Non‐uniform systematic sampling in stereology*

Non‐uniform systematic sampling designs in stereology are studied. Various methods of constructing non‐uniform systematic sampling points from prior knowledge of the measurement function are presented. As an example, we consider area estimation from lengths of linear intercepts. The efficiency of two area estimators, based on non‐uniform sampling of parallel lines, is compared to that of the classical 2D Cavalieri estimator, based on uniform sampling, in a sample of planar profiles from transverse sections of 41 small myelinated axons. The comparison is based on simulations. It is concluded that for profiles of this type one of the non‐uniform sampling schemes is more efficient than the traditional uniform sampling scheme. Other examples where non‐uniform systematic sampling may be used are in area estimation from lines emanating from a fixed point, area estimation from concentric circles or spirals and curve length estimation from sweeping lines. It is shown that proportional‐to‐size sampling is a special case of non‐uniform systematic sampling. Finally, the effect of noise in the observations is discussed.     

Precision of area estimation: a numerical study

After listing some general formulae for sampling in n-dimensional space, the author considers the one-dimensional case: the estimation of the length of a line segment by counting the number of points that happen to fall within the segment. If the points are equidistantly located, the variance of the estimate is a strictly periodic function of the length of the segment. This systematic sample has a higher efficiency than simple and stratified random samples of the same intensity. With some modifications, the results carry over to the two-dimensional case: the estimation of the area of a plane figure by counting the number of sample points falling inside the figure. However, the strict periodicity of the variance in the one-dimensional systematic case is replaced by a ‘Zitterbewegung’. The magnitude of this oscillation is seen to be very different for figures of different shapes. Some results are presented also for the estimation of areas by line transects, and for the estimation of volumes by aid of lattices of points in R³, and R⁴. Some comments are also given on the practical implications of the results for sampling in the plane.     

基于VBA的渐开线齿廓测量尺寸的计算

利用AutoCAD VBA技术,当输入齿轮的基本参数后,采用B样条拟合方法,实现了齿廓曲线的自动生成;在此基础上,用AutoCAD绘制出齿轮全图,从齿廓分度圆处作基圆的切线,并延长到另一轮齿的不同侧,直接从图中数出跨越齿数、量出公法线长度。图解法揭示了公法线长度数学计算的实质,校验了计算数据的可靠性,方法有效快捷。     

Unbiased estimation of capillary length from vertical slices*

Previous stereological approaches to estimate feature length include isotropic sections, which tend to be inefficient for highly anisotropic structures such as skeletal muscle capillaries, and semiparametric model-based methods, which require transverse and longitudinal sections only, but are biased to a variable, unknown degree. The recent method of vertical slices combines the advantages of both approaches, namely it is unbiased, efficient and convenient. This study illustrates for the first time how to apply the vertical slices method in biology by direct light microscopy and intersection counting with a properly orientated cycloid test system. Neither image processing nor confocal microscopy are used. The purpose of the study was to estimate capillary length in the left ventricle of rat heart. Beyond this, a novel histochemical method enables the staining of the venular capillary region in red and the arteriolar capillary region in blue, and hence estimates their separate lengths. The vertical slices method to estimate feature length seems to be a promising approach for biology.     

小波变换轮廓术测量精度影响因素的研究

对影响小波变换轮廓术测量精度的部分因素进行了研究。首先,为了考察光栅频率和入射角对测量精度的影响,获得合适的光栅频率和入射角,对不同光栅频率和入射角下的半圆形轮廓进行了研究;然后,为了研究不同的小波母函数对相位提取精度的影响,使用不同的小波母函数对半圆形轮廓进行小波变换,分析不同的小波母函数对半圆形轮廓测量精度的影响;最后,通过对钢板进行重建验证,研究发现了决定小波变换轮廓术测量精度的一些因素,重建钢板的厚度误差不超过 0.05 mm 。     

A practical sampling method for profile measurement of complex blades

Blade is one of the most important parts in turbine machinery. The complex geometry of blades not only makes them difficult to fabricate, but also leads them difficult to inspect. Typically, the surface of blades is measured by using coordinate measuring machine (CMM). Since the measurement time and cost increase proportionally as the increase of measurement points, it is essential to sample measurement points which can represent entire blade with sufficient confidence and accuracy. In order to achieve a certain allowable deviation with a suitable set of points, a practical sampling method for surface measurement of blades was studied. Firstly, the leading edge curve and trailing edge curve were supposed to represent the twisted and bend information of blades. A sampling method based on maximum chordal deviation for leading edge curve and trailing edge curve was researched. Further, a fusion approach for sampling points on both edge curves, which determine the cross-sections, was proposed. Secondly, the inspection points sampling method for sectional curves were investigated. Finally, two simulation and one experimental examples were used to demonstrate the sampling methodology. The results indicated that the approach of this study can ensure the measurement precision at high curvature potion by measuring a small number of points.     

Stereological length estimation using spherical probes

Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell‐to‐cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (L_v) of linear objects based on random intersections with a two‐dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object–probe intersections, including isotropic‐uniform‐random (IUR) sections, vertical‐uniform‐random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate L_v and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.     

滑动轴承偏心距的动态测量系统设计

依据两点测量法的原理,设计由光电转换模块、采样保持放大电路模块、模数转换模块、数据读取及处理模块、数据显示模块组成两通道信号测量系统,此系统测得的轴颈在X、Y方向跳动值传输给PC机,由PC机处理后得到动态测量曲线。     

Estimating the length of a bounded curve in three dimensions using total vertical projections

A new method is proposed to estimate the total length of a bounded, isolated linear feature in three dimensions from ‘total vertical projections’, obtained by rotating the curve about a fixed axis (arbitrarily called ‘vertical’) and projecting it onto a fixed vertical plane. No sections are required. Properly stained and embedded neuron dendrites, mycelial trees, fluorescent cytoskeletal filaments within a cell, etc., are candidate specimens for the method, especially in combination with the new devices for non-invasive three-dimensional microscopy. It is necessary that the specimen curve is rigid (i.e. of constant shape), that its length density is not too high (so that overlapping effects are not important) and that the embedding medium is fairly transparent. Given these requirements, the method can be very accurate and convenient to use, as is exemplified here.     

Bias of a length density estimator based on vertical projections

Attention is paid to the stereological estimator of the length density of lineal features in three-dimensional space. In Gokhale ( J. Microsc. (1990) 159 , 133–141), the estimator based on measurements performed on a projection of the content of vertical slices with a given thickness Δ was derived. The aim of this paper is to show that using five vertical slices with systematically chosen orientations yields a reliable result, i.e. the bias of the estimator is smaller than 5%. In the case of a choice of the vertical axis such that most lineal features are not perpendicular or nearly perpendicular to the vertical axis, a reliable result can be obtained using only three systematic orientations of vertical slices.     

小半圆直径的高精度测量方法及其应用

在对弓高弦长法进行深入研究的基础上,推导出当弓高或弦长在微小范围内变化时,小半圆直径(或半径)的测量方程是有关弓高变化量或弦长变化量的线性方程.该测量方程充分表明,弓高变化量或弦长变化量的放大系数是与角度有关的"灵敏系数",从而指明了提高测量精度的方向;并给出了灵敏系数是特殊值的几个测量点.用本方法测量小半圆的直径(或...     

Application of design‐based stereology for estimation of absolute volume and surface area of the articular and calcified cartilage compartments of undecalcified human femoral heads

Design‐based stereological methods using systematic uniform random sampling, the Cavalieri estimator and vertical sections are used to investigate undecalcified human femoral heads. Ten entire human femoral heads, obtained from normal women and normal men, were systematically sampled and thin undecalcified vertical sections were obtained. Absolute volumes and surface areas of the entire femoral head, the articular cartilage and the calcified cartilage compartments were estimated. In addition, the average thickness of the articular cartilage and the calcified cartilage were calculated. The stereological procedures applied to the human femoral heads resulted in average coefficient of errors, which were 0.03–0.06 for the volume estimates and 0.03–0.04 for the surface area estimates. We conclude that design‐based stereology using the Cavalieri estimator and vertical sections can successfully be used in large undecalcified tissue specimens, like the human femoral head, to estimate the absolute volume and surface area of macroscopic as well as of microscopic tissue compartments. The application of well‐known design‐based stereological methods carries potential advantage for investigating the pathology in inflammatory and degenerative joint diseases.     

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.     

New variance expressions for systematic sampling: the filtering approach

We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n‐boxes in general, on a bounded domain in n‐dimensional Euclidean space ?ⁿ, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted ? termed the filtering approach ? is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.     

Surface areas of basolateral membranes in renal distal tubules estimated by vertical sections

The surface area of the Na, K-ATPase-rich basolateral membranes in the medullary thick ascending limb of the distal tubule in the rabbit kidney was determined stereologically using the method of ‘vertical sections’, whereby unbiased surface estimates are obtained by imposing a cycloid test-lattice on micrographs of ultrathin sections cut roughly longitudinal to the tubular axis. The unbiased estimate of the surface area of basolateral membranes per tubule length in the thick ascending limb was 1.45 × 10⁶ μm²/mm. The results are compared with previous surface area measurements in this segment of the tubule and discussed with respect to the contributions from all sampling levels to the real biological variation. An optimized sampling scheme with a roughly fourfold reduction in workload is suggested.     

Spatial distribution of curve length: Concept and estimation

The length of a curvilinear feature, such as a dendrite tree of a neuron, can, in principle, be estimated by the recent, non-invasive method of total vertical projections (TVPs). Curve length is a measure of size, but it reports nothing about curve shape. The shape of a tree-like structure can be described to some extent by the distribution of branch length in properly defined regions of three-dimensional (3-D) space. A definition of curve length distribution in three dimensions is proposed and implemented here on a human neuron. The relevant 3-D regions overlap after projection, and therefore the TVPs method cannot be used directly to estimate the corresponding feature lengths. However, using the ANALYZE^TM software system running on a SUN®SPARC workstation, dendrite subsets sitting in predefined regions of space were rendered in different colours and measured separately by the TVPs method using a cycloid test system. In combination with non-invasive image acquisition and processing techniques, the length distribution concept is likely to be useful in the metrical analysis of either microscopic or macroscopic arborizations in a wide variety of contexts, including living cells and organisms.     

Stereological estimation of mean linear intercept length using the vertical sections and trisector methods

A method for estimating the mean linear intercept length of anisotropic microstructures using vertical sections is presented. A test system of cycloids and points is overlaid on the vertical sections, and the mean linear intercept length is estimated from a simple counting procedure — no length measurements are required. The vertical direction is either arbitrarily chosen or chosen perpendicular to most of the surface boundaries of the objects of interest. A design with the latter choice of the vertical direction and three vertical sections — a trisector — will optimize the precision of the estimate from only three vertical sections. The method was applied to two metallic structures, but it may also be used in a biomedical context.     

电磁场的分解在纵向非均匀光栅分析中的应用

介绍利用电磁场的分解来分析纵向非均匀光栅的方法。在此方法中,通过光栅后的散射场被按照两个相反方向分解为两个部分,然后利用传输格林函数近似求解。模拟计算结果证明该方法能较好地计算此类光栅。     

Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction

Displacement measuring interferometry has high resolution and high dynamic range, which is widely used in displacement metrology and sensor calibration. Due to beam leakage in the interferometer, imperfect polarization components, and ghost reflections, the displacement measurement suffers from periodic error, whose pitch is multiple harmonics of the Doppler frequency. In dynamic measurements, periodic error is usually on the order of nanometers, which impacts the dynamic measurement accuracy. This paper presents an approach to estimate and correct periodic error in real time based on an extended Kalman filter, which has the capability to deal with both constant and non-constant velocity motions. This algorithm is implemented on an application-specific hardware architecture in an FPGA, which has advantages in throughput and resource usage compared with conventional implementations. The measurement validation shows that this approach can effectively eliminate the periodic error for both constant and non-constant velocity motion, and the residual error reaches to the level of the background noise of the interferometer.     

A Machine Vision Approach for Detecting and Inspecting Circular Parts

In this paper, we present a machine vision approach for detecting and inspecting circular parts and parts with circular arcs on the contours. The method uses the Hough transform technique and uses the directional information of a normal to the circle at each boundary point. Cubic polynomial curve fitting is used to estimate the normal, and determine the concavity of the fitted curve at each given boundary point. The proposed Hough transform method is a two-stage pro-cedure. The first stage uses a 2D accumulator array to detect circle centres. The second stage uses a 1D accumulator array to detect the radii of circles. The proposed method is robust for detecting circular parts with partial occlusion such as peripheral defects or burrs. For an image of size N × N, the storage requirements are N 2 and the time complexity is bounded by (N+m)n, where m is the number of circle centres detected in the first stage and n is the number of boundary points in the image.