Sensitivity analysis applied to an improved Fourier-transform profilometry

Optics and Lasers in Engineering 49 (2011) 210–221

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Sensitivity analysis applied to an improved Fourier-transform profilometry Giorgio Busca, Emanuele Zappa n Politecnico di Milano, Dipartimento di Meccanica, via La Masa, 1 20156 Milano, Italy

a r t i c l e in f o

a b s t r a c t

Article history: Received 3 June 2010 Received in revised form 6 September 2010 Accepted 7 September 2010 Available online 16 October 2010

Every estimation concerning model parameters has to deal with uncertainty, but its quantification is a complex task to solve, especially for the identification of the different uncertainty sources that affect the system. In this paper we provide an extensive uncertainty analysis on the calibration model of a Fourier-transform profilometry, studying the effect of the uncertainty parameters estimation on the final measuring result. The methods used for the classification are discrete derivatives and the global sensitivity analysis based on Monte Carlo simulations. The experimental results show that the uncertainty propagation of the system parameters to the output strongly depends on different system setups that may be chosen. This dependency is analysed and interpreted. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Sensitivity analysis Uncertainty analysis Fourier-transform profilometry Calibration

1. Introduction Fringe projection techniques are very popular thanks to their possible application in a lot of fields, such as industrial inspection, manufacturing, computer and robot vision, reverse engineering and medical diagnostics. The main qualities of these profilometry methods are non-contact and full field measurement, low cost and speed in obtaining the 3D information [1–7]. One of the most used techniques is Fourier transform profilometry, which is based on the projection of a grid onto a surface and then viewed from another direction by a camera, which acquires the image [8]. The object topography deforms the fringe pattern: the corresponding image is acquired on the camera sensor plane and then processed to obtain depth information. The depth is extracted, through the Fourier-transform, from the phase difference between the grid projected on a reference plane and the same grid projected on the object surface. In comparison with other fringe techniques, which require more than one image for the 3D measurement, the advantages of the FTP are elaboration speed and need of only one deformed image [9–11]. On the other hand, it needs to resolve the projected grid lines individually, and consequently has a strong requirement on the pixel spatial resolution of the recording device. Moreover, FTP requires frequency domain filtering whose consequence is fine detail reduction and resolution loss [12,13]. Phase-to-depth conversion is possible by means of a suitable formula that depends on the geometric model of the acquisition system, i.e. its geometric parameters and the carrier frequency of

n

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the grid. Obtaining the correct depth distribution is possible only if the geometric parameters are known. Theoretically the task is easy to be achieved, but in practice the estimation of these parameters is quite complex: for example the relative position between the projector and the camera cannot be fixed without a certain degree of uncertainty, but also the evaluation of the carrier frequency of the grid and the reference plane position is affected by errors. A calibration procedure is consequently necessary to overcome these limitations. Calibration methods proposed in literature till now may be distinguished in three categories: model-based, polynomial and neural networks. The model-based methods try to define the correct phase-to-depth conversion formula by means of the indirect determination of the system parameters [14–18]. The polynomial methods otherwise use planes, placed at known positions, that are acquired and then processed to get phase distribution. The polynomial coefficients of the function that best fits the phase-depth data are estimated by a least squares algorithm to produce a calibration map [19–26]. Finally the last approaches apply several neural network methods to define the relationship between the inputs of the system (i.e. phase distribution) and the outputs of the system (i.e. depth distribution) with a black-box philosophy that is totally free from the geometric configuration [27]. In a previous paper we proposed a hybrid calibration method, between a model-based and a polynomial calibration process, based on an exhaustive geometric model, which describes the system with a generic relative position of the projector and the camera [28]. The parameters estimation is achieved by a minimization algorithm of the mean squared error between the nominal depth of some planes placed at well defined positions and the result of the conversion formula applied to the phase

G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

obtained from the same planes. This calibration method was chosen because it has many advantages: (a) The model is based on a well defined theory. (b) It is based on a complete geometric model that has a physical correspondence with the real measurement setup; the main consequence is the direct comparison between the parameters estimation and the geometric quantities to avoid macroscopic errors. (c) Both the geometric model and the pin-hole camera model include the main non-linearities, and then the errors due to the model simplifications are reduced. (d) The calibration method is easy to apply. A calibration method based on Mao’s geometric model had already been presented in a previous work [29]. The aim of this paper is to improve the problem analysis, studying the effect of the uncertainty parameters estimation on the final result. The question is how the uncertainty concerning the geometric parameters of the system (input parameters) can be extended to depth estimation (measurement output). The answer may only come out from a sensitivity analysis of the calibration model. It must be noted that the results will be applicable for all the calibration methods based on Mao’s geometric model (and its simplified version that corresponds to Takeda’s model). The aim is the definition of a guide to uncertainty analysis about all the most common system setups to use in the design stage for the optimization of the measure uncertainty.

2. Measurement geometric model and calibration Mao et al. [28] recently proposed a new phase mapping formula based on a complete geometric model of the projection profilometer, where the projector and the camera can be set freely as long as the full-field fringe pattern can be obtained. As it will be clarified later, the traditional FTP formula, which converts phase to depth, is just a special case of this general formula and is suitable only when the pupils of the camera and the projector are aligned [8]. The application of this geometric model is the main difference between our calibration method and most of the others proposed so far which are based on a simplified geometric model that do not consider geometric misalignment [29]. The widest formulation of the problem concerns a projector and a camera, which are not aligned, as shown in Fig. 1(top). The camera is translated along the three spatial directions with respect to the point G, which corresponds to the theoretical aligned position of the camera in the geometric setup proposed by Takeda and Mutoh [8]. The model is characterized by the carrier frequency of the projected grid and five geometric parameters: the distance E20O1 between the camera pupil and the reference plane R, the distance between the projector pupil E1 and the point G, the vertical translation HE20 of the camera and finally the distance between points O and O0 along both x and y directions. However, phase-to-depth conversion formula depends only on the first three parameters (E20O1, E1G, HE20) because, as declared by Mao himself, compared with the case in which the imaging axis crosses the projecting axis at the same point on the reference plane, moving the camera along the direction O1O just causes the movement of the image location on the camera sensor array. This means that the translation of the camera along x and y directions is irrelevant to define the conversion formula between the extracted phase and the depth of the object, whereas the translation along z direction is essential. For this reason it is possible to face the problem with an equivalent two-dimensional

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geometric model, obtained by projecting the three-dimensional model along the O1O direction on the plane E1GO. Fig. 1(bottom) shows the geometric model with the projector and the camera lying on the same plane, where the point E2 is the correspondent to the point E20 along O1O projection. Consequently the angle HE^ 1 E20 and the distance E1E20 between the pupils (Fig. 1(top)) respectively corresponds to the angle GE^ 1 E2 and to the distance E1E2. The angle GE^ 1 E2 and the distance E1E2 will respectively be named a and s. Thanks to these two geometric quantities, it is possible to quantify both the distance E1G and the vertical translation GE2. L is the distance between the entrance pupil of the imaging system and the reference plane R. In this geometric setup the optical axis of the projector lens still crosses the optical axis of the camera lens at point O on the reference plane. The angle between the optical axis of the projector and the camera is y. D is an arbitrary point on the measured object and its coordinates are (x,y) on the reference plane. The points A and C represent the well-known effect of the fringe displacement due to the presence of the object instead of the reference plane. When a sinusoidal fringe pattern is projected onto the reference plane, the captured fringe pattern on the reference plane may be expressed as [30] ( " !#) 2x sin2 y I1 ðx,yÞ ¼ I0 1 þcos 2pfx cos y 1 , ð1Þ s cos a where

y ¼ tan1



 s cos a , L þ ssin a

and f is the spatial frequency of the grid projected on the plane normal to the projector optical axis and crossing the point O. In the same way, the deformed fringe pattern can be expressed as ! ( " #) 2x sin2 y cðx,yÞ , I2 ðx,yÞ ¼ I0 1 þcos 2pfx cos y 1 ð2Þ s cos a where c(x,y) is the phase distribution caused by the depth variation h(x,y), which is [28] given as

cðx,yÞ ¼ 2pfCA cos y, and can be expressed through triangular similarity as [28]   BD xs sin a s cos a : cðx,yÞ ¼ 2pf cos y Lþ s sin aBD LBD

ð3Þ

The two signals I1(x,y) and I2(x,y) are processed with the wellknown method proposed by Takeda to obtain the phase distribution c(x,y) [8]. The method proposed by Takeda is one of the possible ways to remove the non-linear carrier frequency. Other methods are least-squares fitting and series expansion approach [31]. The new relationship between the phase distribution c(x,y) and the depth distribution h(x,y) is the core of this geometric model and its expression is [28] BD ¼

fDC LðL þs sin aÞ , 2pf0 Ls cos a þLfDC fC s sin a

ð4Þ

where L, s, a are geometric quantities and f0 ¼ f cos y is the carrier frequency of the projected grid in the case of telecentric lens of the camera and the projector. From a theoretical point of view these parameters could be measured directly, but in practice, the accurate evaluation of these parameters with direct methods is extremely difficult. A calibration process is consequently required to overcome this problem. Moreover, fDC represents the phase difference between the object and the reference planes, and fC is the phase at point C on the reference plane. The phase difference fDC can be calculated in the same way of the traditional FTP

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Fig. 1. System setup.

proposed by Takeda, whereas the phase fC is the phase of a grid projected onto the object with the same carrier frequency f0 but using a telecentric projector. The calibration process has been already developed in a previous work, so we will focus only on how the uncertainty of the measuring technique can be defined, starting from the uncertainty of the estimated input parameters, i.e. the four parameters on which the phase-height conversion depends.

3. Sensitivity analysis Saltelli et al. [34] defines the sensitivity analysis as: ‘‘The study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input’’. A related practice is ‘uncertainty analysis’, which focuses rather on quantifying uncertainty in model output.

The input of a model is what is allowed to vary in order to study its effect on the output. Generally the input parameters of a model are estimated from some experimental data obtained through simulations or experiments, as in this case. Applying a sensitivity analysis, the modellers will be conscious of the relative importance of the inputs in determining the output. An obvious consequence is the incapacity of determining the importance of those variables, which have been kept fixed. It must be noticed that the choice of the variables is of course a task of the modeller. Most of the sensitivity analyses met in the literature are based on derivatives. The derivative Yj/Xi of an output Yj versus an input Xi is in a certain way a mathematical definition of the sensitivity of Yj versus Xi. The derivative-based approach has the advantage of being very efficient in computing time and simple. Moreover if the model equations are known, derivatives are easily computable from a mathematical point of view. However, the great limitation of a derivative-based approach is that it is unwarranted when the

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model input is uncertain and when the model has unknown linearity. This means that derivatives are only useful at the base point where they are computed (usually for very small variations of the parameters around their nominal values) and do not provide for an exploration of the rest of the space of the input factors. This limitation would not matter for linear systems, but it would matter greatly for nonlinear ones. To overcome these limitations it is necessary to focus on methods for the quantitative uncertainty and sensitivity analysis in the presence of uncertain inputs. In Section 3.2 we will present a statistical analysis of the output data of the system, based on the Monte Carlo simulation. The aim is to quantify how the uncertainty concerning the input parameters can be spread to the depth estimation.

3.1. Discrete derivatives analysis The purpose of this analysis is a description of the dependence of the result on the variation of a single parameter around its nominal value. It is clear that such analysis is incomplete because it is related to the direct influence of the single parameter, without considering joint contributes of multiple parameters varying at the same time. The results obtained with derivatives analysis cannot be generalized to a single parameter variation because it is impossible to omit the contribution of the other parameters without introducing a significant error. Moreover the necessity of simplifying the equations requires a linearization. However the advantage of this simple computation is the possibility to take confidence with the measurement model, plot the trend of the measurement error as a function of the variation of a single parameter around its nominal value and have a first approximation of the dependence of the system output on the input parameters. In the following we will put into practice the discrete derivative applied to the geometric model based on Mao’s equation. As previously declared, the accurate measurement in a direct way of the parameters L, s, a and the grid carrier frequency f0 is actually impossible because of several difficulties in the quantification of relative position between the camera and the projector pupils and the position of the reference plane. This problem is solved through an estimation process called calibration, but an estimation process cannot avoid uncertainty. In the following, Le, se, ae and f0e will represent the estimates of L, s, a and the

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grid carrier frequency f0, whereas DL/L¼(Le  L)/L, Ds/s¼(se  s)/s, Da/a ¼(ae  a)/a, and Df0/f0 ¼ (f0e  f0)/f0 will indicate the relative discrepancy with respect to the nominal values. The relative error Dz(i,j)/z(i,j) can be expressed by the following generic approximation:

Dzði,jÞ zði,jÞ

¼g



Dzk ði,jÞ



ðk ¼ L,s, a,f0 Þ,

zði,jÞ

ð5Þ

where Dzk(i,j) is the error component due to the relative discrepancies DL/L, Ds/s, Da/a or Df0/f0 . The four terms could be summed together only if the parameters are mutually independent and the superimposition principle is valid. The nominal values chosen for the computation of the discrete derivatives are Le ¼ 1000 mm, se ¼300 mm, ae ¼101 and f0e ¼0.1 mm  1, which represent a typical measurement setup. As it will be shown further, the dependence of the relative errors on the nominal values of the four parameters cannot be neglected in some cases.

3.1.1. Influence of the parameter L The term DzL(i,j) is derived as the difference between the depth values calculated by substituting the two values Le and L into Eq. (4):

DzL ði,jÞ ¼

fDC ði,jÞLe ðLe þ s sin aÞ 2pf0 Le s cos a þLe fDC ði,jÞfC ði,jÞs sin a 

fDC ði,jÞLðL þ s sin aÞ : 2pf0 Lscos a þ LfDC ði,jÞfC ði,jÞs sin a

ð6Þ

With some simple steps, the relative error DzL(i,j)/zL(i,j) can be linearized (sin a  0 and cos a  1) as shown in the Appendix:

DzL ði,jÞ zL ði,jÞ

"  1þ2

DL L

2 #

 þ

DL L

L DL : 1 ¼ L þ DL L

ð7Þ

Eq. (7) shows that the relative error DzL(i,j)/zL(i,j) is a linear function of the relative error DL/L. In this case the nominal values of the geometric parameters are not influential. Fig. 2 shows the linear dependence between the two variables with a variation of the L parameter in the interval 720%.

Fig. 2. Discrete derivative for parameters L, s and f0.

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G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

3.1.2. Influence of the parameter s Term Dzs(i,j) is derived as the difference between the depth values calculated by substituting into Eq. (4) the two values se and s:

Dzs ði,jÞ ¼

fDC ði,jÞLðLþ se sin aÞ 2pf0 Lse cos a þ LfDC ði,jÞfC ði,jÞse sin a fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Ls cos a þLfDC ði,jÞfC ði,jÞs sin a

The relative error Dza(i,j)/za(i,j) is simplified as follows and explained in the Appendix:

Dza ði,jÞ za ði,jÞ



Da Qa1 ði,jÞ þ Qa2 ði,jÞ

a

1 Daa Qa2 ði,jÞ

,

ð11Þ

where QS ði,jÞ ¼ ðLfDC ði,jÞ=ð2pf0 L cos aÞÞ. Eq. (9) shows that the relative error Dzs(i,j)/zs(i,j) is a hyperbolic function of the error Ds/s, where the term Qs(i,j) represents a sensitivity coefficient that depends on the other parameters. In this case the limits of a derivative analysis are clear: the effects of the other parameters must be simplified as a constant whose value depends on a specific geometric configuration (associated to the nominal values of the parameters). This limitation must be overcome through global sensitivity analysis that is unaffected by non-linearities and the parameters’ nominal values. For common arrangement of the camera and the projector, the quantity Qs(i,j) takes values below one and is almost irrelevant for the definition of the function trend. Fig. 2 shows the hyperbolic dependence between Z and s with a variation of the s parameter in the interval 720% with the other parameters at fixed values, namely.

where Qa1 ¼ aððs cos aÞ=ðLþ s sin aÞÞ and Qa2 ¼ ð2pf0 Ls sin aDafC s cos aDaÞ=ð2pf0 Ls cos a þLfDC fC s sin aÞ. Eq. (11) shows that the relative error Dza(i,j)/za(i,j) has a complex relationship with the parameter a. The dependence on the other parameters, enclosed into the two factor Qa1(i,j) and Qa2(i,j), cannot be simplified. In this case the two terms cannot be considered irrelevant constants because their values are susceptible to the other parameters, which represent the sensibility coefficients of the function. The relationship between Dza(i,j)/za(i,j) and Da/a is hyperbolic; however Fig. 3 (case 1) shows that the dependence between the two variables with a variation of the parameter a is almost linear inside the interval 720%. Finally, Fig. 3 shows an example of the variation of the relation between Dza(i,j)/za (i,j) and Da/a due to the other parameters. The two curves (case 2 and 3) are originated from the two combinations of the parameters L , s and f0 that minimizes (case 2) and maximizes (case 3) the two values Qa1(i,j) and Qa2(i,j). The definition of the case 2 and 3 depends on the parameters variation in the following ranges: L from 500 to 2000 mm, s from 100 to 500 mm, a from 451 to 451 and f0 from 0.1 to 0.05 mm  1. For these two combination the maximum values for Qa1(i,j) and Qa2(i,j) are 0.0354 and 0.0828, whereas the minimum are 1.1290e 004 and 2.3650e 004, respectively. The maximum condition is verified when L¼600 mm, a ¼10.121, f0 ¼0.03 mm  1 and s¼ 405 mm, whereas the minimum condition is verified when L¼1350 mm, a ¼4.51, f0 ¼0.06 mm  1 and s¼180 mm. It must be noticed that the relative variation between the two curves is important, but the correspondent variation of the height is always below 2%.

3.1.3. Influence of the parameter a Term Dza(i,j) is derived as the difference between the depth values calculated by substituting the two values ae and a into Eq. (4):

3.1.4. Influence of the parameter f0 Term Dzf0(i,j) is derived as the difference between the depth values calculated by substituting the two values f0e and f0 in Eq. (4):

ð8Þ

As shown in the Appendix, the relative error Dzs(i,j)/zs(i,j) may be simplified (sin a  0 and cos a  1) in the following form:

Dzs ði,jÞ zs ði,jÞ



Dza ði,jÞ ¼

Ds

1 , s 1 þ ðDs=sÞ þ Qs ði,jÞ

fDC ði,jÞLðL þ s sin ae Þ 2pf0 Ls cos ae þ LfDC ði,jÞfC ði,jÞs sin ae fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Ls cos a þLfDC ði,jÞfC ði,jÞs sin a

ð9Þ

Dzf0 ði,jÞ ¼ ð10Þ

fDC ði,jÞLðL þs sin aÞ 2pf0e Ls cos a þ LfDC ði,jÞfC ði,jÞs sin a fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Lscos a þ LfDC ði,jÞfC ði,jÞs sin a

Fig. 3. Discrete derivative for parameter a.

ð12Þ

G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

The relative error Dzf0(i,j)/zf0(i,j) may be simplified as follows (Appendix):

Dzf0 ðx,yÞ zf0 ðx,yÞ



Df0

1 , f0 1 þ Df0 þQf f0

ð13Þ

where Qf ¼ ð1=f0 ÞððLfDC fC s sin aÞ=2pLs cos aÞ. Eq. (13) expresses the hyperbolic function that represents the relative error Dzf0(i,j)/zf0(i,j) connected to the parameter variation Df0/f0. Note that Eq. (13) is formally identical to Eq. (9). As in the previous case, the quantity Qf has an almost negligible effect on the value of Eq. (13) for common values of the geometric parameters.

3.1.5. Considerations about the discrete derivatives analysis The evaluation of the discrete derivatives, in correspondence to the 20% variation of the parameters around the nominal values, allows to draw some preliminary considerations about the influence of the input parameters on the linear model output. Fig. 2 shows that the parameter L generates a variation of the result substantially equivalent to the parameter variation itself ( 720%), whereas a 20% variation of the parameters s and f0 entails a variation on the result in the range between 15% and 25%. In other words the slopes of the three derivates are approximately equal to 1. However there is a difference, because the simplified relation between DL/ L and Dz/z is actually linear, whereas the other two parameters have a hyperbolic relation. The effect of the parameter a, for the same range of variation around the nominal value, is different. Fig. 3 shows a limited impact on the system output due to the angle, in fact the slope of the curve is approximately 0.05. The curve slope shows significant percentage variation (from 0.035 to 0.08) for different nominal values of the parameters (L, s, f0, a), as shown in Fig. 3. Nevertheless, the amount of variation Dz/z associated to the angle a is definitely slower than the effects of the other parameters. In conclusion, the discrete derivatives analysis provides an approximated sensitivity analysis which outlines a roughly equivalent contribution of the parameters L, s and f0 to the variation of z. On the contrary the contribution of the parameter a is less important. All these considerations are suitable only for the linearized model.

3.2. Global sensitivity analysis The necessity to overcome the limits of the discrete derivative analysis entails the application of more complex methodologies like the sensitivity analysis. The first step is to consider a description model of the system and to proceed with the parameter estimation through a calibration procedure. Repeating the calibration procedure several times, it is possible to define the probability density function of every single parameter (L, s, a and f0) and the uncertainty accordingly associated. Alternatively, it is possible to fix a priori a probability density function, based on the knowledge about the specific problem. After the parameter estimation process, the model may be considered defined and an uncertainty analysis may be performed propagating the uncertainty in the parameters through the model, all the way to the model output. In this case the estimated parameters become the factors of the analysis. Supplement 1 to the ‘‘Guide to the expression of uncertainty in measurement’’ declares that in case of non-linear systems the estimation of the uncertainty propagation by means of discrete derivatives of variances is non-applicable [32,33]. A more complex procedure is consequently needed.

215

One way of doing the uncertainty propagation for non-linear systems is through Monte Carlo analysis, in which we look at the distribution functions of the input parameters, as derived from the estimation [34]. For example, we may have the following scheme:

 We start from a factor a  Nða, sa Þ, which reads: after  



estimation a is known to be normally distributed with mean a and standard deviation sa. Likewise for factors L, s and f0. For each of these factors, we draw a sample from the respective distributions, i.e. we produce a set of row vectors ðjÞ ðaðjÞ ,LðjÞ ,sðjÞ ,f0 Þ, where j¼ 1, 2, 3,y, N denotes the jth sample taken from the distribution of the parameter. ðjÞ We can compute the model for all vectors ðaðjÞ ,LðjÞ ,sðjÞ ,f0 Þ, thereby producing a set of N values of a model output Zj: 2 ð1Þ 3 z 6 ð2Þ 7 6z 7 6 7 6... 7 6 7 6 ðN1Þ 7 4z 5 zðNÞ

These steps constitute the uncertainty analysis. Then we can calculate the average output, its standard deviation, the quantiles of its distribution, confidence bounds, plot the distribution itself and so on. Having performed this uncertainty analysis we can move on to a sensitivity analysis, in order to determine which input parameters are more important in influencing the uncertainty of the model output, i.e. a factor prioritization. Sensitivity analysis may be performed through different methods. In this case the parameters are only four and an analysis of variance is appropriate. The main indexes used to express the global sensitivity analysis are basically two, both based on the variance of the data. One is the first-order sensitivity index of the generic input parameter Xi over the output Y [34]: Si ¼

VXi ðEX  i ðY=Xi ÞÞ , VðYÞ

ð14Þ

where VXi ðEX  i ðY=Xi ÞÞ is the conditional variance obtained by the variance of the K averages, each of them obtained by fixing the variable Xi at a given value xi (with i¼1yK) and letting the other parameters to vary randomly according to their probability distribution. Si is a number always between 0 and 1. A high Si value signals an important variable, i.e. a variable that, if modified, strongly affects the value of the system output. However a small value of Si does not represent an unimportant variable, because the first order index quantifies only the direct effect of the parameters, dropping the mutual effect with the other parameters. The necessity of another index for the mutual effect quantification is clear. The total effect index of factor Xi is obtained by conditioning all factors but Xi [34]:   VðEðY9X  i ÞÞ : ð15Þ STi ¼ 1 VðYÞ STi is made of all the terms of any order that include xi. Note that Si and STi carry a different type of information; in particular it can be demonstrated that Si ¼0 is a necessary but insufficient condition for fixing factor Xi. Even if Si ¼0, the factor Xi might be involved in interactions with other factors so that, although its first-order term is zero, there might be nonzero higher-order

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terms. On the other hand, if STi ¼0 is a necessary and sufficient condition for Xi being uninfluential. In the next sections we will present the results of the global sensitivity analysis of the calibration method described in Section 3. The aim is to define how the uncertainty in the evaluation of the parameters of the Eq. (4) affects the result of the measurement of an object shape. Some assumptions have been taken to perform the analysis. First, the system input variables are the four parameters of the model, and also the phase difference fDC, which is directly related to the object depth distribution. The phase distribution does not derive from the calibration procedure; however the sensitivity analysis has been evaluated in correspondence of different nominal values of the phase difference, in order to estimate the possible dependence of the factor prioritization on this input (Section 3.2.2). Moreover we assumed the probable distribution of the parameter as rectangular with an amplitude of 75% in comparison with the nominal value, because the aim is the a priori uncertainty analysis of the calibration method. In this way it is possible to quantify the uncertainty dispersion on the final result, if all the four parameters are estimated with a maximum discrepancy of 5% with respect to their nominal values. Our previous work demonstrated that all the parameters can be estimated with an error lower than 2% [29], then a choice of a simulated error of 5% is a reasonable range of variation.

3.2.1. Nonalignment effect for a fixed geometry ratio The limit of an analysis based on discrete derivatives evaluation was described in Section 3.1. The linearization of the symbolic equations necessarily entails a simplification of the problem and, consequently, a loss of information about the effect of the non-linearity of the geometry (see Fig. 1 and Eq. (4)). If the angle a is set to zero, the projector and camera pupils are aligned at the same Z value (distance GE2 in Fig. 1(bottom) becomes zero). In this case Eq. (4) is simplified and the output BD becomes almost linear with the input fDC [28]. On the contrary as a increases Eq. (4) becomes strongly non-linear. Due to this consequence about the optics alignment, a specific analysis of the effect of nonalignment on parameter sensitivity is given in this section. The geometric parameter that defines the vertical translation between G and E2 is the angle a, but the value of the nonalignment depends on the distance s as well. For this reason to quantify the nonalignment it is necessary to define some specific geometric setup proportions, i.e. a particular ratio between the distances L and s, because, as explained in our previous work [29], the measurement model is scale independent. The aim of this analysis is a description of the behaviour of a non-aligned setup, in terms of the influence of the geometric parameters on the measurement result. Consequently a common geometry configuration, characterized by the ratio s/L was chosen (s/L¼0.3 with L¼ 1000 mm) and the frequency of the grid f0 was fixed to 0.1 mm  1. Once the geometric arrangement was fixed, a wide variation of the parameter a (between  451 and 451) was imposed and consequently the indices Si and STi were calculated for each combination, as defined in Eqs. (14) and (15). The test has been repeated in correspondence to three different levels of the input phase difference fDC, which are associated to the two extremes of the measurable field (z¼ 7500 mm) and the midpoint (z¼0 mm). Table 1 shows the results obtained from the sensitivity analysis for the first-order sensitivity index Si when the input phase corresponds to a plane coincident with the reference plane (z¼0). The most important consideration is about the indexes SL, Ss and Sf0 that have a limited variation in a range between 30% and 35%, when the absolute value of a is no larger than 151. The indexes are almost equivalent around the 30% within this range of variation, i.e. the uncertainty on the estimation of these three

Table 1 First-order sensitivity index for different values of the parameter a. L (mm)

s (mm)

a (deg.)

f0 (1/mm)

SL

Ss

Sa

Sf 0

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

 45  40  35  30  25  20  15  10 5 5 10 15 20 25 30 35 40 45

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.24 0.28 0.30 0.31 0.31 0.32 0.33 0.32 0.34 0.35 0.35 0.35 0.34 0.35 0.35 0.34 0.34 0.34

0.32 0.34 0.35 0.37 0.35 0.36 0.34 0.35 0.34 0.32 0.31 0.32 0.31 0.30 0.29 0.27 0.26 0.24

0.16 0.08 0.04 0.02 0.01 0.003 0.001 0.001 0.001 0.001 0.002 0.005 0.01 0.02 0.04 0.07 0.11 0.18

0.28 0.29 0.31 0.31 0.32 0.33 0.31 0.33 0.33 0.34 0.33 0.34 0.34 0.34 0.32 0.31 0.30 0.26

Fig. 4. First-order sensitivity index for a fixed geometry, z ¼0 mm and values of angle a from  451 to 451.

parameters has nearly the same influence on the uncertainty of the measurement result of the profilometer. On the contrary the parameter a has a weight only between 0.1% and 0.5% on the output of the measurement. On the other hand, if the absolute value of the parameter a is higher than 151, the effect of the non-linearity of the system increases considerably. In this case the indices Si related to the parameters L, s and f0 vary in a larger range (between 24% and 37%), whereas the index related to the parameter a increases up to 18%. This change of behaviour of the system, as a increases, can be called ‘‘nonalignment effect’’. Fig. 4 summarizes the data of Table 1 and the considerations stated before. It should be noticed that values of the angle a greater than 151 are very uncommon in practice; for most of the setup geometries, the nonalignment is absolutely evident and unacceptable. The practice [29] confirms that the ideal geometry configuration proposed by Takeda [8] has a lot of advantages, in terms of measurement accuracy and signal elaboration simplicity (1D Fourier transform combined with band pass filter applied to one single image). For this reason the aim during the realization of a 3D measurement instrument, based on Fourier-transform profilometry, should be, in most of the applications, an almost aligned geometry setup. Obviously this condition is ideal and the aim of this paper is to define the effect of an error in the calibration parameters estimation onto the output of the

G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

instrument, when the geometry is not aligned. The nonalignment effect is still evident when a plane placed at z¼  500 mm is measured, as shown by Fig. 5. In this case the main difference

Fig. 5. First-order sensitivity index for a fixed geometry, z ¼  500 mm and values of angle a from  451 to 451.

Fig. 6. First-order sensitivity index for a fixed geometry, z ¼ 500 mm and values of angle a from  451 to 451.

217

with respect to Fig. 4 is about index SL, which reveals that parameter L is down by half. Other parameters maintain more or less the same trend. However Fig. 6 shows that in correspondence to depth z ¼500 mm, the importance of the parameters uncertainty changes a lot with respect to the previous cases and the effect of the nonalignment is no more evident. The parameters a, s and f0 lose importance, whereas the parameter L acquires most of the influence on the output uncertainty. This means that the influence of the parameter L is much more important as the object is close to the camera (z 40 mm), i.e. directly dependent on the relative distance camera–object. In general, when the alignment condition is barely satisfied, the influence of the uncertainty in the estimation of the angle a is lower than the effect of the uncertainty on the other parameters. However, to increase the accuracy of the measurement, the correct estimation of this parameter is also substantial. It must be noticed that the values of the total effect index STi have not been reported in this section, as well as in the next one, because the results are very close to the first order indexes Si. This means that the four parameters are totally independent to each other, which could have also been assumed from the results of the first order indexes, whose variations cover 100% of the uncertainty of the final result. For example the values of the total effect indexes STi obtained for the nominal values of the parameters L¼ 1000, s¼300, a ¼51 and f0 ¼0.1 mm  1, in correspondence to z ¼0 mm, are: STL ¼ 0.33, STs ¼0.34, STa ¼0.001 and STf0 ¼0.34. In the same way the results obtained for L¼1000, s ¼300, a ¼301 and f0 ¼0.1 mm  1 are: STL ¼0.37, STs ¼0.29, STa ¼0.04 and STf0 ¼0.31. It is clear that the analysis of the total effect indexes is basically equivalent to the ones obtained with the first order indexes Si.

3.2.2. Nonalignment effect for a different geometry ratio In Section 3.2.1 the effect of the angle a uncertainty was investigated in the case of fixed setup geometry. In this condition the value of the vertical displacement of the camera with respect to the reference plane depends only on the value of the parameter a, because the distance s is fixed. However, the quantity GE2 (Fig. 1) is a function both of the parameter s, i.e. the geometric proportions of the system, and the angle a, i.e. the nonalignment

Fig. 7. First-order sensitivity index for different geometries, four nominal values of the angle a (51, 101, 151 and 251) and z ¼0 mm.

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G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

of the pupils. In this section it will be described the reciprocal influence of the distance L and the nonalignment of the optics GE2. For a fixed value of L (1000 mm) and the grid frequency f0 (0.1 mm  1), four values of the ratio s/L (0.2, 0.4, 0.6 and 0.8) and four values for the angle a (51, 101, 151 and 251) were chosen. Note that, since the measurement system is scale independent [29], fixing L, the results do not loose generality. Fig. 7 shows the results obtained from the sensitivity analysis in correspondence

to the measured plane z¼0 mm. When the angle a is small (a ¼51), the conclusions drawn in Section 3.2.1 are confirmed because the nonalignment effect is negligible. In particular, with a ¼51, whatever value for the ratio s/L is chosen, the indices Si are steady in a range between 30% and 35%. The nonalignment effect occurs with the increment of the angle a. For example, if we consider a ¼251, the indices SL, Ss and Sf0 are stable in the range 30–35% only for the smallest value of the ratio s/L (s/L¼ 0.2),

Fig. 8. First-order sensitivity index for different values of the ratio GE2/L and z¼ 0, z¼  500 and z¼ 500 mm.

G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

whereas the nonalignment effect becomes significant increasing the s/L ratio. The nonalignment effect depends both on the parameter a and s, because the real quantity that describes the nonalignment is the displacement GE2 ¼ s sinðaÞ. If the values of the indices Si, reported in Fig. 7, are represented as a function of the vertical displacement GE2 =L, the trend dependence of the four indexes Si becomes clear. Fig. 8 shows the trends of the Si values as a function of GE2 =L for each of the three levels of the input phase, respectively, related to z¼ 0, z¼  500 and z¼500 mm. For values of GE2 =L below 10%, the values of the Si indexes are stable for all the three values of z and the indexes are almost constant around the values shown in Figs. 4–6. When the vertical displacement exceeds this limit, the nonalignment effect arises and the index Sa gains importance in comparison to the other parameter indexes.

3.3. Discussion The application of global sensitivity analysis to Fourier-transform profilometry revealed the dependence of the measuring output uncertainty from the system parameters uncertainty in several geometric conditions. Some considerations may be drawn from the previously presented results, with particular attention to three topics: the misalignment effect, the geometric proportions and finally the measurable volume. If only the effect of angle a is considered, Figs. 4–6 show that the uncertainty linked to this parameter barely affects the measurement uncertainty when the geometric model proposed by Takeda is applied (which correspond to a E01 for Mao’s geometric model). These statements may be extended to a misaligned model, if angle a module is maintained below 101. On the contrary, Fig. 8 reveals that the effect of the misalignment on the measurement uncertainty depends both on the parameters s and a (GE2 ¼ s sinðaÞ). It must be noted that a value different from zero for the angle a is a practical consequence of the impossibility of indentifying the optic pupils position, whereas the choice of the parameter s depends on the design requirements. The geometric proportion represented by the ratio s/L defines indeed the possible application of the system. If the ratio s/Lb 1, the profilometer is more sensitive to the object height variation, because the angle y between the optical axes of the camera and the projector increases (Fig. 1), i.e. the greater is the angle y, the larger is the phase difference between the reference and the deformed grid for a fixed height. Although, a high sensitive system incurs in several problems if the object surface is complex and has undercuts. Otherwise, a scanner characterized by a ratio s/L51 is more suitable for the acquisition of complex surfaces with undercuts, but is less sensitive to height variations. Moreover the choice of the system setup has a direct consequence on the parameter prioritization. The results of Fig. 8 show that the system behaviour is almost linear if the relative misalignment GE2 =L is below 10%, independently from the Z value. When this limit is exceeded, the relevance of the uncertainties of angle a and frequency f0 becomes more influential for the measure uncertainty definition. On the other hand, the uncertainties of the parameters s and L contribute less to the system output uncertainty. Generally, the more sensitive the scanner, the more the importance of parameter f0. Consequently the phase to height conversion formula derived from Takeda’s geometric model could be used to calibrate a misalign scanner without introducing a significant error, but only if the misalignment GE2 =L is controlled and limited to negligible values. This means that the angle a must be very low to minimize

219

the misalignment for a profilometer with s/Lb1, whereas the condition on the angle could be weaker if s/L51. Another interpretation of the data could be drawn if we consider the measurable volume. Fig. 8 reveals that the reciprocal importance of the parameters uncertainties changes according to the object position in the measurable volume. If the object is near to the reference plane, the three parameters L, s and f0 have an almost equal relevance (about 35% ) and the possible variations depend only on the misalignment effect. In the case of the object near to camera (Z40 ), the most important parameter is the distance L, on which the output uncertainty depends for a range between 55% and 70%. On the other hand, when the object is far from the camera (Zo0 ), the most important parameters are s and f0. The consequence of these assumptions is the possibility to fix a geometric setup and a workload with particular attention to the estimation performance of the adopted calibration procedure. For example, if a sensitive scanner is needed (the grid frequency f0 is one of the most critical parameter, as shown in Fig. 7) and the calibration procedure privileges the distance estimation, it could be possible to fix the measurable volume far from the reference plane and near to the camera. In this case the aim is to increase the importance of the parameter L and to decrease the importance of the parameter f0. Consequently an improvement of the measure uncertainty will be obtained. The strong assumption that could be drawn is that the system uncertainty is mainly controlled by two causes: the percentage misalignment GE2 =L and the measured object position.

4. Conclusion In this paper we presented a sensitivity analysis of a Fouriertransform profilometry, performed both with the simple discrete derivative method and the more complex global sensitivity analysis, based on Monte Carlo simulations. Global sensitivity analysis is a powerful method that permits the evaluation of the uncertainty distribution from the input parameters to the output. The quantification of the single contribution of every input parameters uncertainty onto to the uncertainty distribution of the system output gives the opportunity to define a factor prioritization, i.e. which factor deserves further analysis or measurement, in order to improve the uncertainty of the estimated height distribution of the measured object. In this work we presented an extensive analysis to reasonable possible cases, in terms of system setup and measured height distribution. The result is a substantial description of the uncertainty propagation from which is possible to derive all the information related to a specific application of the system. The aim obtained is the definition of a guide useful to quantify the measure uncertainty according to the application, i.e. a different geometric disposition and the chosen calibration procedure.

Appendix In this Appendix the steps to obtain Eqs. (7), (9), (11) and (13) are shown.

A.1. Parameter L The term DzL(i,j) of Eq. (7) is derived as the difference between the depth values calculated by substituting the two values Le and

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G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

L into Eq. (4):

Consequently, the ratio Dzs ðx,yÞ=zs ðx,yÞ is

fDC ði,jÞLe ðLe þ s sin aÞ DzL ði,jÞ ¼ 2pf0 Le s cos a þ Le fDC ði,jÞfC ði,jÞs sin a fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Ls cos a þLfDC ði,jÞfC ði,jÞs sin a

Dzs ðx,yÞ zs ðx,yÞ

¼

ðfDC LðL þ se sin aÞ=ð2pf0 Lse cos a þ LfDC fC se sin aÞÞ 1 ðfDC LðL þs sin aÞ=ð2pf0 Ls cos a þ LfDC fC s sin aÞÞ

¼

fDC LðL þ se sin aÞ 2pf0 Ls cos a þ LfDC fC s sin a U 1: fDC LðL þ s sin aÞ 2pf0 Lse cos a þ LfDC fC se sin a

ð16Þ

Consequently, the ratio DzL ðx,yÞ=zL ðx,yÞ is

DzL ðx,yÞ zL ðx,yÞ

¼

ðfDC Le ðLe þ s sin aÞ=ð2pf0 Le s cos a þ Le fDC fC s sin aÞÞ 1 ðfDC LðLþ s sin aÞ=ð2pf0 Ls cos a þ LfDC fC ssin aÞÞ

¼

fDC Le ðLe þ ssin aÞ 2pf0 Ls cos a þ LfDC fC s sin a U 1: fDC LðL þ s sin aÞ 2pf0 Le s cos a þ Le fDC fC s sin a ð17Þ

Substituting Le ¼L+ DL, the first term of Eq. (17) can be rewritten as follows:

fDC Le ðLe þs sin aÞ fDC ðL þ DLÞðL þ DL þ s sin aÞ ¼ fDC LðL þ ssin aÞ fDC LðL þ s sin aÞ LðL þ s sin aÞ þ LDL þ DLðL þ DL þ ssin aÞ ¼

LðL þ ssin aÞ

:

ð18Þ

Simplifying the term L þ ssina, Eq. (18) can be rewritten as   LðLþ s sin aÞ þ LDL þ DLðL þ DL þ s sin aÞ DL DL þL ¼ 1þ þ1 : LðLþ s sin aÞ L L þ s sin a ð19Þ If we consider an almost aligned geometric model (sina  0 and cosa  1), Eq. (19) may be simplified as follows:  2 fDC Le ðLe þs sin aÞ DL DL þ ¼ 1þ2 : ð20Þ L L fDC LðL þ ssin aÞ Substituting Le ¼L+ DL, the second term of Eq. (17) can be rewritten in this way: 2pf0 Ls cos a þ LfDC fC ssin a 2pf0 Le s cos a þ Le fDC fC s sin a 2pf0 Ls cos a þLfDC fC s sin a 2pf0 ðLþ DLÞs cos a þðL þ DLÞfDC fC s sin a 1 : ¼ 1þ ðð2pf0 DLs cos a þ DLfDC Þ=ð2pf0 Ls cos a þ LfDC fC s sin aÞÞ ¼

ð21Þ If we consider an almost aligned geometric model (sina  0 and cosa  1), Eq. (21) may be simplified as follows: 1 1 þðð2pf0 DLs cos a þ DLfDC Þ=ð2pf0 Ls cos a þLfDC fC s sin aÞÞ ¼

1 L : ¼ Lþ DL 1þ DLL

Substituting Eqs. (20) and (22) into Eq. (17) we obtain "  2 # DzL ðx,yÞ DL DL L DL þ :  ¼ 1þ2 1 ¼ zL ðx,yÞ L L Lþ DL L

ð25Þ Substituting se ¼s+ Ds, the first term of Eq. (25) can be rewritten as follows:   fDC LðL þ se sin aÞ Lþ ðsþ DsÞsin a Ds sin a ¼ ¼ 1þ : ð26Þ fDC LðL þs sin aÞ Lþ s sin a Lþ s sin a If we consider an almost aligned geometric model (sina  0 and cosa  1), Eq. (26) may be simplified as follows:   fDC LðL þ se sin aÞ Ds sin a s ¼ 1þ  1: ð27Þ s fDC LðL þs sin aÞ Lþ s sin a Consequently Eq. (25) can be rewritten as 2pf0 Ls cos a þ LfDC fC s sin a 1 2pf0 Lse cos a þ LfDC fC se sin a ¼

¼

Ds2pf0 L cos aDsfC s sin a 2pf0 Lðs þ DsÞcos a þ LfDC fC ðs þ DsÞsin a  Dss ð2pf0 L cos afC s sin aÞ Ds ð2pf

0 L cos

s

ð28Þ

2pf0 Ls cos a þLfDC fC s sin a 1 2pf0 Lse cos a þLfDC fC se sin a



ðDs=sÞ 1 þ ðDs=sÞ þ ðLfDC =sð2pf0 L cos aÞÞ

¼ ðDs=sÞð1=1 þðDs=sÞ þ ð1=sÞQs Þ,

ð29Þ

where Qs ¼ ðLfDC =ð2pf0 L cos aÞÞ. A.3. Parameter f0 The term Dzf0 ðx,yÞ=zf0 ðx,yÞ can be derived with the same steps used for the parameter s.

ð22Þ A.4. Parameter a

ð23Þ

The term Dza(i,j) of Eq. (11) is derived as the difference between the depth values calculated by substituting the two values ae and a into Eq (4):

Dza ði,jÞ ¼ The term Dzs(i,j) of Eq. (9) is derived as the difference between the depth values calculated by substituting into Eq. (4) the two values se and s:

fDC ði,jÞLðLþ se sin aÞ 2pf0 Lse cos a þ LfDC ði,jÞfC ði,jÞse sin a fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Ls cos a þLfDC ði,jÞfC ði,jÞs sin a

:

Remembering that the almost aligned condition (sina  0 and   cosa  1) was introduced, and dividing for 2pf0 LcosafC ssina , Eq. (28) may be simplified as follows:

A.2. Parameter s

Dzs ði,jÞ ¼

afC s sin aÞ þ2pf0 L cos afC s sin a þ LfsDC

fDC ði,jÞLðL þs sin ae Þ 2pf0 Ls cos ae þ LfDC ði,jÞfC ði,jÞs sin ae fDC ði,jÞLðLþ s sin aÞ  : 2pf0 Lscos a þ LfDC ði,jÞfC ði,jÞs sin a

ð30Þ

The ratio Dza(i,j)/za(i,j) is

Dza ði,jÞ ð24Þ

za ði,jÞ

¼

fDC ði,jÞLðL þ s sin ae Þ 2pf0 Ls cos a þ LfDC ði,jÞfC ði,jÞs sin a 1: U fDC ði,jÞLðLþ s sin aÞ 2pf0 Ls cos ae þ LfDC ði,jÞfC ði,jÞs sin ae ð31Þ

G. Busca, E. Zappa / Optics and Lasers in Engineering 49 (2011) 210–221

Substituting ae ¼ a + Da, the first term of Eq. (31) becomes

fDC LðLþ s sin ae Þ fDC LðLþ s sinða þ DaÞÞ ¼ fDC LðLþ s sin aÞ fDC LðL þ ssin aÞ Lþ s sin a cos Da þ scos a sin Da ¼

ðL þ s sin aÞ

,

ð32Þ

whereas the second term of Eq. (31) becomes 2pf0 Lscos a þ LfDC fC s sin a 2pf0 Ls cos ae þ LfDC fC s sin ae 2pf0 Ls cos a þLfDC fC s sin a ¼ : 2pf0 Ls cos a cos Da þ LfDC fC s sin a cos Da2pf0 Ls sin a sin DafC s cos a sin Da

ð33Þ If we consider Da C 0 (cosDa  1 and sinDa  Da), Eq. (32) may be simplified as follows: L þs sin a cos Da þ s cos a sin Da ðLþ s sin aÞ   Dascos a Da scos a Da  1þ ¼ 1þ Q , ¼ 1þ a L þs sin a a Lþ s sin a a a1

ð34Þ

where Qa1 ¼ aðs cos aÞ=ðL þ s sin aÞ. In the same way, the second term of Eq. (31) can be simplified as 2pf0 Ls cos a þ LfDC fC s sin a 2pf0 Ls cos a cos Da þ LfDC fC s sin a cos Da2pf0 Ls sin a sin DafC s cos a sin Da ¼

1 1ðDa=aÞðð2pf0 Ls sin aDafC s cos aDaÞ=ð2pf0 Ls cos a þLfDC fC s sin aÞÞ

¼

1 , 1ðDa=aÞQa2

ð35Þ

where Qa2 ¼ ðð2pf0 Ls sin aDafC s cos aDaÞ=ð2pf0 Ls cos a þLfDC  fC s sin aÞ. Substituting Eqs. (34) and (35) into Eq. (31), we obtain   Dza ði,jÞ  1 Da Qa1 þ Qa2  1 þ ðDa=aÞQa1 : 1 ¼ za ði,jÞ 1ðDa=aÞQa2 a 1ðDa=aÞQa2 ð36Þ References ¨ zuurel UD, Bulut K, Inci MN. Vibration amplitude analysis with a [1] Yilmaz ST, O single frame using a structured light pattern of a four-core optical fibre. Opt Commun 2005;249(4–6):515–22. [2] Osten W, editor. Interferometry XI: Applications; 2002. 10 July 2002 through 11 July 2002. [3] He X, Sun W, Zheng X, Nie M. Static and dynamic deformation measurements of micro beams by the technique of digital image correlation. Key Eng Mat 2006;326–328 I:211–4. [4] Huang PS, Jin F, Chiang F-. Quantitative evaluation of corrosion by a digital fringe projection technique. Opt Lasers Eng 1999;31(5):371–80. [5] Zhang Q-, Su X-. An optical measurement of vortex shape at a free surface. Opt Laser Technol 2002;34(2):107–13. [6] Hui T-, Pang GK. Solder paste inspection using region-based defect detection. Int J Adv Manuf Technol 2009;42(7-8):725–34.

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