Classification of spatial textures in benign and cancerous glandular tissues by stereology and stochastic geometry using artificial neural networks

Stereology and stochastic geometry can be used as auxiliary tools for diagnostic purposes in tumour pathology. Whether first-order parameters or stochastic-geometric functions are more important for the classification of the texture of biological tissues is not known. In the present study, volume and surface area per unit reference volume, the pair correlation function and the centred quadratic contact density function of epithelium were estimated in three case series of benign and malignant lesions of glandular tissues. The information provided by the latter functions was summarized by the total absolute areas between the estimated curves and their horizontal reference lines. These areas are considered as indicators of deviation of the tissue texture from a completely uncorrelated volume process and from the Boolean model with convex grains, respectively. We used both areas and the first-order parameters for the classification of cases using artificial neural networks (ANNs). Learning vector quantization and multilayer feedforward networks with backpropagation were applied as neural paradigms. Applications included distinction between mastopathy and mammary cancer (40 cases), between benign prostatic hyperplasia and prostatic cancer (70 cases) and between chronic pancreatitis and pancreatic cancer (60 cases). The same data sets were also classified with linear discriminant analysis. The stereological estimates in combination with ANNs or discriminant analysis provided high accuracy in the classification of individual cases. The question of which category of estimator is the most informative cannot be answered globally, but must be explored empirically for each specific data set. Using learning vector quantization, better results could often be obtained than by multilayer feedforward networks with backpropagation.     

Boolean random functions

A Boolean function f in Rⁿ is the supremum of upper semi-continuous random functions f'i which are almost surely positive, bounded with compact support and centred at the Poisson points (i). They generalize to functions of classical Boolean model for sets. The Boolean function f may be studied via its subgraph, i.e. as a random set in Rⁿ x R. The key notion is then the functional Q(Bt), i.e. the probability that a compact set Bt centred at altitude t misses the subgraph of f. The general expression of Q(Bt) is given, and followed by a series of important derivations (volumes, gradients, numbers of summits, etc). Theorems of structure are given: they concern the properties of infinite divisibility for the sup, and domains of attraction for Boolean functions. The last sections are devoted to the study of two particular Boolean functions; emphasis is put on the stereological implications of the approach. A critical example illustrates the theory.     

Uncertainty analysis on the wear coefficient of Archard model

This paper proposes a probabilistic model for the wear of surfaces in contact. An initial value problem (IVP) is formulated from the particularization of Archard model for a contact in line. Based on this problem, two mathematical formulations for this model are presented. In the former, the wear coefficient is modeled as a random variable, while in the latter this coefficient is assumed as a stochastic process. The Karhunen–Loeve series is employed to represent the wear coefficient stochastic process. The solution of the IVP is the worn height stochastic process (WHSP). From this result, the functions of expectation and covariance are obtained. The results of mathematical formulations are compared with the simulations made by Monte Carlo and Latin Hypercube methods. The stochastic process presented better results, regarding the expectation and covariance functions. In relation to the propagation of uncertainty of wear coefficient through Archard model it was observed that in both presented problems, the variance of WHSP increased as the sliding time increased.     

Stochastic analysis of Poisson impact series using discrete form,spectrum analysis and time correlation

A stochastic model based on Poisson impact series is developed from a random pulse train model. Using the discrete form of the expectation function, first- and second-order time correlation functions are generated. Then, the force and power spectral density functions are generated by the Fourier transform of the first- and second-order correlation functions, respectively. Experimental force and power spectra are compared with those from the developed model and an existing model using single, multiple and ensemble averaged impact samples. Overlapping and consistent matching occur in most of the regions in these comparisons. The effects of noise and pulse shape deviation are discussed. It is shown theoretically and experimentally that the signal-to-noise ratio increases with the impaction rate. Using ensemble averages of a 100 samples, the impact magnitude and its arrival time is validated to be Gaussian and Uniformly distributed. The signal-to-error ratio increases with the number of ensemble averages. The established model is utilised to carry out modal tests of a space truss. Comparing modal parameters, such as mode shapes, natural frequencies, and damping ratios, consistent matching is observed with those obtained from the random noise test.     

Two methods of random seed generation to avoid over-segmentation with stochastic watershed: application to nuclear fuel micrographs

A stochastic version of the watershed algorithm is obtained by choosing randomly in the image the seeds from which the watershed regions are grown. The output of the procedure is a probability density function corresponding to the probability that each pixel belongs to a boundary. In the present paper, two stochastic seed-generation processes are explored to avoid over-segmentation. The first is a non-uniform Poisson process, the density of which is optimized on the basis of opening granulometry. The second process positions the seeds randomly within disks centred on the maxima of a distance map. The two methods are applied to characterize the grain structure of nuclear fuel pellets. Estimators are proposed for the total edge length and grain number per unit area, L _A and N _A, which take advantage of the probabilistic nature of the probability density function and do not require segmentation.     

Grinding force and power modeling based on chip thickness analysis

The ability to predict grinding force and power is important to many aspects of grinding process optimization, monitoring, and control. This paper presents the predictive modeling of grinding force and power based on the probabilistic distribution of undeformed chip thickness as a function of the kinematic conditions, material properties, wheel microstructure, and dynamic effects. The chip thickness is the main random variable and it is expected to assume a Rayleigh probability density function. The model takes into account the microstructure of the grinding wheel given by the grain geometry and the static grain density in terms of the radial depth into the wheel. The dynamic cutting edge density was calculated incorporating the effects of kinematic and dynamic phenomena such as the kinematic hidden grains and the local grain deflection. The elastic deformation of the grinding contact length was also considered. The model was used to predict the total tangential and normal forces in surface grinding and the total grinding power in cylindrical grinding. In both cases experimental measurement data in the context of chip thickness probability density, tangential force, normal force, and power have been presented and compared to model calculations.     

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.     

Characterization of planar local arrangement by means of the Delaunay neighbourhood

A suitable procedure is presented to characterize the neighbourhood in disperse micro-structures by measuring the Delaunay neighbour distances and neighbour numbers. Results are discussed from simulation experiments with three germ-grain models, regular arrangements and a seam-like model. A plot of variation coefficients is suggested to differentiate between various random distributions of particles.     

Stereological analysis and modelling of gradient structures

Gradient structures are inhomogeneous along a particular gradient direction but homogeneous perpendicular to that direction. Consequently, structural parameters such as volume fraction or surface area density are local characteristics which depend on the 'vertical' coordinate with respect to the 'vertical' gradient axis.
Analogously, models for gradient structures have model parameters depending on the vertical coordinate z . For example, a Voronoi tessellation with a gradient is generated by a gradient point process with a local intensity which is a function of z . Similarly, a gradient germ grain model is obtained from a gradient point process where the grain size distribution may also depend on z . For a gradient Boolean model, local volume fraction V_V ( z ) and local surface area density S_V ( z ) can be calculated from the model parameters.
Stereological methods for gradient structures are based on vertical sections parallel to the gradient direction. Estimation of V_V ( z ), S_V ( z ) and local length density L_V ( z ) is done by lineal analysis using horizontal test lines with vertical coordinate z . Similarly, lineal analysis is used to estimate local mean cell volume of gradient tessellations. For the estimation of local particle number density and size in the spirit of the Wicksell problem the use of kernel methods and distributional assumptions is required.     

Diffusion as a model of formation and development of surface topography

Changes of surface topography in tribological systems are due to inelastic processes as plastic deformation, detaching of wear particles and their reintegration into the surfaces. Due to these processes, the material particles are transported either along a surface or from one tribological partner to the other. Both processes are due to random interactions between surface asperities and are stochastic processes. The stochastic mass transfer between the surfaces is interpreted and described in the paper as a random deposition, the transport along a surface as a `diffusion' processes with some effective diffusion coefficient. We consider the development of the surface topography due to the described two kinds of random processes. There exist some stationary (in statistical sense) random surface topography with a power spectrum (spectral density) typical for many real frictional surfaces. The parameters of the model can either be obtained from comparison with measured topography or extracted from simulations at a lower space scale. The proposed mass transport model further allows to determine the wear rate in the system.     

Stochastic modeling of friction force and vibration analysis of a mechanical system using the model

The squeal noise generated from a disk brake or chatter occurred in a machine tool primarily results from friction-induced vibration. Since friction-induced vibration is usually accompanied by abrasion and lifespan reduction of mechanical parts, it is necessary to develop a reliable analysis model by which friction-induced vibration phenomena can be accurately analyzed. The original Coulomb’s friction model or the modified Coulomb friction model employed in most commercial programs employs deterministic friction coefficients. However, observing friction phenomena between two contact surfaces, one may observe that friction coefficients keep changing due to the unevenness of contact surface, temperature, lubrication and humidity. Therefore, in this study, friction coefficients are modeled as random parameters that keep changing during the motion of a mechanical system undergoing friction force. The integrity of the proposed stochastic friction model was validated by comparing the analysis results obtained by the proposed model with experimental results.     

Reconstruction of pore space from a pore connectivity network via morphological transformations

Pore and grain regions were separated via thresholding techniques from sandstone images. A mathematical morphology-based framework was followed to pack the random pore space with overlapping and nonoverlapping disks. This framework has several advantages in implementation and is generally applicable to multiscale images. The random pore space was reconstructed in two ways from the minimum morphological information through: (a) overlapping and (b) nonoverlapping disks of various shapes and sizes. The structuring elements employed to carry out this analysis included octagon, square and rhombus templates. The results achieved through these two types of reconstruction of sandstone pores are compared. These results provided the basis on which to test the accuracy of these techniques. Reconstruction recovery was tested by computing shapiness indices for the reconstructed pores achieved through the two methods.     

A stochastic finite element dynamic analysis of structures with uncertain parameters

The real life structural systems are characterized by the inherent uncertainty in the definition of their parameters in the context of both space and time. In the present study a stochastic finite element method has been proposed in the frequency domain for analysis of structural dynamic problems involving uncertain parameters. The harmonic forces as well as earthquake-induced ground motion are treated as random process defined by respective power spectral density function. The uncertain structural parameters are modelled as homogeneous Gaussian stochastic field and discretized by the local averaging method. The discretized stochastic field is simulated by the Cholesky decomposition of respective covariance matrix. By expanding the uncertain dynamic stiffness matrix about its reference value the Neumann expansion method is introduced in the finite element procedure within the framework of Monte Carlo simulation. This approach involves only single decomposition of the dynamic stiffness matrix for entire simulated structure. Thus a considerable saving of computing time and the facility that several stochastic fields can be simultaneously handled are the basic advantages of the proposed formulation. Numerical examples are presented to elucidate the accuracy and efficiency of the proposed method with the direct Monte Carlo simulation.     

基于球切多面体和光密度的砂轮建模与测量

针对金刚石砂轮磨粒形状的不规则性、尺寸的变化性以及空间位置随机分布的特点,采用基于球坐标的随机点产生的空间平面切分实体的方法,建立更接近实际的虚拟多面体磨粒,并建立金刚石砂轮模型;从类似机床刀具切削刃的磨刃二面角出发,运用仿生物学中光密度以及计算机数字图像处理的方法,提出了基于光密度、积分光密度、平均积分光密度的一种新的砂轮形貌评价方法,并通过试验测量研究了磨刃二面角与积分光密度值之间的关系。研究结果表明,磨刃二面角越大,积分光密度值越大,在0~200之间变化。最后结合砂轮表面面积密度、磨粒平均个数、出刃高度、积分光密度值等参数验证了所建立的虚拟多面体磨粒和砂轮模型的有效性。     

Stochastic model updating—Covariance matrix adjustment from uncertain experimental modal data

With deterministic methods finite element model parameters are updated by using a single set of experimental data. As a consequence the corrected analytical model only reflects this single test case. However, test data are inherently exposed to uncertainty due to measurement errors, different modal extraction techniques, etc. Even a more relevant factor for variability originates from production tolerances and consequently the question arises, how to describe model parameters from the stochastic point of view? Therefore it would be desirable to use statistical properties of multiple sets of experimental and to consider the update parameters as random variables. This paper presents an inverse approach how to identify a stochastic finite element model from uncertain test data. In detail, this work demonstrates a method to adjust design parameter means and their related covariance matrix from multiple sets of experimental modal data. Results are shown from a numerical example.     

弧齿锥齿轮接触特性的概率分析

弧齿锥齿轮齿面接触特性受到传动系统中许多随机因素的影响,概率分析可对齿面接触中的随机特性进行量化描述。本文针对某航空发动机弧齿锥齿轮传动系统,将传递功率、转速、转子不平衡量及支承刚度等作为基本随机变量,通过对系统进行动态概率计算,获得了齿轮安装基点变形的统计规律。将概率分析引入齿面接触分析技术中,研究了在齿轮安装基点随机变化过程中齿面接触域和传动误差的概率特性,计算了齿面稳定接触的可靠性。     

Limits of applicability of elastic contact models of rough surfaces

Recent stochastic models for analyzing the contact of rough surfaces assume that the asperities are microhertzian, i.e. that they can be approximated as second-order surfaces in the vicinity of contact points, and that the asperities deform elastically. Using a plane strain solution from the literature for a sinusoidally corrugated half-space, the range of validity of these assumptions is shown to be related to the mean square surface slope and the macrocontact pressure. By extension to random surfaces characterized by a one-dimensional spectral density function an interval on the surface spatial frequency is found over which the asperities deform elastically but without completely flattening. A numerical example is given.     

局部应变可靠性分析方法

根据随机振动试验的有关理论，确定产品局部应变载荷的概率密度函数。基于Coffin-Manson律，将曲线近似表示为均值和均方差循环应变-寿命曲线的形式。利用应变-寿命转换关系，获得循环寿命均值和标准差函数，在此基础上确定给定寿命下的应变强度概率密度函数。建立了随机循环强度-载荷动态干涉可靠性分析模型。实例分析说明了方法的有效性和实用性。     

A method for determining void arrangements in inverse opals

The periodic arrangement of voids in ceramic materials templated by colloidal crystal arrays (inverse opals) has been analysed by transmission electron microscopy. Individual particles consisting of an approximately spherical array of at least 100 voids were tilted through 90° along a single axis within the transmission electron microscope. The bright‐field images of these particles at high‐symmetry points, their diffractograms calculated by fast Fourier transforms, and the transmission electron microscope goniometer angles were compared with model face‐centred cubic, body‐centred cubic, hexagonal close‐packed, and simple cubic lattices in real and reciprocal space. The spatial periodicities were calculated for two‐dimensional projections. The systematic absences in these diffractograms differed from those found in diffraction patterns from three‐dimensional objects. The experimental data matched only the model face‐centred cubic lattice, so it was concluded that the packing of the voids (and, thus, the polymer spheres that composed the original colloidal crystals) was face‐centred cubic. In face‐centred cubic structures, the stacking‐fault displacementvector is . No stacking faults were observed when viewingthe inverse opal structure along the orthogonal <110>‐type directions, eliminating the possibility of a random hexagonally close‐packed structure for the particles observed. This technique complements synchrotron X‐ray scattering work on colloidal crystals by allowing both real‐space and reciprocal‐space analysis to be carried out on a smaller cross‐sectional area.     

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.