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。     

隐式曲面上两相图像分割的变分水平集方法

在平面图像分割的Chan-Vese模型基础上,提出隐式曲面上两相图像分割模型。用静态水平集函数的零水平集表达图像所在的闭合曲面,用另一动态水平集函数的零水平集与静态水平集函数零水平集的交线表达静态曲面上图像分割的动态轮廓线。所研究模型的能量泛函的数据项即为曲面上两分割区域的图像强度与对应区域平均图像强度的差的平方,其轮廓线长度项为两水平集函数的零水平集交线的长度。为避免动态水平集函数的重新初始化,在能量泛函中引入水平集函数为符号距离函数的约束惩罚项。通过变分方法得到图像分割空间轮廓线演化的梯度降方程。通过显式差分格式对演化方程进行离散。实验结果表明,该模型能有效实现复杂封闭曲面上图像的两相分割。     

三维图像多相分割的变分水平集方法

变分水平集方法是图像分割等领域出现的新的建模方法,借助多个水平集函数可有效地实现图像多相分割.但在区域/相的通用表达、不同区域内图像模型的表达、通用的能量函的设计、高维图像分割中的拓展研究等方面仍是图像处理的变分方法、水平集方法、偏微分方程方法等研究的热点问题.文中以三维图像为研究对象,系统地建立了一种新的三维图像多相分割的变分水平集方法.该方法用n-1个水平集函数划分n个区域,并基于Heaviside函数设汁出区域划分的通用的特征函数;其能量泛函包括通用的区域模型、边缘检测模型和水平集函数为符号距离函数的约束项3部分;最后,针对所得到的曲面演化方程,采用半隐式差分格式进行离散,并对多种类型三维图像进行分割验证了所提出模型的通用性和有效性.     

曲面扩散的变分水平集方法

基于变分水平集方法提出了一种通用的曲面扩散变分模型,其数据项为演化曲面与原曲面的水平集函数Heaviside函数差的平方,规则项为基于整体曲率的通用函数,通过图像扩散模型中的总变差与该模型中的总曲率类比设计该规则项,以实现曲面扩散的任务。为了避免水平集函数的重新初始化,在本文的能量泛函中增加了水平集函数为符号距离函数的惩罚项。所得到的演化方程为4阶偏微分方程,对其对流项采用经典迎风差分格式离散,对其中的扩散项采用中心差分格式。最后通过数值算例验证了模型用于曲面光滑、边缘保持与边缘增强的可行性。     

基于PM模型的曲面去噪变分水平集方法

PM（perona-malik）模型是一种经典的非线性图像扩散模型,该模型能根据设定的阈值对图像光滑区域进行扩散,并能自适应地保持图像边缘。本文将曲面法矢量与一般灰度图像的强度进行类比,将经典的图像扩散的PM模型转化为曲面几何噪声处理的自适应扩散变分模型,在使曲面光滑的同时,能够保持曲面边缘。曲面采用隐函数的零水平集表达,能量泛函中的数据项用初始水平集函数的Heaviside函数与演化后水平集函数的Heaviside函数差的平方表达,能量泛函中的光滑项基于几何曲率定义。此外,在能量泛函中增加了水平集函数为符号距离函数的惩罚项,避免了水平集函数需要不断重新初始化的问题。数值实验验证了所提出模型的曲面噪声去除及曲面边缘保持效果。     

水平集网格简化算法在遗址场景中的应用

将水平集方法引入到三维模型网格简化中,构造符号距离函数,函数的零集定义为初始曲面;引入一个能量泛涵,通过对其极小化诱导出一个水平集形式的二阶几何偏微分方程,从而将网格简化过程转化为隐式模型的体素扩散过程。该方法目前已经用于文化遗产数字化的大场景和文物的模型简化中。对水平集网格简化算法和现常用的基于点对收缩的网格简化算法在视觉质量和几何误差方面做了比较和分析,实验表明该方法适用于任意拓扑形状的网格模型,使得模型大规模简化后,在保持较低误差的同时,仍然能够保持相当多的重要几何特征和较好的整体视觉效果。     

基于多层水平集函数的三维多相图像分割

多相图像分割通常利用多个水平集函数分别定义不同区域的特征函数,其极值求解问题需要对多个函数分别求极值,计算效率较低。针对三维多相图像,提出一种改进的变分水平集模型,采用一个多层水平集函数的n层水平集隐式曲面,将图像划分为n个区域,通过对一个水平集函数求极值,实现三维多相分段常值图像的快速分割与重建。将能量泛函表达为数据项和规则项,借助规则化Heaviside函数设计区域划分的通用特征函数,采用Split-Bregman投影方法进行能量最小化求解。实验结果表明,该模型可以有效地实现三维多相图像分割,与Chan-Vese模型相比,其迭代步数较少,分割速度较快。     

基于变分水平集方法的体素模型修复

针对体素着色方法所重建出的模型存在大量空洞与缺损的问题,提出了一种模型修复方法。首先采用改进的变分水平集方法恢复出封闭完整的体素模型曲面;再利用原始体素模型的颜色信息,对新生成的模型曲面进行重着色并滤除杂色,最终完成体素模型的修复。其中,改进的变分水平集方法在水平集能量函数中引入了符号距离函数自动规整项,避免了重初始化操作;添加了新的进化加速项,防止了隐式曲面穿透模型而造成模型侵蚀。实验表明,修复好的体素模型较原模型相比具有完整的外型和平滑的色彩,且曲面恢复效率更高。     

基于形状保持的CV变分水平集矩形目标分割EI北大核心CSCD

目标在被局部遮挡、与背景灰度信息相似以及纹理比较明显等情况下,传统CV模型无法进行准确分割.为此,将模型中活动轮廓曲线的水平集函数用先验形状的水平集函数来代替,使得曲线在演化过程中始终保持某一类特定形状,从而实现了目标分割过程中的形状保持.根据形状保持的CV变分水平集分割模型建立适用于矩形目标分割的能量函数模型,推导出一组Euler-Lagrange常微分方程;通过水平集函数的不断迭代演化最终实现了矩形目标的分割;最后演化得到的水平集函数是对矩形目标的定量描述.3组实验结果证明,该模型能够解决复杂情况下的矩形目标分割问题,且具有计算量小、鲁棒性强的优点.     

基于形状保持的CV变分水平集矩形目标分割

目标在被局部遮挡、与背景灰度信息相似以及纹理比较明显等情况下,传统CV模型无法进行准确分割.为此,将模型中活动轮廓曲线的水平集函数用先验形状的水平集函数来代替,使得曲线在演化过程中始终保持某一类特定形状,从而实现了目标分割过程中的形状保持.根据形状保持的CV变分水平集分割模型建立适用于矩形目标分割的能量函数模型,推导出一组Euler-Lagrange常微分方程;通过水平集函数的不断迭代演化最终实现了矩形目标的分割;最后演化得到的水平集函数是对矩形目标的定量描述.3组实验结果证明,该模型能够解决复杂情况下的矩形目标分割问题,且具有计算量小、鲁棒性强的优点.     

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 New Level-Set Model for the Representation of Non-Smooth Geometries

In Starinshak et al. (J Comput Phys 262(1):1–16, 2014), we proposed a new level-set model for representing multimaterial flows in multiple space dimensions. Rather than associating each level-set function with the boundary of a material, the new model associates each level-set function with a pair of materials and the interface that separates them. In this paper, we extend the model to represent geometries with non-smooth boundaries. The model uses multiple level-set functions to describe the shape boundary, typically with one level-set function per smooth boundary segment. Sign information is collected from all level-set functions and a voting algorithm is used to determine the interior/exterior of the geometric shape. The model is well suited for representing boundaries with singularities; it offers significant improvement over standard level-set approaches, both in shape preservation and area conservation; and it eliminates the need for costly redistancing of the level-set function. Numerical examples illustrate the superior performance of the proposed model.     

Stochastic Level Set Dynamics to Track Closed Curves Through Image Data

We introduce a stochastic filtering technique for the tracking of closed curves from image sequence. For that purpose, we design a continuous-time dynamics that allows us to infer inter-frame deformations. The curve is defined by an implicit level-set representation and the stochastic dynamics is expressed on the level-set function. It takes the form of a stochastic partial differential equation with a Brownian motion of low dimension. The evolution model we propose combines local photometric information, deformations induced by the curve displacement and an uncertainty modeling of the dynamics. Specific choices of noise models and drift terms lead to an evolution law based on mean curvature as in classic level set methods, while other choices yield new evolution laws. The approach we propose is implemented through a particle filter, which includes color measurements characterizing the target and the background photometric probability densities respectively. The merit of this filter is demonstrated on various satellite image sequences depicting the evolution of complex geophysical flows.     

A Self-Referencing Level-Set Method for Image Reconstruction from Sparse Fourier Samples

We address an ill-posed inverse problem of image estimation from sparse samples of its Fourier transform. The problem is formulated as joint estimation of the supports of unknown sparse objects in the image, and pixel values on these supports. The domain and the pixel values are alternately estimated using the level-set method and the conjugate gradient method, respectively. Our level-set evolution shows a unique switching behavior, which stabilizes the level-set evolution. Furthermore, the trade-off between the stability and the speed of evolution can be easily controlled by the number of the conjugate gradient steps, thus avoiding the re-initialization steps in conventional level set approaches.     

A streaming narrow-band algorithm: interactive computation and visualization of level sets

Deformable isosurfaces, implemented with level-set methods, have demonstrated a great potential in visualization and computer graphics for applications such as segmentation, surface processing, and physically-based modeling. Their usefulness has been limited, however, by their high computational cost and reliance on significant parameter tuning. This paper presents a solution to these challenges by describing graphics processor (GPU) based algorithms for solving and visualizing level-set solutions at interactive rates. The proposed solution is based on a new, streaming implementation of the narrow-band algorithm. The new algorithm packs the level-set isosurface data into 2D texture memory via a multidimensional virtual memory system. As the level set moves, this texture-based representation is dynamically updated via a novel GPU-to-CPU message passing scheme. By integrating the level-set solver with a real-time volume renderer, a user can visualize and intuitively steer the level-set surface as it evolves. We demonstrate the capabilities of this technology for interactive volume segmentation and visualization.     

A level-set approach for the metamorphosis of solid models

We present a new approach to 3D shape metamorphosis. We express the interpolation of two shapes as a process where one shape deforms to maximize its similarity with another shape. The process incrementally optimizes an objective function while deforming an implicit surface model. We represent the deformable surface as a level set (iso-surface) of a densely sampled scalar function of three dimensions. Such level-set models have been shown to mimic conventional parametric deformable surface models by encoding surface movements as changes in the grayscale values of a volume data set. Thus, a well-founded mathematical structure leads to a set of procedures that describes how voxel values can be manipulated to create deformations that are represented as a sequence of volumes. The result is a 3D morphing method that offers several advantages over previous methods, including minimal need for user input, no model parameterization, flexible topology, and subvoxel accuracy     

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.