Evaluation of form data using computational geometric techniques—Part I: Circularity error

The present work deals with evaluation of form error from the measured profiles obtained using a form tester, namely roundness/cylindricity measuring instrument. In Part I, details of circularity evaluation are presented. Due to eccentricity in component setting and radius-suppression inherent in the measurement, circularity error has to be evaluated with reference to a limacon. A computational geometry-based algorithm is proposed for establishing minimum circumscribed, maximum inscribed and minimum zone limacons. A new type of control hull for directly constructing equi-angular diagrams and a new procedure for updating are introduced. Validation has been done with bench-mark data set and corresponding results available in the literature. Being geometry-based algorithm, it is simple to follow and each iteration can be visualized and interpreted geometrically. On comparison with simplex search method, the proposed algorithm is found to be computationally efficient in terms of accuracy and time taken. The proposed methods can be easily implemented in computer-aided roundness measuring instruments. Extension of this work for evaluation of cylindricity error has been dealt in Part II.     

A Hu moment invariant as a shape circularity measure

In this paper we propose a new circularity measure which defines the degree to which a shape differs from a perfect circle. The new measure is easy to compute and, being area based, is robust—e.g., with respect to noise or narrow intrusions. Also, it satisfies the following desirable properties:

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it ranges over (0,1] and gives the measured circularity equal to 1 if and only if the measured shape is a circle;

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it is invariant with respect to translations, rotations and scaling.

Compared with the most standard circularity measure, which considers the relation between the shape area and the shape perimeter, the new measure performs better in the case of shapes with boundary defects (which lead to a large increase in perimeter) and in the case of compound shapes. In contrast to the standard circularity measure, the new measure depends on the mutual position of the components inside a compound shape.Also, the new measure performs consistently in the case of shapes with very small (i.e., close to zero) measured circularity. It turns out that such a property enables the new measure to measure the linearity of shapes.In addition, we propose a generalisation of the new measure so that shape circularity can be computed while controlling the impact of the relative position of points inside the shape. An additional advantage of the generalised measure is that it can be used for detecting small irregularities in nearly circular shapes damaged by noise or during an extraction process in a particular image processing task.     

狭义相对论中常见问题的图解法

介绍了狭义相对论时空的一种新图示法，并用此图示法解释了诸如长度收缩、时间延缓等相对论效应；本方法具有简洁、直观的特点。     

Improving angular error via systematically designed near-circular Gaussian-based feature extraction operators

In image filtering, the ‘circularity’ of an operator is an important factor affecting its accuracy. For example, circular differential edge operators are effective in minimising the angular error in the estimation of image gradient direction. We present a general approach to the computation of scalable circular low-level image processing operators that is based on the finite element method. We show that the use of Gaussian basis functions within the finite element method provides a framework for a systematic and efficient design procedure for operators that are scalable to near-circular neighbourhoods through the use of an explicit scale parameter. The general design technique may be applied to a range of operators. Here we evaluate the approach for the design of the image gradient operator. We illustrate that this design procedure significantly reduces angular error in comparison to other well-known gradient approximation methods.     

Measure of circularity for parts of digital boundaries and its fast computation

This paper focuses on the design of an effective method that computes the measure of circularity of a part of a digital boundary. An existing circularity measure of a set of discrete points, which is used in computational metrology, is extended to the case of parts of digital boundaries. From a single digital boundary, two sets of points are extracted so that the circularity measure computed from these sets is representative of the circularity of the digital boundary. Therefore, the computation consists of two steps. First, the inner and outer sets of points are extracted from the input part of a digital boundary using digital geometry tools. Next, the circularity measure of these sets is computed using classical tools of computational geometry. It is proved that the algorithm is linear in time in the case of convex parts thanks to the specificity of digital data, and is in O(nlogn) otherwise. Experiments done on synthetic and real images illustrate the interest of the properties of the circularity measure.     

砼环梁内撑在软土深基坑支护中的应用

在软土深基坑中，侧向土压力较大。采用混凝土环梁内支撑体系，可以将侧向荷载转化为环梁的轴力，以充分发挥混凝土的受压特性；同时中间环梁形成的大空间可以方便土方开挖和地下室施工。环梁应有足够的截面刚度，并设置足够的立柱以保证垂直面内的稳定。     

FORM ERROR EVALUATION OF CIRCLES BASED ON A FINELY-DESIGNED GENETIC ALGORITHM

A genetic algorithm(GA)-based new method is designed to evaluate the circularity error of mechanical parts. The method uses the capability of nonlinear optimization of GA to search for the optimal solution of circularity error. The finely-designed GA (FDGA) characterized dynamical bisexual recombination and Gaussian mutation. The mathematical model of the nonlinear problem is given. The implementation details in FDGA are described such as the crossover or recombination mechanism which utilized a bisexual reproduction scheme and the elitist reservation method; and the adaptive mutation which used the Gaussian probability distribution to determine the values of the offspring produced by mutation mechanism. The examples are provided to verify the designed FDGA. The computation results indicate that the FDGA works very well in the field of form error evaluation such as circularity evaluation.     

Asymptotic distribution of circularity coefficients estimate of complex random variables

In this correspondence we mainly consider the asymptotic distribution of the estimator of circularity coefficients of scalar and multidimensional complex random variables. A particular attention is paid to rectilinear RV. After deriving new properties of the circularity coefficients, the maximum likelihood estimate of the circularity coefficients in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given. Finally, an illustrative example is presented in order to strengthen the obtained theoretical results.     

Development of a measuring method for several types of programmed tool paths for NC machine tools using a laser displacement interferometer and a rotary encoder

Many circular motion measuring methods for NC machine tools have been proposed, however, the drawback common to many of these methods is the restriction on the radius size due to the short measuring range of the displacement transducers used. Moreover, most of these measurement tools are specialized, and can only perform circular test path measurements. A circularity test method using a laser displacement interferometer and a rotary encoder has been developed. The measuring method features a much longer range of motion than ordinal circular test methods such as the double ball bar (DBB) method and, therefore, the radius restriction on these measurements is greatly reduced. Moreover, this measuring system can also be used for the evaluation of positioning accuracy and other more complex test paths.

The proposed device consists primarily of a laser displacement interferometer and a rotary encoder. The holders for the interferometer head and the retroreflector are connected with a stainless steel rod. The retroreflector holder has a synthetic resin linear bearing allowing it to move relative to the interferometer head so that both optical components are always facing each other. The laser interferometer measures the change in distance between the interferometer head and the retroreflector, and the rotary encoder measures the rotation angle of the stainless steel rod.

In this paper, the background, measuring principle and apparatus structure are briefly described. The experimental setup is also presented. The apparatus was employed in several measuring experiments, including circularity tests for a vertical machining center. The results from these experiments support the validity of this measurement apparatus.     

Inline method of droplet and particle size distribution analysis in dilute disperse systems

In the present article a measurement method of particle size distributions (PSD) in industrial installations which use a dispersed phase of low concentration (like spray dryers or spray scrubbers) is introduced. A new type of inline-measurement system has been developed and designed to work in spray drying conditions. A standard digital camera is used to record shadows of flowing particles inside the spray drying chamber. Collected images were analyzed by a newly developed software which recognizes particles only in the focus area and eliminates several types of artifacts. The constructed prototype of the PSD inline-analyzer was installed and used to monitor large laboratory scale spray dryer. All data collected by the designed system during the spray drying experiments were compared with data measured with an offline reference system to show accuracy of the new measurement technique.