Centred contact density functions for the statistical analysis of random sets A stereological study on benign and malignant glandular tissue using image analysis

Real structures investigated in the material and biological sciences, such as minerals or tissues, can often be reduced to two phases. In a stochastic approach, the components of such binary structures may be considered as the union of grains — random sets implanted with their centres at random points — and their complementary space, which is called the pore space. The simplest stochastic germ-grain model is the Boolean model of random sets, which we use here instrumentally as a null model (reference model) for comparison with our biological material. After a brief review of basic properties of the Boolean model and related statistical methods, we introduce centred contact density functions as a new approach. Empirical contact density functions are estimated from the empirical contact distribution functions with an image analyser by dilation of the grain phase. Theoretical contact density functions are then predicted from a set of image parameters, under the assumption that the Boolean model holds. A centred contact density function is the difference between the estimated and the predicted contact density function. Apart from a random error term, centred contact density functions amount to zero irrespective of the area fraction of the grain phase, when the germ-grain model is Boolean. As a section of a spatial Boolean model is a planar Boolean model, the method is also applicable in stereological studies where digitized images are obtained from sections of a three-dimensional structure. Centred contact density functions were determined for mastopathic tissue as compared to mammary cancer, and for tumour-free prostatic tissue as compared to prostatic cancer. For each category of specimens, twenty cases with 10 images each were analysed. Benign and malignant glandular tissue of the aforementioned types deviates significantly from the Boolean model. Centred contact density functions show that malignant transformation is connected with profound geometric remodelling of the pore space.     

Stereology for pores in wheat bread: statistical analyses for the Boolean model by serial sections

A simple approximative stereological method is proposed for obtaining the fundamental parameters of a three-dimensional Boolean model. It uses serial sections for the determination of N⁺_V, the specific convexity number. This procedure was applied to two different examinations of bread, which can be very well described by Boolean models.     

基于布尔运算的轻量化模型建立技术研究

为了解决基于三维模型的机加工艺设计中工序模型的建立问题,提出了基于布尔运算的轻量化模型建立方法。通过对轻量化模型格式的分析,建立了轻量化模型的数据结构,在此基础上对轻量化模型的建立方法进行了深入地研究,给出了模型建立的实现过程并对其中涉及的相交测试及交点计算、点及面的重组算法进行了详细的讨论。最后,通过建模实例对该算法进行了验证。     

Directional analysis of planar fibre networks: Application to cardboard microstructure

This paper discusses the problem of determining suitable roses of intersections for systems of planar thick fibres which cross and overlap. A planar Boolean model with long rectangular (or similar) grains is suggested as an appropriate mathematical model. It leads to a statistical estimator for the rose of intersections of the system of fibre spines. The method is used to analyse the microstructure of the outer layer of two samples of cardboard.     

基于隐函数转化的点取样模型布尔运算

提出一种基于点取样模型的布尔运算算法。该算法首先将点模型隐式化，得到点模型的隐函数表示；然后采用隐式曲面的布尔运算方法对点模型进行布尔操作。提出了隐式曲面布尔运算的简单形式。点模型的隐式化采用散乱点径向基函数（RBF）变分插值，采用该算法可由多个点模型构造复杂的点模型。所提出的点模型布尔操作算法可用于一般网格模型，实现网格模型的剪切／粘贴编辑处理。     

Random models for morphological analysis of powders

Models to estimate powder size distributions or fractions of components in a mixture of powders were developed and tested for various specimens. The models derived from the Boolean model and from the dead leaves models can be implemented on rough secondary electron scanning electron microscope images, obtained after minimal sample preparation. With the dead leaves tessellation, the size distributions of spherical particles or short fibres can be estimated either from the size distribution of intact grains, or from the area fraction measurement after binary erosions. With the dead leaves random function, there is no need for image segmentation. It provides data on the size and shape of a population of particles from an estimation of the distribution of grey-level images after erosions by convex structuring elements of increasing size. A version of these models for long fibres is developed for estimating their diameter distribution. Allowing for superposition of particles, the proposed methods enable an unbiased estimation of the size distribution and characterization of the shape of complex particles. The approach is illustrated by applications to spherical particles obtained by simulations and from scanning electron microscope micrographs.     

Improved estimation of the pair correlation function of random sets

The texture of binary spatial structures can be characterized by second-order methods of spatial statistics. The pair correlation function, which describes the structure in terms of spatial correlation as a function of distance, is of central importance in this context. Conventionally, the pair correlation function of stationary and isotropic random sets is estimated as the ratio of the covariance to the square of volume fraction of the phase of interest. In the present paper, an improved estimator of the pair correlation function is presented, where the covariance is divided by the square of a distance-adapted estimator of volume fraction. The new estimator is explained mathematically and applied to simulated images of the Boolean model and to microscopic images from neoplastic and non-neoplastic human glandular tissues. It leads to a considerable reduction of bias and variance of estimated pair correlation functions, in particular for large distances.     

Expectation formulas and isoperimetric properties for non-isotropic Boolean models

Boolean models Y in R² and R³ are considered and formulas connecting the quermass densities of Y with the mean values of certain functional of the primary grain are given in the non-isotropic case. Under symmetry conditions, the mean functionals are interpreted as mixed volumes of convex bodies which are associated with Y. Extremal properties of isotropic Boolean models are deduced from the isoperimetric inequality.     

Boolean operations of STL models based on loop detection

For the data processing of rapid prototyping manufacturing, Boolean operation can offer a versatile tool for editing or modifying the STL model, adding the artificial construction, and creating the complex assistant-support structure to meet special technical requests. The topological structure of STL models is built firstly in order to obtain the relationship among the triangular facets of the object. The intersection test between two triangles picked, respectively, from two solids is taken to get the intersection triangle pairs and the intersection segments array, from which the intersection segments loops are detected. The intersection surfaces are divided into several surface patches along the intersection loops. The inclusion prediction is taken by testing the candidate point whether inside or outside the solid region of the solid slice. Detecting the loops for determination of the valid intersection lines greatly increases the efficiency and the reliability of the process. An erratum to this article can be found at     

Using biased image analysis for improving unbiased stereological number estimation – a pilot simulation study of the smooth fractionator

The smooth fractionator was introduced in 2002. The combination of a smoothing protocol with a computer‐aided stereology tool provides better precision and a lighter workload. This study uses simulation to compare fractionator sampling based on the smooth design, the commonly used systematic uniformly random sampling design and the ordinary simple random sampling design. The smooth protocol is performed using biased information from crude (but fully automatic) image analysis of the fields of view. The different design paradigms are compared using simulation in three different cell distributions with reference to sample size, noise and counting frame position. Regardless of clustering, sample size or noise, the fractionator based on a smooth design is more efficient than the fractionator based on a systematic uniform random design, which is more efficient than a fractionator based on simple random design. The fractionator based on a smooth design is up to four times more efficient than a simple random design.     

Estimating mean particle volume and number from random sections by sampling profile boundaries

We describe some new shape-independent stereological estimates of particle mean volume and surface area. Finding volumes or surface areas of cell nuclei, from electron micrographs of random thin sections, is a central problem of biological stereology. The well-known point-sampled intercept (PSI) method samples profile interiors to find the volume-weighted mean volume. This can be used in place of the true mean volume, but to do so introduces bias when volumes vary a great deal, as they do in fixed specimens. Jensen and Gundersen quite recently extended the PSI estimator to provide particle surface area, with no bias in the case of uniform surface areas. Here we extend the PSI volume estimator in a different way, sampling profile boundaries rather than their interiors. We obtain a boundary-sampled intercept (BSI) volume estimator, simpler than the PSI surface area estimator, but also unbiased for uniform surface areas. Both of these estimators are attractive, for example, in measuring and counting cell nuclei, where membrane surface area varies less than volume. Furthermore, they have no shape bias whatsoever. This paper also examines the general relationship between boundary- and area-sampled estimates, and we clarify the formal connection between our volume estimator and the PSI surface area estimator. We also calculate and compare their theoretical efficiencies.     

Vibration and reliability of a rotating beam with random properties under random excitation

In this paper, vibration and reliability of a rotating beam with random properties under random excitations are studied. The rotating beam is under a stochastic load modeled as a stationary white noise. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus, damping coefficient, mass density and rotational speed are modeled as random variables. To develop the equations of motion, the finite element method and space state analysis are applied. In order to consider the randomness of properties, a second order perturbation method is used. The effects of rotational speed, setting angle, hub radius, variances of random properties, correlation of random variables and damping matrix forms on the vibration and reliability of rotating beams, are studied completely.     

Morphological modelling of three‐phase microstructures of anode layers using SEM images

A general method is proposed to model 3D microstructures representative of three‐phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross‐)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three‐phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross‐covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three‐phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.     

Maximum-likelihood estimation for discrete Boolean models using linear samples

An observation of a Boolean model consists of a set of covered points. In the one-dimensional discrete case, the likelihood function of an observation can be expressed via the lengths of sequences of covered points and points not covered, called black and white runlengths, respectively. The black and white runlengths are independent random variables whose respective distributions determine the one-dimensional discrete Boolean model completely. Under certain conditions, a two-dimensional discrete Boolean model induces a one-dimensional discrete Boolean model, thereby allowing the likelihood function of a one-dimensional observation to be expressed in terms of the parameters of the two-dimensional model. This relationship enables maximum likelihood estimation to be performed on the two-dimensional model using linear samples. Examples are given including an application involving micrographs of toner particles.     

布尔运算在压缩机主机设计中应用

布尔运算是计算机图形学中的基本算法。在计算机几何造型技术与CAD设计软件建模过程中,布尔操作作为一个常用工具发挥着重要的作用。从压缩机主机机身油池容量及压缩缸端余隙容积的计算2个方面,阐述了布尔运算在压缩机主机设计中的运用。通过使用该方法,可以高效快捷的得到上述空间容积,获得主机的主要性能参数,同时有效地解决了传统计算方法工作量大、效率低等问题,并提高了计算结果的准确性。     

A simple and efficient alternative to implementing systematic random sampling in stereological designs without a motorized microscope stage

When properly applied, stereology is a very robust and efficient method to quantify a variety of parameters from biological material. A common sampling strategy in stereology is systematic random sampling, which involves choosing a random sampling relevant objects start point outside the structure of interest, and sampling at sites that are placed at pre‐determined, equidistant intervals. This has proven to be a very efficient sampling strategy, and is used widely in stereological designs. At the microscopic level, this is most often achieved through the use of a motorized stage that facilitates the systematic random stepping across the structure of interest. Here, we report a simple, precise and cost‐effective software‐based alternative to accomplishing systematic random sampling under the microscope. We believe that this approach will facilitate the use of stereological designs that employ systematic random sampling in laboratories that lack the resources to acquire costly, fully automated systems.     

一种伪随机信号发生器的研制

本文介绍了一种纯硬件实现的伪随机信号发生器的研制工作。首先给出了设计方案，继而介绍了这一信号发生器的基本原理和具体电路的设计．给出了电路原理图，最后给出了性能测试的结果。     

Reconstruction of random heterogeneous media

Stochastic reconstruction is a technique to generate samples of random structures with prescribed distributional properties in the sense that certain of their statistical summary characteristics match target values or forms. This technique can be used to produce structures of any wanted size for further statistical analysis starting from small samples, which may be even only lower dimensional, for instance, when three‐dimensional imaging techniques are not available. In this review we explain the main ideas of stochastic reconstruction, concentrating on the most important case of digitized binary media and with a particular emphasis on stereological reconstruction.     

Feature-based multi-resolution modeling of solids using history-based Boolean operations — Part I: Theory of history-based boolean operations —

The requirements of multi-resolution models of feature-based solids, which represent an object at many levels of feature detail, are increasing for engineering purposes, such as analysis, network-based collaborative design, virtual prototyping and manufacturing To provide multi-resolution models for various applications, it is essential to generate adequate solid models at varying levels of detail (LOD) after feature reanangement, based on the LOD criteria However, the non-commutative property of the union and subtraction Boolean operations is a severe obstacle to arbitrary feature rearrangement To solve this problem we propose history-based Boolean operations that satisfy the commutative law between union and subtraction operations by considering the history of the Boolean operations Because these operations guarantee the same resulting shape as the original and reasonable shapes at the intermediate LODs for an arbitrary rearrangement of its features, various LOD criteria can be applied for milti-resolution modeling in different applications     

Second-order stereology of benign and malignant alterations of the human mammary gland

The purpose of the present study was a quantitative characterization of the three-dimensional arrangement of the epithelial component of benign and malignant alterations of the female breast by combining stereology with stochastic geometry. Twenty cases of fibrous mastopathy and 20 cases of invasive ductal mammary cancer were studied at the light microscopic level. Segmentation of the epithelial tissue component was performed with an image analyser. From the resulting binary images, unbiased estimates of the covariance C(r) and the intensity V_v of the epithelial volume component were obtained automatically by computer. From these data, estimates of the correlation function k(r), of the pair correlation function g(r), of the radial distribution function RDF(r) and of the reduced second moment function K(r) of epithelial volume were determined. The estimates of C(r) and RDF(r) differed between groups, but these functions depend on spatial pattern and V_v. As carcinomas showed a significantly higher epithelial volume density V_v than mastopathies, estimation of C(r) and RDF(r) alone did not permit a safe distinction between possibly different types of spatial arrangement of epithelium in the benign and malignant lesions. Analysis of the estimates of k(r), g(r) and K(r), which are not influenced by V_v, showed definite interaction between epithelial volume elements, with clustering at short distances and repulsion at long distances. In both groups, the null hypothesis of purely random arrangement of epithelium had to be rejected. The clearest distinction between groups was obtained by estimation of g(r), which showed that short-range, tubular pattern as well as long-range, lobular architecture are better preserved in benign than in malignant lesions. The low interindividual scatter of k(r), g(r) and K(r) indicates a high biological significance of spatial pattern, which is presumably under strict genetic control. Potential uses of the method are: (i) the identification of biomathematical models which could contribute to a better understanding of the growth processes involved, (ii) conditional simulation of the underlying three-dimensional structures by computer, and (iii) supporting the diagnosis of mammary lesions with borderline histopathological appearance.