorithms: single level set method tracking for root segmentation to handle the complex image conditions as well as the root branching problem, and coupled level set method tracking for crown segmentation in order to separate the touching teeth and create the virtual common boundaries between them. Furthermore, we improve the variational level set method in several aspects: gradient direction is introduced into the level set framework to prevent catching the surrounding object boundaries; in addition to the shape prior, intensity prior is introduced to provide adaptive shrinking or expanding forces in order to deal with the topological changes. The test results for both tooth segmentation and 3D reconstruction show that the proposed method can visualize individual teeth with high accuracy and efficiency.     

A level set method based on the Bayesian risk for medical image segmentation

This paper proposes an alternative criterion derived from the Bayesian risk classification error for image segmentation. The proposed model introduces a region-based force determined through the difference of the posterior image densities for the different classes, a term based on the prior probability derived from Kullback-Leibler information number, and a regularity term adopted to avoid the generation of excessively irregular and small segmented regions. Compared with other level set methods, the proposed approach relies on the optimum decision of pixel classification and the estimates of prior probabilities; thus the approach has more reliability in theory and practice. Experiments show that the proposed approach is able to extract the complicated shapes of targets and robust for various types of medical images. Moreover, the algorithm can be easily extendable for multiphase segmentation.     

Level set method for two-phase incompressible flows under magnetic fields

This article presents a level set method for two-phase, incompressible flows under magnetic fields. The magnetic force caused by the difference in magnetic permeability of the fluids are derived and incorporated into the momentum equation using the level set function. The governing equation for magnetic fields (magnetostatic equation) is also obtained by considering the discontinuity in the spatial distribution of magnetic permeability using the level set function. To test the method, falling droplet, droplet oscillation and stretching, and rising bubble problems in the presence of a magnetic field are simulated.     

Structure topology optimization: fully coupled level set method via FEMLAB

This paper presents a procedure which can easily implement the 2D compliance minimization structure topology optimization by the level set method using the FEMLAB package. Instead of a finite difference solver for the level set equation, as is usually the case, a finite element solver for the reaction–diffusion equation is used to evolve the material boundaries. All of the optimization procedures are implemented in a user-friendly manner. A FEMLAB code can be downloaded from the homepage www.imtek.de/simulation and is free for educational purposes.     

多目标轮廓Mumford-Shah水平集提取

目标轮廓的快速检测进而提取其几何形状,在图形图像处理中有着重要的作用.提出了一种多目标轮廓的水平集提取方法,对基于Mumford-Shah模型的C-V方法从两方面进行了改进:增加梯度矢量场和曲线法方向的融合作为边界吸引场,生成可以驱动主动轮廓向边缘进化的双向几何变形流,保留原图像分布信息作为区域进化能,解决未考虑局部几何信息造成的区域能量捕捉信息不全,或边缘梯度场和演化曲线法线方向正交时无法实现拓扑结构变化的缺陷;对水平集函数进行修正,使得它在收敛过程中能自动进行调整,确保其满足符号距离函数的要求,扩大初始化前迭代搜索区域,减少初始化次数,提高收敛效率;最后给出所提方法的数字化求解方案.实验表明该方法可行且具有较好的鲁棒性.     

Retinal vessels segmentation based on level set and region growing

Retinal vessels play an important role in the diagnostic procedure of retinopathy. Accurate segmentation of retinal vessels is crucial for pathological analysis. In this paper, we propose a new retinal vessel segmentation method based on level set and region growing. Firstly, a retinal vessel image is preprocessed by the contrast-limited adaptive histogram equalization and a 2D Gabor wavelet to enhance the vessels. Then, an anisotropic diffusion filter is used to smooth the image and preserve vessel boundaries. Finally, the region growing method and a region-based active contour model with level set implementation are applied to extract retinal vessels, and their results are combined to achieve the final segmentation. Comparisons are conducted on the publicly available DRIVE and STARE databases using three different measurements. Experimental results show that the proposed method reaches an average accuracy of 94.77% on the DRIVE database and 95.09% on the STARE database.     

Active contours with selective local or global segmentation: A new formulation and level set method

A novel region-based active contour model (ACM) is proposed in this paper. It is implemented with a special processing named Selective Binary and Gaussian Filtering RegularizedLevel Set(SBGFRLS) method, which first selectively penalizes the level set function to be binary, and then uses a Gaussian smoothing kernel to regularize it. The advantages of our method are as follows. First, a new region-based signed pressure force (SPF) function is proposed, which can efficiently stop the contours at weak or blurred edges. Second, the exterior and interior boundaries can be automatically detected with the initial contour being anywhere in the image. Third, the proposed ACM with SBGFRLS has the property of selective local or global segmentation. It can segment not only the desired object but also the other objects. Fourth, the level set function can be easily initialized with a binary function, which is more efficient to construct than the widely used signed distance function (SDF). The computational cost for traditional re-initialization can also be reduced. Finally, the proposed algorithm can be efficiently implemented by the simple finite difference scheme. Experiments on synthetic and real images demonstrate the advantages of the proposed method over geodesic active contours (GAC) and Chan–Vese (C–V) active contours in terms of both efficiency and accuracy.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.