Flowers and wedges for the stereology of particles

Recently, a new decomposition has been found for the motion invariant density of straight lines in, with applications in stereology. The new principle, called the invariator, leads to new rotational formulae which express the surface area and the volume of a bounded subset (called a 'particle') in terms of an observable functional defined in an isotropically oriented section (called a pivotal section) through a fixed point (called the pivotal point). The results have been extended to intrinsic volumes of manifolds in general space forms. The purpose of this paper is to present new results and computational formulae for three-dimensional particles. Explicit estimators are obtained for a convex polyhedral particle with a pivotal point in its interior, in terms of the coordinates of the vertices of the pivotal section. The results are applied to a population of polyhedral grains from a cemented carbide which was studied earlier by alternative methods.     

A new stereological principle for test lines in three-dimensional space

A new principle is presented to generate isotropic uniform random (IUR) test lines hitting a geometric structure in three-dimensional space (3D). The principle therefore concerns the estimation of surface area, volume, membrane thickness, etc., of arbitrary structures with piecewise smooth boundary. The principle states that a point-sampled test line on an isotropic plane through a fixed point in 3D is effectively an invariant test line in 3D. Particular attention is devoted to the stereology of particles, where an alternative to the surfactor method is obtained to estimate surface area. An interesting case arises when the particle is convex. The methods are illustrated with synthetic examples.     

Sampling theory and automated simulations for vertical sections,applied to human brain

In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images – although convenient manual application is always an option – and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the ‘lambda method’) is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable – one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.     

表面织构对角接触球轴承润滑性能的影响

为研究油气润滑状态下表面织构对角接触球轴承润滑性能的影响,在角接触球轴承的内圈滚道上设计表面织构,基于气液两相流理论,采用多重参考坐标系(MRF)模型和流体体积(VOF)法,对高速角接触球轴承腔内润滑油流动特性进行数值分析,对比分析长沟槽、短沟槽、矩形、圆柱形4种表面织构对角接触球轴承润滑性能的影响,发现长沟槽形的表面织构对改善角接触球轴承的润滑性能效果最为显著,并进一步分析长沟槽形表面织构结构参数对轴承润滑性能的影响。结果表明：轴承腔内油相体积分数随着沟槽深度和面积率的增大先增大后减小,随着沟槽宽度的增加逐渐增大,存在最佳织构参数使得轴承腔内油相体积分数最大。因此,合理地设计织构参数能有效提高角接触球轴承润滑性能,该研究结果对角接触球轴承表面织构设计具有一定的参考价值。     

The laser ball bar: a new instrument for machine tool metrology

Current techniques for mapping the volumetric positioning errors of machine tools are time and labor intensive. To address these shortcomings, a linear displacement measuring device is introduced to rapidly and easily determine tool positions via trilateration. The laser ball bar (LBB) consists of a laser interferometer aligned within a telescoping ball bar. The design of the device is discussed and an error budget is developed to estimate its predicted accuracy. Results of repeatability and linear accuracy test of the assembled prototype LBB are given. The LBB is used to map the volumetric errors of a two-axis turning center. Comparison of the LBB error map and the error map obtained through parametric error measurements shows the LBB can accurately map the machine errors in a timely manner.     

A two-dimensional isotropic spline filter

The work deals with the development of a two-dimensional isotropic spline filter with application to the separation of components of the surface topography, i.e. roughness, waviness and form. An appropriate variational problem was formulated to define the mapping of primary surface data into filtered surface data with a transfer function of a circularly symmetric low-pass Butterworth filter of a given order. The variational problem was solved by approximating the filtered surface data by means of two-dimensional B-spline functions. The Fourier transform of the filter impulse response was determined to estimate the quality of approximation of the Butterworth filter and to select the appropriate filter cutoff. The paper presents one application of the designed filter to determine 3D roughness of an inner ring race of a ball bearing.     

Estimation of individual feature surface area with the vertical spatial grid

The area of an individual bounded surface (e.g. the boundary of a properly sampled cell) can be estimated from an isotropic uniform random stack of parallel sections, or of non-invasive planar scans, using the well-known spatial grid. A standing problem was to estimate the area of an individual bounded surface with an arbitrary degree of accuracy from a vertical (i.e. not isotropic) stack of sections or scans. A new tool to do this, called the ‘vertical spatial grid’, is presented.     

Automatic measurement of frontal area and volume of fluted ingots using image-based instrumentation

The development of an image-based measurement system for area and volume estimation is reported here with a novel idea of area projection with partial information. The measurement system has been put into a field trial in one of the Integrated Steel Plants in India. The estimated error was limited to ±0.5% with a high degree of repeatability.     

Stereology for anisotropic cells: Application to growth cartilage*

A number of either new or recently available stereological methods are described for estimating volume, surface area and number of anisotropic cells. The methods are illustrated with direct reference to the epiphyseal growth plate. Different estimates of a given quantity are obtained by applying alternative methods to the same set of sections, in order to compare the relative merits of the methods. For instance, the surface area of the cells is estimated via the Dimroth–Watson model (which gives a measure of the degree of anisotropy in addition to the surface area estimate) and from vertical sections using cycloid test systems. Cell number is estimated by traditional unfolding methods and by the new disector method. Also, volume-weighted mean cell volume is estimated from vertical sections via point-sampled intercepts using two different kinds of rulers to classify intercept lengths. Finally, nested design statistics is applied to a set of data from twelve animals in order to compare the relative impacts of biological and stereological (sampling) variations on the observed coefficient of error of a group mean estimate. The preferred methods are listed in the final section.     

Vertical LM sectioning and parallel CT scanning designs for stereology: application to human lung

Practical, unbiased stereological methods are described to estimate lung volume and external surface area, and total volume and surface area of relatively large and anisotropic structures (bronchi and arteries) inside the lung. The volume of each of five lung strata was estimated first by fluid displacement and then by computed tomography (CT) using Cavalieri's method; the reliability of CT was assessed through a calibration procedure, and image thresholding criteria for an accurate volume estimation using CT were established. The parallel, perfectly registered CT section images were also used to estimate the external surface area of each stratum by the spatial grid method. Unbiased estimation of internal surface areas in lung is a long-standing problem: since the structures are large and essentially void, large sections are needed; to facilitate identification, thin sections have to be used for light microscopy, and since such structures are anisotropic, the sections should be vertical. A practical stereological design is demonstrated here on an infant lung, which fulfils all these requirements. This study illustrates the potential of using unbiased stereology to characterize infant pulmonary hypoplasia.     

Global spatial sampling with isotropic virtual planes: estimators of length density and total length in thick, arbitrarily orientated sections

Existing design-based direct length estimators require random rotation around at least one axis of the tissue specimen prior to sectioning to ensure isotropy of test probes. In some tissue it is, however, difficult or even impossible to define the region of interest, unless the tissue is sectioned in a specific, nonrandom orientation. Spatial uniform sampling with isotropic virtual planes circumvents the use of physically isotropic or vertical sections. The structure that is contained in a thick physical section is investigated with software-randomized isotropic virtual planes in volume probes in systematically sampled microscope fields using computer-assisted stereological analysis. A fixed volume of 3D space in each uniformly sampled field is probed with systematic random, isotropic virtual planes by a line that moves across the computer screen showing live video images of the microscope field when the test volume is scanned with a focal plane. The intersections between the linear structure and the virtual probes are counted with columns of two dimensional disectors.
Global spatial sampling with sets of isotropic uniform random virtual planes provides a basis for length density estimates from a set of parallel physical sections of any orientation preferred by the investigator, i.e. the simplest sampling scheme in stereology. Additional virtues include optimal conditions for reducing the estimator variance, the possibility to estimate total length directly using a fractionator design and the potential to estimate efficiently the distribution of directions from a set of parallel physical sections with arbitrary orientation.
Other implementations of the basic idea, systematic uniform sampling using probes that have total 3D × 4π freedom inside the section, and therefore independent of the position and the orientation of the physical section, are briefly discussed.     

四边形网格等体积法的螺旋曲面近似展开

提出了以螺旋曲面为例的不可展曲面近似展开的方法，其步骤为曲面网格化，展开中心的确定，以展开前后曲面片体积不变作为基本展开原则的四边形网格的等体积展开，并在计算机中实现此方法。     

在机测量激光测头位姿的线性标定

针对在机激光扫描测量中激光测头安装位置和姿态引起的测量误差,提出了一种适用于在机激光测量的测头标定方法。构造了在机激光扫描测量原型系统,建立了激光测头随机床运动的测量模型;通过多角度扫描标准球球面拟合球心,给出了一种线性求解测头安装位姿参数的算法,避免了非线性优化求解中的大量计算和不稳定问题。分析了测量过程中机床各个轴的运动误差对测量结果的影响,建立了误差模型,并给出补偿机床系统误差的方法。实验显示,对直径已知的标准球进行测量时,测头在不同摆角测得的标准球直径误差小于0.05 mm,误差补偿后球心位置误差减小了83%。实验结果验证了该标定方法的可行性,以及机床误差对测量精度影响的模型及补偿方法的正确性。     

Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging

The classical methods for estimating the volume of human body compartments in vivo (e.g. skin-fold thickness for fat, radioisotope counting for different compartments, etc.) are generally indirect and rely on essentially empirical relationships — hence they are biased to unknown degrees. The advent of modern non-invasive scanning techniques, such as X-ray computed tomography (CT) and magnetic resonance imaging (MRI) is now widening the scope of volume quantification, especially in combination with stereological methods. Apart from its superior soft tissue contrast, MRI enjoys the distinct advantage of not using ionizing radiations. By a proper landmarking and control of the scanner couch, an adult male volunteer was scanned exhaustively into parallel systematic MR ‘sections’. Four compartments were defined, namely bone, muscle, organs and fat (which included the skin), and their corresponding volumes were easily and efficiently estimated by the Cavalieri method: the total section area of a compartment times the section interval estimates the volume of the compartment without bias. Formulae and nomograms are given to predict the errors and to optimize the design. To estimate an individual's muscle volume with a 5% coefficient of error, 10 sections and less than 10min point counting (to estimate the relevant section areas) are required. Bone and fat require about twice as much work. To estimate the mean muscle volume of a population with the same error contribution, from a random sample of six subjects, the workload per subject can be divided by √6, namely 4 min per subject. For a given number of sections planimetry would be as accurate but far more time consuming than point counting.     

角接触球轴承端面凸出量测量仪结构设计

利用ANSYS软件,对角接触球轴承端面凸出量测量仪的箱盖和挡盘3种设计方案进行了有限元分析,得出箱盖和挡盘的位移等值线图,并根据角接触球轴承端面测量仪箱盖和挡盘的位移量及其分布形态对测量误差的影响,确定出较合理的改进设计方案。     

On semiautomatic estimation of surface area

In this paper, we propose a semiautomatic procedure for estimation of particle surface area. It uses automatic segmentation of the boundaries of the particle sections and applies different estimators depending on whether the segmentation was judged by a supervising expert to be satisfactory. If the segmentation is correct the estimate is computed automatically, otherwise the expert performs the necessary measurements manually. In case of convex particles we suggest to base the semiautomatic estimation on the so‐called flower estimator, a new local stereological estimator of particle surface area. For convex particles, the estimator is equal to four times the area of the support set (flower set) of the particle transect. We study the statistical properties of the flower estimator and compare its performance to that of two discretizations of the flower estimator, namely the pivotal estimator and the surfactor. For ellipsoidal particles, it is shown that the flower estimator is equal to the pivotal estimator based on support function measurements along four perpendicular rays. This result makes the pivotal estimator a powerful approximation to the flower estimator. In a simulation study of prolate and oblate ellipsoidal particles, the surfactor also performs well for particles which are not extremely elongated. In particular, the surfactor is not very much affected by the singularity in the surfactor formula or by possible inaccuracies in the necessary angle measurements. We also assess the performance of the semiautomatic procedure in a study of somatostatin positive inhibitory interneurons from mice hippocampi. Only 35% of the cells needed to be analysed manually and an important decrease in workload was obtained by using the semiautomatic approach.     

新型大工作空间仿生被动球面铰链构型及工作空间分析

被动球面铰链是构成机构的基本单元之一,但是,较小的工作空间制约了其在机械结构设计中的应用。以普通被动球面铰链为原型,在对其结构及工作空间进行分析的基础上,采用运动与约束单项分解、交叉匹配的方法,提出并设计完成了具有较大工作空间的双耳支架固定型被动球面铰链和球/窝换位型被动球面铰链;基于仿生设计理论,对照人体肱骨结构,提出偏置输出的设计新思想,进而,设计完成具有大工作空间的偏置输出的横向剖分球窝型被动球面铰链和偏置输出的球/窝换位型被动球面铰链;建立两空间映射的函数方程。分析发现,随着铰链输出杆偏置角的增大,铰链的工作空间随之增大,并且形状发生相应变化。所提出的新型铰链,解决了被动球面铰链因工作空间小而在工程应用中受到限制的问题,对新型被动球面铰链的研制具有理论指导意义和工程实践价值。     

直线滚动导轨反向器回珠曲线曲率半径的优化设计

直线滚动导轨中钢球的运动是很复杂的,一般情况下,滚动体绕自身轴线转动,同时又绕反向器沟道轴线公转.在滚动的同时,滚动体沿滚道还伴随有一定的滑动.通常直线滚动导轨中钢球的运动是三维的,若接触角不等于零.钢球会有绕垂直于接触面的滑动,成为自旋滑动.高速运转时,钢球还可能有因陀螺力矩而产生的运动.直线滚动导轨的结构参数,工作条件,润滑状况以及制造精度都会影响其内部的运动学关系.综合考虑结构参数以及润滑状况对直线滚动导轨运动的影响,对反向器的回珠曲线进行多目标函数的优化设计,使得钢球能从受载区向非受载区平稳过渡.     

On the empirical variance of a fractionator estimate

The fractionator consists of several sampling stages with systematic sampling at each stage; data are collected only at the last stage. Therefore, predicting the error variance of a fractionator estimator is a non-trivial problem, because the observations are correlated in a complicated, unknown way. Gundersen proposed to split the material sampled at the first stage into two subsets, and to compute the variance of the pooled estimate empirically using the corresponding pair of observations made in these two subsets. The idea is very effective, but the estimator thus proposed needed some corrections. The purpose of this paper is to present an improved estimator of the coefficient of error of a fractionator estimator using Gundersen's design.     

成形法加工新型环面蜗杆的原理性误差分析

新型环面蜗杆副由蜗杆、钢球和蜗轮构成,利用失配共轭技术,提高了其承载能力和适应性.在分析蜗杆廓面结构特征的基础上,给出了蜗杆廓面的准线方程,提出了基于准线的成形法加工方法,着重分析了加工过程中的原理性误差,推导出了廓面法向误差公式.以一组数值参数为例,绘制了齿形误差和齿向误差图,并在OpenGL平台上进行误差可视化处理.结果表明:成形法加工的蜗杆廓面齿形和齿向误差值较小,能够保证获得高精度的传动廓面.