measurement of the size and spacing of standard wire...

Measurement of the Size and Spacing of Sieves Using an Image Processing & Wavelt Transform

15

Measurement of the Size and Spacing of StandardWire Sieves Using an Image Processing System

and Wavelet Transform

K.P. CHAUDHARY*, CHANDRA SHAKHER1 and SHASHI K. SINGH2National Physical Laboratory (NPLI)

Council of Scientific and Industrial Research (CSIR)Dr. K.S. Krishnan Marg, New Delhi-110 012

1Indian Institute of Technology , New Delhi2IIMT College of Engineering, Greater Noida

*e-mail: [email protected]

[Received: 22.09.2010 ; Revised: 12.11.2010 ; Accepted: 15.11.2010]

AbstractA new methodology using image processing and wavelet transform is investigated to measure the sizeof the wire sieves and their spacing. Wire sieves are used in pharmaceuticals/chemical industries forfiltering the grains of chemical powder. Experimental results show measurement that the diameter andspacing of the wire in the sieves can be measured with the accuracy of 1um and uncertainty of 2 μm at95% confidence level. The method is found suitable for detecting any missing wire or defect likebending or kink in the wire. In this techniques, wavelet transform (Symlet wavelet) analyses the imageof sieve in such a way that the discontinuity (cracks, defects, nonuniformity) can be detected moreprecisely and the spacing/ wire diameter can be measured accurately. This method provides quite canfast and accurate results with ease, in comparison to the existing methods.

© Metrology Society of India, All rights reserved 2011.

1. Indroduction

Standard wire sieves are immensly importantindustrial product, particularly in pharmaceuticals/chemical industries, where the size of the particle ofchemical powder matters. These are used for filteringthe particular size of the grain of the powder. Themeasurement of standard wire sieves involves thedetermination of three average size i.e. pitch, aperturewidth and wire diameter, which is normally a tedious,time consuming and offline process.

As per ASTM standard E-11 and IS standard 460,wire mesh sieves are calibrated for their diameter, holediameter, frame and skirt diameter are measured [1].Wire diameter and spacing are measured with acalibrated optical projector while spacing is measuredindirectly. In this case, the standard uncertainty ofthe measuring scale is around 10 um over 100mm oftotal travel giving a standard relative pitch uncertaintyof 0.01% [2-4].

The traditional methods do not give the completepicture of the sieve measurement. Recently, theon-line measurement of the size of the standard sieveshas been described by using optical Fourier transform

MAPAN - Journal of Metrology Society of India, Vol. 26, No. 1, 2011; pp. 15-27ORIGINAL ARTICLE

,

K.P. Chaudhary, Chandra Shakher and Shashi K. Singh

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[5]. This system has several advantages such asoperating efficiently and precisely online and in realtime in comparison to other traditional methods. Themeasuring system consists of a laser, an opticalFourier transform system, a charge coupled device(CCD) camera, image acquisition system and apersonal computer [6,7]. The system is capable ofmeasuring averaged size and percentage of apertureexceeding the specified size of standard wire sieve intwo dimensional directions.

Fourier method, however do not preserve thedetails of the test objects, e.g. position of holes, cracksand discontinuity in pattern etc. It is basically due tothe reason that the Fourier method expands theoriginal functions in terms of orthogonal basisfunctions of the sine and cosine waves of infiniteduration, and while transforming to the frequencydomain, the time (space) information is lost [8-10]. Toavoid the problem in the analysis due to Fourieranalysis we have investigated a method ofmeasurement of the size of wire of the standard wiresieve using an image processing system and wavelettransform.

In this paper, we present the analysis of the wiresieve i.e. the measurement of aperture and spacing inhorizontal and vertical directions. Experimentalresults obtained from image processing and wavelettransform are compared with the results obtained byoptical profile projector and it is observed that theresults obtained by image processing and wavelettransform are more accurate and precise. Measuringwire diameter optically is difficult because ofdiffraction effect at the edge of the wire. Diameter variesquite widely depending on the type of lightening(direction, coherence) and the quality of optics. Wehave studied several techniques using back lighting,front lighting, diffused and collimated lighting anddifferent optical systems. For these measurements,both the stage micrometer and the calibrated wire wereused to compare the system. It is seen that these resultsagree well within 2 μm. Having no clear theoreticalreasons to choose one method over the other, we havetaken this spread as the uncertainty of the opticalmethods. Taking the value of 2 μm as a half width ofrectangular distribution, we have estimated thestandard uncertainty to ± 1 μm.

2. Wavelet and Description of Filtering

Wavelets are new families of orthonormal basisfunctions, which do not need to have of infinite duration.

It has oscillating wavelike characteristic and ability toallow simultaneous time (space and frequency analysiswith a flexible mathematical foundation [11]. Whenwavelet decomposition is dilated, it accesses lowerfrequency information, and when contracted, it accesseshigher frequency information. The fundamental ideabehind wavelets is to analyze according to scale. Waveletalgorithm processes data at different scale or resolutions.If we look at a signal with a large "window", we wouldnotice gross features. Similarly, if we look a signal with asmall window, we would notice small features. Imageprocessing provides the fastest and non contactmeasurement of wire diameter and spacing of wire sieve.It is a real time measurement and computationally fasttechnique. In the case of wire sieve, there is a low frequencyalong wave direction and high frequency event acrossweaving direction. It contains holes, cracks, and diameterirregularities. Also the spacing between wire (weft andwrap) are very small so it should be expanded so that thespacing, wire diameter and the defects or discontinuitiescan be detected. It needs multiresolution analysis to givegood time (space) resolution and poor frequency resolutionat high frequencies and good frequency resolution andpoor time (space) resolution at low frequencies.

In order to isolate discontinuities, we have very shortbasis functions. At the same time, in order to obtain detailsfrequency analysis, we have some very long basisfunctions. The conventional filters and image processingtools lack these properties. Fourier methods however donot preserve details of the object (in practice object usuallycontain holes, cracks, shadow in the image field), whichalso get effected by filtering. It is basically due to the reasonthat the Fourier transform expands the original functionin terms of orthogonal basis function of sine and cosine ofinfinite duration. When looking at Fourier transform of asignal, it is impossible to tell when a particular event tookplace. Also, the size of analyzing window cannot bevaried. Wavelets are little wave whose average value iszero. Wavelets are the new families of orthonormal basisfunctions, which do not need to have infinite duration.This is a "small wave", which has its energy concentratedin time to give a tool for the analyses of the transient nonstationary or time-varying, phenomena. It still hasoscillating wavelike characteristic but also has the abilityto allow simultaneously time and frequency analysis witha flexible mathematical foundation.


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The fundamental idea behind wavelets is to analyzeaccording to scale. Wavelets are the functions that satisfycertain mathematical requirements and are used inrepresenting data or other functions. Wavelet algorithmsprocess data at different scale or resolution. There aredifferent types of wavelets ( Haar, Daubechies, Coifletsand Symlets).The image is loaded in wavelet tool box inMatlab platform. The Program made for Symlet waveletin graphical user interface of Matlab Wavelet tool box,the amplitude calculation function is able to amplitudedistribution against pixel coordinate.

We have used Symlet Wavelet made by I, Daubechies[5,12]. It is quite effective in boundaries of image in texture.The wavelet decomposition function decomposes theimage into different level of intensities, which makes ableto compute the wire spacing and wire diameter in bothwarp and weft directions in the sieve. It detects defectivesieve. Daubechies proposes modifications of her waveletsuch that their symmetry can be increased while retainingsimplicity. The idea consists of reusing the function m0introduced in the dbN11 considering the |m0(ω)|2 asfunctionω of z= eiω. Then we can factor ω in several different

ways in the form of 1( ) ( ) ( )z zz

ω = U U

The roots of ω with modulus not equal to 1 go in pairs. If

one is, 11

1 and z

z is also a root. By selecting U such that

the modulus of all its root is strictly less than 1, we buildDaubechies wavelets dbN. The U filter is minimum phasefilter. By making another choice, we obtain moresymmetrical filters; these are Symlets. Symlets arecompactly supported wavelets with least asymmetry andhighest number of vanishing moments for a given supportwidth. Associated scaling filters are near linear-phasefilters having support width 2N-1 and filters length 2NDaubechies introduced scaling function

= −∑ n( ) 2 (2 )n

x h x nφ φ for wavelet dbN ( are the

coefficients associated to a 'standard' multiresolutionanalysis and the corresponding orthonormal basis).However, more symmetric wavelet filters make easierto deal with the boundaries of the image [13-15].

Symmetric filters are linear phase filters. Moreprecisely, a filter with filter coefficients an is calledlinear phase if the phase of the function

ξ=∑ -inn( ) n

a a eξ is a linear function of ξ , i.e., if for some

l∈ Z , a(ξ) = e-i/ξ|a(ξ)|. This means that an are symmetricaround l, an=a2/-n .

The phase introduced by I. Daubechies [5] for Symletwavelet is given below

∞− −

=

= -2j-1 2 j 20 0j 1

[ ( 2 ) (2 ) ], ξ ξm m (1)

where = ∑ -in0 nn

1 ( ) 2

ξξm h e

The phase 1Φ of the Symlet wavelet is closer to linear

phase than that of dbN11,∞

=

= -j0j 1

( ) (2 )mξ ξΦ .

The continuous wavelet transform decomposes afunction f(x) at several scales. This decomposition isperformed by convolving the function with the dilationsand translations of a special function ψ called motherwavelet:

∞+

−∞

−= Ψ ∈ ∈∫

1( , ) ( ) * ( ) , ,x bW a b f x dx a R b Raa (2)

The function f(x) can then be recovered as

∞

Ψ−∞

−= Ψ∫ 2

1( ) ( , ) ( ) ,x b dadbf x W a baC a (3)

where, Cψ

is a constant depending only on ψ .

The discrete version of this transform results fromthe sampling of the parameter space (a, b). One of themost well known discrete wavelet transform algorithmsis multi-resolution analysis. In the multiresolution

=1 0 0 0 0( ) ( /2) ( /4) ( /8) ( /16) .. ξ ξ ξ ξ ξΦ m m m m


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analysis, a sequence of embedded function subspaces.........⊂Vi ⊂ Vi-1 ⊂ ....V0 is spanned by the dilationsΦ(2-i x) of a function Φ (x) called the scaling function,

such that if f (x) ∈Vi-1 , then ∈x ( )2

f Vi. Therefore, the

sampled parameters are a = 2-i, and b = k 2-i. A continuoussignal f (x) is projected on each subspace by means of thescalar product with the scaling function dilated andtranslated to integer positions. We can see the initialdiscrete signal C0 (k) as the projections of f(x) on V0 ,

∞

∞

= 〈 〉 = ∫0-

( ) ( ), ( - ) ( ) ( - ) d .C K f x x k f x x k xΦ Φ (4)

The projection on a subspace Vi ,

=i i1 x( ) ( ), ( - )

22C K f x kΦ

is then an appropriation of C0 at scale or resolution i.The greater the i, the coarser the approximation will be.For scaling function fulfilling certain conditions, thedifference between two successive approximationsCi-1 and Ci is a discrete signal belonging to new spaceWi ⊂Vi-1, which is the orthogonal complement of Vi inVi-1 ; that is Vi-1 = Vi⊕Wi. These subspaces Wi aresimilarly spanned by the dilations and translationsof a mother wavelet function ψ .

Note that Eq. (3 ) includes a decimation; the numberof coefficients at level i is half the number of coefficient atlevel i-1. Thus the full decomposition of a signal C0 of Nsamples is a set of N-1 wavelet coefficient plus oneapproximation coefficient. Because of orthogonalityproperty, there is no redundancy among the coefficientsthus; this is very suitable for compression purposes.However, this decomposition scheme lags an importantproperty for image analysis, namely, spatial invariance.Therefore, we have applied a decomposition algorithm 'atrous', which does not produce a non-redundantrepresentation [15].

=i i i1( ) ( ), ( - ) ,2 2

xC K f x kΦ (5)

However, we need some expression that enables thecalculation of details and approximation coefficient of ascale 'i' from those of the proceeding scale, because we donot know the continuous function f (x) but only its discreteapproximation C0. In the 'a trous' algorithm, despite ofthe non-orthogonality of the sub-spaces Vi and Wi, westill have a sequence of embedded sub-spaces. As thetranslation of (x) span V0 and φ (x/2) ∈ V1 ⊂ V0,

= −∑1 ( ) ( ) ( ),2 2 nxΦ h n Φ x n (6)

where h(n) is the kernel filter associated with thescaling function φ(x). Then from Eq. (5-6), we see thatthe approximation coefficient at scale i+1 arecalculated by means of the discrete convolution ofcoefficient at scale i with a filter h.

+ = +∑ ii 1 in

( ) ( ) ( 2 ).C k h n C k n (7)

The wavelet or deleted coefficients Wi are computed asthe difference between two consecutive scales

=i i i-1( ) ( ) - ( )W k C k C k (8)

The reconstruction stage of this algorithm is simplythe sum of all the coefficient and the coarsestapproximation.

=

= +∑0 N ii 1

( ) ( ) ( )N

C k C k W k (9)

In this paper, we use a 2D version of this algorithm.The matrix of initial values Co (k, l) is the image of sieveand the number of levels is N = 4. The scaling function in2D case has been chosen as the separable function[16-20]

[ ]

[ ]

( , ) ( ) ( ), which is

1- if -1,1 ( )

0 if -1,1

Ф x y Ф x Ф y

x xФ x

x

= ×

⎧ ∈⎪

= ⎨⎪ ∉⎩

(10)


19

3. Experimental

The system for the image acquisition is developedaround National instruments image Acquisition Card,PCI 1408 (Fig. 1), installed on the Pentium III. This cardcan acquire the monochrome images with a maximumtransfer rate of 132 Mbytes/sec on 32 bit wide bus. Imagegrabbing window is configured to acquire the image size640 x 480 and pixel depth of 8 bits. The image istransferred from the camera to the computer at frame rateof 30 frames per second. To achieve the optimum results,the sieve under study was illuminated from back andfront. The image was recorded by Pulnix TMC-76 CCDcamera and image acquisition system. The image wasstored in the two dimensional array [21-28].

Discrete wavelet transform is used to analyse theacquired image. The 2D wavelet transform decomposesthe image in horizontal, vertical, and diagonalcomponents at different level of intensities containingtexture information content. The processing is madethrough Labview 5.1 software and Matlab 5.2. Two sieveswere analysed by using machine vision system describedabove and Symlet wavelet transform.

Figure 2(a) shows the original image of sieve 1. Theseoriginal images were processed by using filter functionand filtering scheme as described in the last section.Figures 2(b) and 2(c) give the vertical component or weftand horizontal component or warp of the two sieves,

respectively. A flow diagram of the analysis andmeasurement of wire Sieves is shown in Fig. 3

4. Measurements

Measurement of sieve1 parameters (diameter andspacing per unit cm per sieve for sieve1): NPL ID. no.98 DM 284. Diameters of horizontal-wires withinconsecutive unit of 1.0 cm along horizontal directionand mean/coefficient of variation along unit area andunit wire are given below. The whole image is dividedinto 8 parts and each part, the measurement of pixelvalue (by 'pix val' command and necessary functionsin program) is done consecutively. So here, unitdescribes parts of the image or portion in image.Diameter of the each wires is measured in eachportion, separately [29-31].

5. Conclusion

Optical method using wavelet transform is usedto analyze the sieve dimension. Its results have beencompared with the standard wire sieve calibrated atNPL and it was found that the results lie within theuncertainty limit. Intensity of light on the surface isimportant parameter in this study [32-35].

Acknowledgment

The author is thankful to the Director, NationalPhysical Laboratory for his support and permissionto publish this paper.

PC with Image Grabber

Card (IMAQ1408) & the software

Wire sieve

Fig. 1. The schematic of experimental set up

Lens CCD Camera


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Fig. 2 a. Original image of sieve 1 NPL ID.No. 98 DM 284, (b) Horizontal component analyzed by Wavelet of Sieve 1,(c) Vertical component analysed by Wavelet of Sieve 1

Fig. 3. Flow diagram of the analysis and measurement of wire Sieves

Image Grab

Decomposition of intensity Component by wavelet transform

Decomposition in warp component

Decomposition in weft component

Resolve and decimate the image data in the

level of ½, ¼, etc.

Resolve and decimate the image data in the

level of ½,¼, etc.

Reconstruction of the image

Display of the image in different levels

Measurement of diameter and spacing in warp and weft

a b

c


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Table 1Spanned and scaled by wavelet transform through image processing

Sieve Dia.in Dia.in Dia.in Dia.in Dia.in Dia.in Dia.in Dia.inNo. 1st unit 2nd unit 3rd unit 4th unit 5th unit 6th unit 7th unit 8th unit

1 0.145 0.145 0.148 0.145 0.147 0.148 0.148 0.1490.145 0.146 0.147 0.145 0.148 0.145 0.147 0.1480.146 0.147 0.147 0.146 0.145 0.146 0.146 0.147

2 0.146 0.148 0.148 0.148 0.144 0.146 0.147 0.1470146 0.147 0.148 0.147 0.144 0.145 0.145 0.1450.148 0.149 0.148 0.146 0.144 0.146 0.148 0.147

3 0.144 0.149 0.143 0.147 0.145 0.147 0.148 0.1470.148 0.147 0.145 0.148 0.145 0.145 0.145 0.1460.148 0.144 0.146 0.146 0.147 0.146 0.147 0.146

4 0.144 0.146 0.147 0.148 0.149 0.145 0.146 0.1470.147 0.147 0.147 0.147 0.148 0.146 0.147 0.1460.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

5 0.149 0.145 0.146 0.146 0.146 0.146 0.145 0.1480.147 0.145 0.145 0.146 0.147 0.146 0.146 0.1470.146 0.146 0.146 0.146 0.147 0.147 0.147 0.146

6 0.145 0.147 0.147 0.146 0.146 0.145 0.145 0.1450.146 0.145 0.147 0.146 0.145 0.145 0.146 0.1450.146 0.146 0.146 0.146 0.146 0.145 0.146 0.146

7 0.147 0.147 0.147 0.145 0.147 0.147 0.146 0.1460.145 0.148 0.148 0.145 0.146 0.146 0.145 0.1470.146 0.146 0.146 0.144 0.146 0.146 0.146 0.146

8. 0.144 0.145 0.145 0.145 0.145 0.144 0.146 0.1450.143 0.144 0.146 0.147 0.145 0.145 0.145 0.1470.144 0.146 0.146 0.146 0.146 0.146 0.147 0.146

9 0.148 0.148 0.148 0.148 0.148 0.15 0.152 0.1500.149 0.148 0.147 0.149 0.149 0.151 0.151 0.1480.146 0.146 0.146 0.146 0.146 0.149 0.146 0.147

10 0.145 0.147 0.146 0.147 0.149 0.147 0.146 0.1480.147 0.145 0.145 0.149 0.148 0.145 0.145 0.1450.146 0.147 0.146 0.146 0.146 0.146 0.146 0.146

11 0.145 0.145 0.145 0.145 0.145 0.145 0.145 0.1460.147 0.148 0.145 0.147 0.147 0.147 0.145 0.1460.146 0.146 0.146 0.146 0.146 0.147 0.145 0.146

12 0.146 0.147 0.148 0.145 0.145 0.145 0.146 0.1480.149 0.145 0.145 0.147 0.147 0.149 0.145 0.1480.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

13 0.148 0.151 0.148 0.148 0.149 0.149 0.148 0.1470.15 0.149 0.147 0.148 0.147 0.147 0.148 0.1470.146 0.148 0.146 0.146 0.146 0.146 0.146 0.146

Contd....


22

14 0.145 0.146 0.146 0.145 0.147 0.145 0.148 0.1450.146 0.147 0.146 0.149 0.145 0.145 0.147 0.1450.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

15 0.149 0.145 0.148 0.147 0.145 0.145 0.148 0.1450.145 0.147 0.147 0.147 0.145 0.147 0.145 0.1480.146 0.146 0.146 0.146 0.145 0.146 0.146 0.146

16 0.145 0.145 0.145 0.145 0.145 0.145 0.145 0.1450.148 0.149 0.147 0.149 0.148 0.147 0.149 0.1490.146 0.146 0.146 0.146 0.146 0.146 0.146 0.147

17 0.146 0.148 0.146 0.148 0.148 0.149 0.149 0.1480.145 0.148 0.147 0.145 0.145 0.145 0.145 0.1470.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

18 0.149 0.147 0.147 0.148 0.145 0.145 0.145 0.1470.147 0.145 0.145 0.149 0.145 0.145 0.146 0.1470.146 0.146 0.146 0.146 0.144 0.146 0.146 0.146

19 0.145 0.145 0.145 0.145 0.145 0.145 0.145 0.1450.145 0.145 0.145 0.145 0.145 0.145 0.145 0.1450.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

20 0.149 0.148 0.147 0.145 0.146 0.145 0.145 0.1450.145 0.145 0.145 0.147 0.147 0.146 0.145 0.1450.146 0.146 0.146 0.146 0.146 0.146 0.146 0.145

Mean 0.1463 0.1464 0.1462 0.1464 0.1461 0.1461 0.1464 0.1463

Vertical -Thread Dia.in Dia.in Dia.in Dia.in Dia.in Dia.in Dia.in Dia.inNo. 1st unit 2nd unit 3rd unit 4th unit 5th unit 6th unit 7th unit 8th unit

1 0.148 0.148 0.15 0.151 0.148 0.148 0.148 0.1480.144 0.146 0.151 0.144 0.15 0.144 0.147 0.1460.146 0.146 0.15 0.146 0.149 0.146 0.146 0.146

2 0.151 0.148 0.148 0.148 0.149 0.148 0.148 0.1490.15 0.148 0.149 0.148 0.148 0.144 0.147 0.1490.149 0.147 0.146 0.146 0.146 0.146 0.147 0.148

3 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1450.144 0.148 0.146 0.149 0.148 0.149 0.147 0.1440.146 0.146 0.146 0.146 0.146 0.149 0.147 0.146

4 0.148 0.148 0.148 0.149 0.148 0.148 0.148 0.1480.145 0.147 0.148 0.144 0.148 0.148 0.147 0.1480.146 0.146 0.146 0.146 0.146 0.149 0.146 0.148

5 0.148 0.148 0.147 0.147 0.148 0.148 0.148 0.1460.148 0.148 0.147 0.147 0.148 0.147 0.148 0.1460.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

6 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.148 0.147 0.147 0.146 0.145 0.149 0.149 0.1440.146 0.146 0.146 0.146 0.146 0.146 0.146 0.148

Contd....


23

7 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.146 0.147 0.148 0.147 0.147 0.147 0.146 0.1450.147 0.146 0.149 0.146 0.146 0.146 0.146 0.146

8 0.148 0.148 0.148 0.148 0.148 0.148 0.15 0.1520.147 0.146 0.147 0.148 0.147 0.149 0.151 0.1490.147 0.147 0.146 0.146 0.146 0.146 0.146 0.146

9 0.148 0.148 0.147 0.145 0.147 0.148 0.147 0.1480.149 0.148 0.144 0.144 0.146 0.144 0.147 0.1480.146 0.148 0.146 0.146 0.146 0.146 0.147 0.146

10 0.151 0.148 0.147 0.148 0.148 0.148 0.148 0.1490.154 0.148 0.147 0.145 0.144 0.149 0.149 0.1490.15 0.146 0.146 0.146 0.146 0.149 0.146 0.146

11 0.148 0.149 0.147 0.148 0.148 0.148 0.148 0.1460.149 0.149 0.144 0.146 0.146 0.144 0.144 0.1460.146 0.146 0.146 0.149 0.146 0.145 0.146 0.146

12 0.148 0.148 0.148 0.147 0.145 0.148 0.148 0.150.146 0.148 0.148 0.147 0.145 0.149 0.148 0.1540.146 0.146 0.148 0.146 0.145 0.149 0.146 0.15

13 0.148 0.146 0.144 0.147 0.146 0.148 0.148 0.1480.154 0.144 0.144 0.147 0.144 0.144 0.149 0.1490.15 0.146 0.146 0.146 0.146 0.146 0.146 0.149

14 0.148 0.148 0.148 0.148 0.147 0.148 0.148 0.1480.148 0.148 0.144 0.147 0.144 0.148 0.144 0.1440.148 0.146 0.146 0.146 0.146 0.146 0.146 0.146

15 0.148 0.149 0.148 0.148 0.148 0.148 0.144 0.1480.149 0.146 0.146 0.147 0.145 0.144 0.144 0.1440.149 0.146 0.146 0.146 0.146 0.146 0.146 0.146

16 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.15 0.149 0.148 0.147 0.146 0.148 0.147 0.1460.146 0.146 0.147 0.147 0.146 0.146 0.146 0.146

17 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.144 0.146 0.147 0.148 0.144 0.148 0.144 0.1440.146 0.146 0.146 0.146 0.147 0.146 0.147 0.146

18 0.15 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.152 0.147 0.148 0.147 0.147 0.145 0.149 0.1480.146 0.147 0.146 0.146 0.147 0.146 0.146 0.146

19 0.148 0.148 0.148 0.15 0.148 0.148 0.148 0.1510.149 0.148 0.148 0.15 0.15 0.15 0.148 0.1590.146 0.146 0.146 0.146 0.146 0.146 0.146 0.146

20 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.1480.148 0.15 0.151 0.154 0.144 0.149 0.144 0.1440.146 0.146 0.146 0.146 0.148 0.146 0.151 0.146

Mean 0.1478 0.1472 0.1471 0.1471 0.1467 0.1471 0.147 0.1473


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Table 2Spacing of wire for sieve1: (h- horizontal, v- vertical)NPL.I.D.no. 98DM284

1st unit 2nd unit 3rd unit 4th unit 5th unit

H V H V H V H V H V0.278 0.285 0.277 0.287 0.278 0.281 0.278 0.284 0.276 0.2800.276 0.286 0.276 0.286 0.279 0.285 0.275 0.284 0.276 0.2800.279 0.285 0.278 0.285 0.279 0.285 0.279 0.285 0.274 0.2850.278 0.284 0.278 0.289 0.275 0.281 0.28 0.289 0.278 0.2850.279 0.289 0.28 0.289 0.275 0.279 0.279 0.289 0.278 0.2890.279 0.286 0.279 0.285 0.279 0.28 0.279 0.285 0.274 0.2890.275 0.283 0.274 0.285 0.274 0.28 0.278 0.285 0.275 0.2840.28 0.284 0.28 0.289 0.28 0.281 0.28 0.289 0.285 0.2840.281 0.288 0.282 0.288 0.282 0.279 0.281 0.288 0.281 0.2890.279 0.285 0.279 0.288 0.28 0.279 0.279 0.288 0.279 0.2890.278 0.287 0.278 0.287 0.278 0.285 0.278 0.287 0.278 0.2870.277 0.291 0.277 0.287 0.274 0.283 0.277 0.28 0.275 0.2910.275 0.285 0.275 0.285 0.275 0.282 0.279 0.285 0.275 0.2850.277 0.286 0.276 0.284 0.274 0.281 0.277 0.285 0.275 0.2810.278 0.287 0.279 0.288 0.278 0.28 0.278 0.28 0.278 0.2820.275 0.285 0.275 0.288 0.277 0.279 0.278 0.28 0.275 0.285Mean0.2777 0.286 0.2777 0.2867 0.2773 0.2811 0.2784 0.2849 0.277 0.2853

RepeatibilityType 'A' uncertaintyWire diameter (mm)0.1450.1450.1460 .1450.1450.1450.1450.1460.1460.1450.145

Mean; 0.1454 mmStd. deviation: 0.000516398 mmStd. uncertainty: 0.1632993 mmDegree of freedom: 9

Type 'B'Temperature during calibration- 22.5 °CLeast count of thermometer - 0.10 °CUncertainty of Thermometer - 0.20 °C

Tabel 3Measurement results of an array of standard wire sieve

Parameter Between Between weft Wncertainty ofwarp (mm) (mm) measurements (μm)

At 9% confidence level

Average apertue 2.0177 2.0361 ± 2.24Maximum aperture 2.0420 2.0420Average wire diameter 0.5268 0.5177 ± 2.74


25

Table 4Uncertainty budget

Source of Limits Probability Sensitivity Uncertainty Degrees of U(y)uncertainty ± Δxi Distribution coefficient contribution freedom XI type A or B cI ui (y) μm (±) vi

factor

u1ccd pixel 1/581= Rectangular 1 1.0 μ m ∞ 1μm

(756X581) 0.172 μ m -Type B- √3

u2 0.2 m 2 Normal 1 0.1 μ m ∞ 0.1μmStandard gaugeblocksu3 0.20C 2 Normal α.L 0.10C ∞ 0.6 nmTemperaturemeasuredu4 2.2X10-6 Rectangular L.Δ t 0.20C-1 ∞ 3 nmThermal Expansion -Type BCoefficient. - /2Optical effects 0.5μ m Rectangular 1 0.29 μ m ∞ 0.3μm

-Type B- √3

Software 0.2 μm Rectangular 1 0.1 μ m ∞ 0.1μm-Type B- √3

Repeatibility 0.16 μm 'A' normal 1 0.16 μ m 9 0.16μmType A

Uc= ± 2.0 μm at 95% Confidence level i.e. at k = 2

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