Measurement Uncertainty Limit of a Video Probe in Coordinate Metrology

Seung-Woo Kim, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology - Submitted by Prof. P. A. McKeown (1)

Received on January 9,1996

Abstract

Video probes are increasingly used in recent industrial applications of coordinate measuring machines, since they are found more efficient than conventional ball-tip probes especially in inspecting small-sized features of complex shapes. However, no thorough investigation has yet been accomplished to comprehend the measuring capabilities of video probes. In this paper. analytical and experimental approaches are made to explore how the measuring un- certainty limit of video probes is determined by major design parameters related to imaging optics, coherence of illumination. and edge detection algorithms using a CCD array. Finally an exemplary optimal design is discussed to demonstrate that an ultraprecision measurement of 0.01 pm uncertainty can be practically achieved.

Keywords : CMM, Inspection, Vision

1. Introduction

In recent years, with increasing demand on non- contact coordinate metrology, video probes have been gaining popularity over conventional ball-tip probes that have been used long since the ad- vent of coordinate measuring machines. Video probes are in fact opto-electronical combination of microscopes with computer vision being capa- ble of providing many advantages: multiple data points within a given field of view can be gathered at a time and small-sized objects of complicated shapes can be treated efficiently. However, no much work has been done yet to un- derstand the measuring capabilities of video probes, thus a great deal of misinterpretation arises in assessing their measurement perfor- mances.

This paper in the first place synopsizes how the im- age profile of an edge is formed by adopting wave optics theory. Then the measurement un- certainty limit of a video probe is scrutinized by considering practical design parameters related to objective lens, coherence of illumination, defo- cusing, aliasing of a CCD array, and edge detec- tion algorithms. Finally, a pragmatical uncertainty model is proposed with which the performances of a video prove may be evaluated systematically through a series of well-prepared calibration pro- cedures.

2. Half-Intensity Point Criterion

A video probe comprises three major opto- electronical elements as illustrated in Figure 1: a light source with a condensing lens for illumina- tion, an objective imaging lens of appropriate

Annals of the ClRP Vol. 45/1/1996

magnification, and a CCD array for discrete sam- pling of the image of object. Fundamental func- tion of the video probe in coordinate metrology is to detect the edges constituting the boundary of a geometrical feature of interest. For simplicity of analysis, an edge may be defined to possess the relative amplitude distribution in the object plane of x’-coordinate such as

1, forx’rO 0, for X‘LO

t(X’) =

When the illumination is completely incoherent and the objective lens is free from aberrations, analysis of imaging optics becomes linear and stationary. Then the intensity profile of the edge in the image plane of x-coordinate can be simply determined by the convolution integral of 1’.21

I (x) = I“ t2(x’) h2(x -XI) dx‘ , -m

The function h2(x) in the above is referred to as the point spread function. If the objective lens is con- sidered as a one-dimensional slit aperture of width a, then h(x] can be readily obtained as

h(x) = a sinc[ka(x’/z’ -x/z)]

where sinc(x]=sin(x]/x: k is the wavenumber, i.e., k = 2 d , L is the wavelength: z’ is the object dis- tance: z is the image distance. By substituting Eq.(3), the intensity profile of Eq.(2) can be worked out in a normalized form as121

(41

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where r=kax/(l+m)f , m i s the magnification, i.e., m=z/z’, and Si(r] i s defined as the finite integral of the sinc(r) from 0 to r. Eq.(4) may be graphically plotted as in Figure 2 where it is seen that the edge projected in the image plane is no more a step function of its original form of Eq.(l). In fact the image of edge turns out to be a smoothened profile filtered by the objective lens whose imag- ing characteristics are represented by two optical parameters: one is the magnification m and the other f/a called the f-number of the objective lens.

The true edge is located at x=O. hence its intensity level is calculated as

I I(0) = ; + 0 + 0 = 1 . (51

This result demonstrates that the intensity level at the true edge becomes always 1/2, regardless of the parameters m and f/a. This consequence is referred to as the half-intensity point criterion for

. the tcue edge and still valid even for the case of two-dimensional analysis considering the objec- tive lens as a circular aperture, whose description is omitted here due to space limit.

3. Coherence Effects

The half-intensity point criterion never be strictly authentic in actual instances. This is because the illumination can not be perfectly incoherent but yields some coherence effects that give rise to edge ringing and shifting. The coherence pa- rameter S may be defined as the ratio of the nu- merical aperture of the objective lens to that of the condensing lens, whose value varies from 0 to infinite. The illumination is perfectly coherent in case S O . completely incoherent in case S=m. and partially coherent otherwise. The perfectly coher- ent illumination is an extreme case, in which the image profile of edge is determined by I2.’l

(61

Substituting h(x] of Eq.(3) produces the image pro- file of edge such as

which is plotted in Figure 3 showing edge ringing and shifting. In this perfectly coherent case, the intensity level at the true edge is obtained as I(0) =1/4, not 1 /2. And the half-intensity point is shifted by the amount

Ax = 0.21 2( 1 + m)If/a. (8 )

An extended analysis demonstrated that for par- tially coherent case, i.e., 0 < S < 00 , the shift lies within the limit of Eq.(8). ‘‘I Therefore. the uncer- tainty of the half-intensity point may be prescribed in terms of the object coordinate x’ such as

Thus it is noted that the uncertainty of coherence effects is affected only by the f-number f/a. not by the magnification m, and it reduces with decreas- ing f/a. For an example with f/a=l .O and ic4 .5 prn (this value is taken as the mean wavelength when a white light of wide spectrum is used for illumina- tion.], U, reaches approximately 0.1 pm. It i s how- ever worth noting that this uncertainty is not random but systematic so it can be well compen- sated for by a calibration process which will be described later.

Defocus also affects the intensity profile. It is therefore necessary to assure that the edge is in focus before measurement by using a suitable autofocusing method. An exhaustive analysis done in Ref.4 showed that the intensity profile is further smoothened with increasing amount of de- focus. However, as illustrated in Figure 4, the in- tensity level at the true edge remains the same as 1/2 not being affected by defocus. It is therefore concluded that the half-intensity point criterion is acceptable and the uncertainty U, of Eq.(9) is still valid if defocus is constrained within a few wave- lengths.

4. Edge Detection by CCD Array

The image of edge is discretely sampled by using a CCD array. By using Taylor‘s series expansion, the intensity profile of Eq.(4] may be converted into the polynomial of

1 2 5 /(r) = 1 + ir - k r 3 + -r - ... 2 2251

Neglecting the higher order terms the above equation may be linearly approximated as I(r) z 1/2+(1/x)r. It can be then seen that the intensity variation around the half-intensity point takes place mainly in the range of 4 2 < r < x/7, or in terms of image coordinate x. -( 1 /4) (1 +m])cf/a c x < ( 1 /4) ( 1 +m))cf/a. Therefore. the total span of inten- sity voriation is obtained as

Let p be the pitch of pixel of the CCD array, then the total number of pixels N within Ax i s calculated as N = Ax/p z (0.5rn)cf/a)/p. Since the CCD array is a discrete device, i t becomes important to avoid aliasing effects. The cutoff spatial frequency of the objective lens is given by f,=l /()crnf/a),[’I while that of the CCD array is f,=l /p. Therefore, accord- ing to the Nyquist criterion. the condition f, 1 2f, should be satisfied. i.e.. rn 2 2p/()cf/a). Substituting this into Eq.(l 1 ) gives N 2 1. This result infers that if at least one pixel is used to sample the intensity variation, no aliasing occurs when the illumination is incoherent, regardless of m and f/a. In practice, however. it is necessary to take N to be sufficiently larger than 1 since there exists some degree of co- herence. Furthermore, if N=l, the uncertainty of sampling in the object coordinate x’ is given as Us = Ax’ = Ax/m P 0.5A,f/a , which in fact turns out to be larger than the uncertainty of coherence

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derived in Eq.(9). It i s therefore inevitable to in- crease N to reduce the uncertainty U, , and this can be done by adjusting the magnification m following the relation of

random nature but it can be well eliminated by suppressing undesirable disturbances of electrical noise.

5. Calibration and Discussion rn = Np/(O.5Lf/a).

Once the intensity profile of edge is obtained over a sufficient number of pixels, an appropriate edge detection algorithm is needed which can provide reliable subpixeling aftereffects to improve the un- certainty Us. According to the half-intensity point criterion. the location of true edge may be simply found by fitting the sampled intensities into the polynomial form of Eq.( 10). This thresholding method is however found not immune to electri- cal noises encountered in data sampling. In this case the Gaussian derivative operator is found more effective whose mathematical expression is given as 151

This operator basically detects the steepest slope of the original profile of intensity when convoluted as illustrated in Figure 5. Since the theoretical pro- file of Eq.(4) satisfies the condition ~ 3 ~ l ( O ) / a X ~ = 0, so the slope at the true edge has a maximum and thus this inflection point can be readily localized. For coherent illumination, this inflection point method yields error but its magnitude is approxi- mately the same as the uncertainty U, of Eq.(9).'" Edge detection with the Gaussian operator pro- vides an advantage of eliminating electrical noises by adjusting the parameters of p and 0 in Eq.[13). Now the uncertainty of edge detection may be specified from Eq.[l 1) as

Us = k& = ksrnAx z kS(0.57cf/a) (141

where k, is a constant parameter that turns out to be less than 1 owing to the ability of subpixeling of the edge detection algorithm.

Now the total uncertainty limit of a video probe may be deduced by combining Eq.(9) and Eq.( 14) such as

U = uM/a s U, + Us = 0.7M/a (1 51

where the parameter u i s defined as the uncer- tainty constant. This concludes that the total un- certainty U i s mainly determined by the wavelength and the f-number, not being signifi- cantly influenced by the magnification m. For ex- ample, when ). = 0.5 pm and f/a = 1.0, the total uncertainty limit U i s calculated as 0.35 pm. This specific value, however, simply indicates the maximum limit that is theoretically specified for the worst case. In practice, the achievable ac- tual uncertainty can be improved much less the limit since the coherence error U, is systematic and can be compensated for through appropri- ate calibration. In addition, the error U, is of

The apparatus set up for calibration is shown in Figure 6 . The objective lens is of X20 magnification with 1.25 f/a-value. The light source is a Halogen lamp that is guided by a cable of optical fiber and passed through a diffusing plate to increase incoherence in illumination. The object tested is a ceramic gage block whose width measures 125.00 pm with 0.01 pm uncertainty as precali- brated by using the national length standard laser interferometer. The CCD array has 512x512 pixels with 14.0 pm pitch. Prior to measurement, actual magnification was precalibrated by using a stan- dard specimen.

The intensity profile of edge measured from the gage block is shown in Figure 7. where the slope signal G(x) processed by the Gaussian operator of Eq.(13) is also added. It i s clearly seen that the true edge point detected by the steepest slope nearly coincides with the point of half-intensity, demonstrating that the half-point intensity crite- rion i s correct. The width of the gage block was repeatedly measured and its results are summa- rized in Table 1: the systematic offset error was 0.02 pm and the dispersion of f3a was 0.06 pm. So the total uncertainty error was 0.08 pm. Therefore. the uncertainty constant u defined in the model of Eq.(15) turns out to be about 0.2.

Another test was conducted with a file hole that was fabricated for the connection of optical fi- bers. The diameter of the hole was measured by processing 40 data points sampled along the hole periphery. Table 2 summarizes the results of 10 consecutive measurements. This time the f3a dis- persion was 0.01 pm, which is much less that of gage block. This improvement was owing to the averaging effects 40 data in determining the di- ameter. In this case .the uncertainty constant u defined in the model of Eq.(l5) turns out to be about 0.02.

6. Conclusion

The measurement uncertainty of a video probe has been scrutinized by considering practical de- sign parameters related to objective lens, coher- ence of illumination, defocus, aliasing of a CCD array, and edge detection algorithms. As results, a pragmatical uncertainty model has been pro- posed with the following conclusions: (a) The uncertainty of a video probe is mainly af- fected by coherence effect of illumination and also by edge detection algorithms. On the other hand, defocus and aliasing produce no significant influences. (b] Among many design parameters of the optical system, the f/a value is the most important to be selected to be as much small as possible to re- duce the uncertainty.

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Increasing D e f x u s

(c) Finally, an ultraprecision measurement of 0.01 pm uncertainty can be practically achieved.

References

[ 1 ) Thomson. B.J.. et al. 1989, Phvsical ODtics Note- book: Tutorials in Fourier ODtics, SPlE Optical Engi- neering Press. (2) Considine, P.S., 1966, Effects of Coherence on Imaging Systems. J. Opt. SOC. Am., 56:lOOl-1009. (3) Nyyssonen, D., 1982, Theory of Optical Edge De- tection and Imaging of Thick Layers, J. Opt. SOC. Am.. 72:1425-1436. (4) Park, M.C.. 1996, Edge Detection and Calibra- tion of High Magnification Optical Microscope Vi- sion Systems for Coordinate Metrology. MSc Thesis, Korea Advanced Inst. of Science 8 Technology. (5) Canny, J.. 1986. A Computational Approach to Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-698.

candrnrlna CCD array - lam

Figure 1 : Opto-electronical configuration of a video probe.

true edge of half-intensity

0 A

Figure 2: The ideal image profile of an edge.

I(X1

far caharrnf

0

Figure 3: The image profile of an edge with coherent illumination.

I -I u I J n o 0 5 10

X( rm 1 Figure 4: Variation of edge profile with

increasing defocus.

I I Icol I I

. Figure 5: The Gaussian derivative operator.

Field Image stop CCD Sensor

Back illuminator

Figure 6: The apparatus for calibration.

Distance

Figure 7: The intensity profile of a gage block.

Table 2: Test results of a fine hole.

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