A level-set method for vibration and multiple loads structural optimization

We extend the level-set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads. This method is based on a combination of the classical shape derivative and of the Osher–Sethian level-set algorithm for front propagation. In two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations. The shape derivative is used as an advection velocity in a Hamilton–Jacobi equation for changing the shape. This level-set method is a low-cost shape capturing algorithm working on a fixed Eulerian mesh and it can easily handle topology changes.     

Introducing the sequential linear programming level-set method for topology optimization

This paper introduces an approach to level-set topology optimization that can handle multiple constraints and simultaneously optimize non-level-set design variables. The key features of the new method are discretized boundary integrals to estimate function changes and the formulation of an optimization sub-problem to attain the velocity function. The sub-problem is solved using sequential linear programming (SLP) and the new method is called the SLP level-set method. The new approach is developed in the context of the Hamilton-Jacobi type level-set method, where shape derivatives are employed to optimize a structure represented by an implicit level-set function. This approach is sometimes referred to as the conventional level-set method. The SLP level-set method is demonstrated via a range of problems that include volume, compliance, eigenvalue and displacement constraints and simultaneous optimization of non-level-set design variables.     

Strings: variational deformable models of multivariate continuous boundary features

We propose a new image segmentation technique called strings. A string is a variational deformable model that is learned from a collection of example objects rather than built from a priori analytical or geometrical knowledge. As opposed to existing approaches, an object boundary is represented by a one-dimensional multivariate curve in functional space, a feature function, rather than by a point in vector space. In the learning phase, feature functions are defined by extraction of multiple shape and image features along continuous object boundaries in a given learning set. The feature functions are aligned, then subjected to functional principal components analysis and functional principal regression to summarize the feature space and to model its content, respectively. Also, a Mahalanobis distance model is constructed for evaluation of boundaries in terms of their feature functions, taking into account the natural variations seen in the learning set. In the segmentation phase, an object boundary in a new image is searched for with help of a curve. The curve gives rise to a feature function, a string, that is weighted by the regression model and evaluated by the Mahalanobis model. The curve is deformed in an iterative procedure to produce feature functions with minimal Mahalanobis distance. Strings have been compared with active shape models on 145 vertebra images, showing that strings produce better results when initialized close to the target boundary, and comparable results otherwise.     

Moment-based alignment for shape prior with variational B-spline level set

This paper presents a new shape prior-based implicit active contour model for image segmentation. The paper proposes an energy functional including a data term and a shape prior term. The data term, inspired from the region-based active contour approach, evolves the contour based on the region information of the image to segment. The shape prior term, defined as the distance between the evolving shape and a reference shape, constraints the evolution of the contour with respect to the reference shape. Especially, in this paper, we present shapes via geometric moments, and utilize the shape normalization procedure, which takes into account the affine transformation, to align the evolving shape with the reference one. By this way, we could directly calculate the shape transformation, instead of solving a set of coupled partial differential equations as in the gradient descent approach. In addition, we represent the level-set function in the proposed energy functional as a linear combination of continuous basic functions expressed on a B-spline basic. This allows a fast convergence to the segmentation solution. Experiment results on synthetic, real, and medical images show that the proposed model is able to extract object boundaries even in the presence of clutter and occlusion.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

局部区域拟合的快速水平集图像分割方法

目的 由于灰度不均匀图像在不同目标区域的灰度分布存在严重的重叠,对其进行分割仍然是一个难题;同时,图像中的噪声严重降低了图像分割的准确性。因此,传统水平集方法无法鲁棒、精确、快速地对具有灰度不均匀性和噪声的图像进行分割。针对这一问题,提出一种基于局部区域信息的快速水平集图像分割方法。方法 灰度不均匀图像通常被描述为一个分段常数图像乘以一个缓慢变化的偏移场。首先,通过一个经过微调的多尺度均值滤波器来估计图像的偏移场,并对图像进行预处理以减轻图像的不均匀性;然后,利用基于偏移场校正的方法和基于局部区域信息拟合的方法分别构建能量项,并利用演化曲线轮廓内外图像灰度分布的重叠程度,构建权重函数自适应调整两个能量项之间的权重;最后,引入全方差规则项对水平集进行约束,增强了数值计算的稳定性和对噪声的鲁棒性,并通过加性算子分裂策略实现水平集快速演化。结果 在具有不同灰度不均匀性和噪声图像上的分割结果表明,所提方法不但对初始轮廓的位置、灰度不均匀性和各种噪声具有较强的鲁棒性,而且具有高达94.5%的分割精度和较高的分割效率,与传统水平集方法相比分割精度至少提高了20.6%,分割效率是LIC（local intensity clustering）模型的9倍;结论 本文提出一种基于局部区域信息的快速水平集图像分割方法。实验结果表明,与传统水平集方法相比具有较高的分割精度和分割效率,可以很好地应用于具有灰度不均匀和噪声的医学、红外和自然图像等的分割。     

Efficient liver segmentation using a level-set method with optimal detection of the initial liver boundary from level-set speed images

Automatic liver segmentation is difficult because of the wide range of human variations in the shapes of the liver. In addition, nearby organs and tissues have similar intensity distributions to the liver, making the liver's boundaries ambiguous. In this study, we propose a fast and accurate liver segmentation method from contrast-enhanced computed tomography (CT) images. We apply the two-step seeded region growing (SRG) onto level-set speed images to define an approximate initial liver boundary. The first SRG efficiently divides a CT image into a set of discrete objects based on the gradient information and connectivity. The second SRG detects the objects belonging to the liver based on a 2.5-dimensional shape propagation, which models the segmented liver boundary of the slice immediately above or below the current slice by points being narrow-band, or local maxima of distance from the boundary. With such optimal estimation of the initial liver boundary, our method decreases the computation time by minimizing level-set propagation, which converges at the optimal position within a fixed iteration number. We utilize level-set speed images that have been generally used for level-set propagation to detect the initial liver boundary with the additional help of computationally inexpensive steps, which improves computational efficiency. Finally, a rolling ball algorithm is applied to refine the liver boundary more accurately. Our method was validated on 20 sets of abdominal CT scans and the results were compared with the manually segmented result. The average absolute volume error was 1.25+/-0.70%. The average processing time for segmenting one slice was 3.35 s, which is over 15 times faster than manual segmentation or the previously proposed technique. Our method could be used for liver transplantation planning, which requires a fast and accurate measurement of liver volume.     

A shape design system using volumetric implicit PDEs

Solid modeling based on partial differential equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of elliptic PDEs for boundary value problems to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves and iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space. The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary and additional constraints.     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

A geodesic-active-contour-based variational model for short-axis cardiac MR image segmentation

We propose a novel geodesic-active-contour-based (GAC-based) variational model that uses two level-set functions to segment the right and left ventricles and the epicardium in short-axis magnetic resonance (MR) images. For the right ventricle, the myocardial wall is typically very thin and hard to identify using the resolution of existing MR scanners. We propose to use two level sets to identify both the endocardial wall by pushing away one level-set function from another, in the setting of the edge-driven GAC model with a new edge detection function. Existing edge detection functions have strict restrictions on the location of initial contours. We develop a new edge detection function that relaxes this restriction and propose an iterative method that uses a sequence of edge detection functions to minimize the energy of our model successively. Experimental results are presented to validate the effectiveness of the proposed model.     

Accelerating Liquid Simulation With an Improved Data-Driven Method

In physics-based liquid simulation for graphics applications, pressure projection consumes a significant amount of computational time and is frequently the bottleneck of the computational efficiency. How to rapidly apply the pressure projection and at the same time how to accurately capture the liquid geometry are always among the most popular topics in the current research trend in liquid simulations. In this paper, we incorporate an artificial neural network into the simulation pipeline for handling the tricky projection step for liquid animation. Compared with the previous neural-network-based works for gas flows, this paper advocates new advances in the composition of representative features as well as the loss functions in order to facilitate fluid simulation with free-surface boundary. Specifically, we choose both the velocity and the level-set function as the additional representation of the fluid states, which allows not only the motion but also the boundary position to be considered in the neural network solver. Meanwhile, we use the divergence error in the loss function to further emulate the lifelike behaviours of liquid. With these arrangements, our method could greatly accelerate the pressure projection step in liquid simulation, while maintaining fairly convincing visual results. Additionally, our neutral network performs well when being applied to new scene synthesis even with varied boundaries or scales.     

Shape optimization for a link mechanism

This paper presents a numerical solution for shape optimization problems for link mechanisms, such as a piston-crank mechanism. The dynamic behavior of a link mechanism is described by a differential-algebraic equation (DAE) system consisting of motion equations for each single body and constraints of linkages and rigid motions. In a shape optimization problem, the objective function to maximize is constructed from the external work done by a given external force, which agrees with the kinetic energy of the link mechanism, for an assigned time interval, and the total volume of all the links forms the constraint function. The Fréchet derivatives of these cost functions with respect to the domain variation, which we call the shape derivatives of these cost functions, are evaluated theoretically. A scheme to solve the shape optimization problem is presented using the H ¹ gradient method (the traction method) proposed by the authors as a reshaping algorithm, since it retains the smoothness of the boundary. A numerical example shows that reasonable shapes for each link such that mobility of the link mechanism is improved are obtained by this approach.     

新型基于分层多假设跟踪的冠脉骨架提取算法

为解决大多数脉管骨架提取算法中存在的运算复杂、准确率低以及无法同步获取脉管半径问题,提出了一种新型基于分层多假设跟踪的冠脉骨架提取算法. 首先,提出改进局部形状分析方法用于冠脉预分割,通过引入单连通约束和体积约束和降低非血管型结构及细小类血管型结构误分割率;其次,定义新的中心检测能量函数,增强骨架定位能力,并提出分层多假设策略,避免跟踪过程产生局部最优解和实现脉管半径同步获取;此外,通过生成水平集图,使算法可根据脉管树分支情况自动初始化多条跟踪路径,具有较好的拓扑适应性. 实验表明,与其他骨架提取算法相比,该算法可以同步获取冠脉骨架及半径等信息,且结果精度较高.     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

Visualized CAD models of objects made of a multiphase perfect material

A component, which has a perfect combination of different materials (including homogeneous materials and different types of heterogeneous materials) in its different portions for a specific application, is considered as the component made of a multiphase perfect material. After design of such a component according to the requirements from a specific application, a CAD model for representing it should be built so that its further analysis and manufacturing can be implemented based on the model. According to a new modeling method, this paper further presents the approaches of retrieving and visualizing all the necessary information of each layer of a component from its CAD model. The information includes: (1) 2D geometric boundary of each material region on each layer; (2) volume fraction of each material constituent for each position in each material region; and (3) 2D geometric boundaries of void phases of base cells in material regions with a periodic microstructure. Finally, two examples are given to demonstrate both geometric and material information on the layers, thus verifying the proposed approach.     

基于几何活动轮廓模型的人脸轮廓提取方法

针对在结构性噪声较严重的情况下 ,常规几何活动轮廓模型无法获得理想分割效果的问题 ,提出一种基于几何活动轮廓模型的人脸轮廓提取方法 ,该方法首先将人脸形状的椭圆性约束作为算子嵌入到几何活动轮廓模型中 ,并利用几何活动轮廓模型提取任意轮廓的优势来快速抽取出图象中类似椭圆的目标边缘 ;然后根据图象中人脸的先验知识 ,通过对检测到的椭圆目标进行进一步验证来找出最终人脸轮廓 .由于采用变分水平集方法做数值计算 ,因此该方法不仅能够自然地处理曲线的拓扑变化和能较精确地提取出图象中的人脸轮廓 ,而且同时可以给出人脸水平旋转的大致角度等信息 .实验结果表明 ,该方法是有效的 .     

Globally optimal grouping for symmetric closed boundaries by combining boundary and region information

Many natural and man-made structures have a boundary that shows a certain level of bilateral symmetry, a property that plays an important role in both human and computer vision. In this paper, we present a new grouping method for detecting closed boundaries with symmetry. We first construct a new type of grouping token in the form of symmetric trapezoids by pairing line segments detected from the image. A closed boundary can then be achieved by connecting some trapezoids with a sequence of gap-filling quadrilaterals. For such a closed boundary, we define a unified grouping cost function in a ratio form: the numerator reflects the boundary information of proximity and symmetry and the denominator reflects the region information of the enclosed area. The introduction of the region-area information in the denominator is able to avoid a bias toward shorter boundaries. We then develop a new graph model to represent the grouping tokens. In this new graph model, the grouping cost function can be encoded by carefully designed edge weights and the desired optimal boundary corresponds to a special cycle with a minimum ratio-form cost. We finally show that such a cycle can be found in polynomial time using a previous graph algorithm. We implement this symmetry-grouping method and test it on a set of synthetic data and real images. The performance is compared to two previous grouping methods that do not consider symmetry in their grouping cost functions.