Second- and first-order phase-locked loops in fringe profilometry and application of neural networks for
phase-to-depth conversion
Dinesh Ganotra, Joby Joseph, Kehar Singh*
Photonics Group, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India Received 13 August 2002; received in revised form 1 December 2002; accepted 20 December 2002
Abstract
Second- and first-order phase-locked loops (PLLs) are applied to recover phase information from projected fringes recorded experimentally. The phase recovery is studied in terms of loop gain coefficients for both PLLs. Second-order loop compared to first-order loop recovers larger phase changes. Values of the phases thus recovered are compared with phases recovered by Fourier transform method. Neural networks are used to obtain depth information from phase planes and their known positions. The spatial period of the projected grating is calculated using Fourier transform of the image of the grating. An analysis is presented of second- and first-order loops in terms of performance comparison for extracting the phase information from fringe patterns. The use of neural network and automatic calculation of spatial period brings the measuring system closer to complete automation.
Keywords: Profilometry; Phase-locked loops; Neural networks; Phase recovery
1. Introduction
The technique of using phase-locked loops (PLLs) in profilometry was first introduced by Servin and Rodriguez-Vera [1]. Methods using PLLs have been preferred over the conventional techniques because the problem of phase un- wrapping is not encountered while using PLL- based methods. Servin et al. [2] extended the technique to demodulate noisy fringe patterns.
Cuevas et al. [3] used first-order PLL to estimate the phase of the fringes, and calculated height using neural networks. Gdeisat et al. [4] presented the use of second-order PLL to demodulate fringe patterns. They found that the second-order PLL has better tracking ability and more noise immu- nity than the first-order PLL.
A comparison between phase shifting and PLL profilometry was done by Ochoa et al. [5]. For targets which cannot remain perfectly still for more than a few seconds, for example in robotic vision, the distances keep on changing and the usual tri- angulation methods cannot be directly applied.
Fourier transform profilometry (FTP) is not able
to recover a detailed surface profile from interfer- ograms because a Fourier transform does not exist for the stochastic nature of real surfaces [6]. Be- cause phase measuring profilometry, FTP, and moire profilometry are all based on structured light projection, they are limited by shadow and phase truncating and cannot be used to measure the areas if discontinuous height steps and spatial isolation are present in the object. A survey was conducted by Jarvis [7] on approaches for generalised range finding techniques, and perspective given on their applicability and shortcomings in the context of computer vision studies. They concluded that no one method seems clearly superior to the rest. All methods appear to have some drawbacks as well as merit in their applications. It seems that there is still a need to improve upon these methods and come up with a practical solution.
In the present paper, the phase recovery is studied in terms of loop gain coefficients for both second- and first-order PLLs. The spatial period of the projected grating is calculated from its Fourier transform. Values of the phases recovered are compared with those recovered using FTP. Neural networks [8,9] are used for phase to height con- version. The use of neural network and automatic calculation of spatial period brings the measuring system closer to complete automation.
2. Theory of PLLs
Phase-locked loop is a subject in communica- tion engineering and many results available in the PLL literature could be applied with minor chan- ges to profilometry. The basic components (Fig. 1) of a digital PLL are multiplier, low pass filter, and digital controlled oscillator (DCO). The basic principle is to follow the phase changes of the in-
put signal by varying the value to the DCO signal so that the phase error between the input signal and the DCO signal vanishes.
2.1. First-order PLL
First-order PLL has weak noise immunity than the second-order. The whole algorithm will fail if the first iteration fails. Consider Fig. 1. Here x is the sample index, f is the spatial frequency of the carrier fringes' image, /0ðxÞ is the phase of the fringe pattern and /ðxÞ is the phase of the DCO.
The DCO is initially tuned to fringe pattern an- gular frequency which changes with the low pass filter output. cosð2pfx þ /0ðxÞÞ denotes the high pass filtered input signal, and sinð2pfx þ /ðxÞÞ denotes the DCO output signal (reference signal).
Signals cosð2pfx þ/0ðxÞÞ and sinð2pfx þ /ðxÞÞ give the output at the multiplier (error signal) as
4>e(x) = cos[2n/x + 4>0(x)] sir
= — sin[4rt/x + 4>0(x) -f
(1)
(2) After the error, signal is passed through a low- pass digital filter, the high frequency component term sinð4pfxÞ is eliminated. The signal is pro- portional to the phase difference between the de- modulated signal and the DCO signal
- 0 o W ] ,
Change of /ðxÞ in proportion
W
filtered(3) (4) can be expressed as a first-order differential equation
dx • = - T 0 X - (5)
Fig. 1. Block diagram of digital phase-locked loops.
where Y is the closed loop gain. This relation will work well if the input phase does not have large discontinuities. If modulated fringe pattern has narrow bandwidth (as usually required by direct or Fourier interferometry) s is set to a small value to obtain good noise rejection. If modulated fringe pattern has a wide bandwidth, s should be sub- stantially higher in order to track the wider band signal. If s is small, then it itself acts as a low-pass
D. Ganotra et al. / Optics Communications 217 (2003) 85-96 87
filter and no additional filter is required. Thus Eq. (5) can be expressed as
, = -x</>e(x). (6)
dx
Irradiance in a fringe pattern can be expressed as I(x) = a{x) + b(x) cos[2nfx + (f>0(x)], (7)
where aðxÞ is the background illumination (DC grey level) and bðxÞ is related to fringe amplitude.
It is assumed that the variations of aðxÞ; bðxÞ and /0ðxÞ are slow compared with variations intro- duced by 2pf.
In Eq. (1) the high pass filtered input signal and the DCO output signal (reference signal) were as- sumed to be of unit amplitude with zero back- ground illumination for mathematical simplicity.
In real life situation, the images captured by the camera when digitised by the frame grabber are generally 256 grey levels. They cannot be rescaled to unit amplitude because of many reasons. First, all the objects or planes over which the fringes are projected may not be perfect diffusers. Moreover, in crossed axis geometry the reference plane can- not be conjugate to the camera plane and the grating plane simultaneously. This also brings some illumination variance across the captured image. We tend to reduce it by increasing the contrast of the captured image but this cannot be carried too far as it would mean a loss of infor- mation at the source itself.
The derivative of the signal with respect to x acts as a high pass filter and mostly eliminates aðxÞ. In practice a high pass filter is as simple as the partial derivative of irradiance. Another fil- tering operation is performed by the close loop gain Y. A low value of s acts as low pass filter.
The value of s should be less than one for stability consideration. It is set to small value for a good noise rejection. If modulated fringe pattern is a wide band, it should be substantially higher in order to track the wider band phase signal that modulates the interference fringes. s may be re- duced after each iteration to improve the signal to noise ratio [2]. Thus
dIðxÞ dx
can be substituted instead of /eðxÞ in Eq. (6). As- suming that the PLL behaviour will not be affected by the amplitude variation at its input, and con- sidering bðxÞ as constant,
dI ðx
dx dxx
In discrete form
sin \2nfx
x sin[2nfx
(8)
(9) Eq. (9) is the basic iterative equation for the first- order PLL. The above iteration is carried in for- ward and backward directions through the same line across the image of projected fringes to re- cover the incorrect phase as it takes certain itera- tions before the phase settles to the correct values.
In backward scanning, the last term calculated by using Eq. (9) is taken as the initial condition
x sin[2nfx þ / ð x þ 1)]. ð10Þ
sin[27:/x
It is assumed that the phase change introduced by the object is smooth and continuous in both coordinates. Apart from the limitation on back- ground illumination, and phase modulation, cer- tain other factors for digital simulation should also be taken into account, e.g., the carrier frequency should be at most one-third of the maximum fre- quency represented by the resolution of the digi- tised image. For simulations, the fringe pattern extent needs to be increased by two pixels, and af- terwards the first part, which contains the unwanted portion included for initialisation is dropped.
2.2. Second-order PLL
As mentioned in the introduction, second-order PLL to demodulate fringe patterns in profilometry was presented by Gdeisat et al. [4]. Consider the irradiance distribution in a fringe pattern as given by Eq. (7), taking bðxÞ = 1, and differentiating w.r.t. x to eliminate aðxÞ,
dIðx Þ
dx
— sin[27:/x ð11Þ
Also let the reference signal (DCO output signal) be mðxÞ = cosð2pfx þ /ðxÞÞ: ð12Þ Thus, the multiplier output (error signal) will be the product of Eqs. (11) and (12)
4>e(x) = &in[2nfx + </>0(x)] cos[2nfx + </>(x — 1)]
dIðxÞ
dx • cos[2n/x þ /ðx - 1)]. ð13Þ The change of /ðxÞ in proportion to /eðxÞ can be expressed as a second-order differential equation d2/
Here s1 and s2 are coefficients of the digital filter.
This equation can be split into two first-order differential equations as
ð15Þ ð16Þ dx
dsðxÞ dx
Eqs. (15) and (16) in discrete form can be ex- pressed as
<i>{x)-<f>{x-\)=xl<i>e{x)+s{x), (17) s(x)-s(x-l) = T2fa(x), (18) known as characteristics equations. The values of the coefficient of the digital filter determine the performance of PLL against noise. After substi- tuting /eðxÞ, Eqs. (17) and (18) can be expressed as
<K* + i)-</>(*)
= T l(/(x - 1) - / ( x ) ) COS[2T:/X + 4>(x)] + s(x + 1), ð19Þ sðxþ l)-s(x)
= x2(/(x - 1) - /(x)) cos(2n/x + <j)(x)). (20) In simulations, fringe intensity distribution across each line needs to be scanned more than once, often with a backward pass to extract the phase more accurately. In such a case, the back- ward pass equations can be similarly formulated as
</>(*)-</>(*+1)
cosð2pfx þ /ðx þ
s(x)-s(x+l)
= x2(/(x - 1) - /(x)) COS(2T:/X + 4>{x + 1)). (22) Eqs. (19)-(22) are iteratively used to recover the phase from the image of the irradiance of a fringe pattern.
3. Experiment
When a grating pattern is projected onto an object's surface, the grating is perturbed according to the topography of the object. The perturbed grating, a 2-D pattern, which carries 3-D infor- mation of the object, is then recorded. The ex- perimental set-up for 3-D measurement is shown in Fig. 2. We estimated the surface of a pyramidal object, whose footprint is a square with an area of 81 cm2, and height of 5 cm. The set-up parameters such as grating spacing (0.6 mm), light source (slide projector), and the object size (pyramid with a square base of side 9 cm and height 5 cm) were kept approximately same as used by Cuevas et al.
[3]. The grating was projected over three different reference planes whose positions in space were known. During measurement, we maintained the relative positions of the object, grating and CCD camera by moving projection system in the direc- tion of the projecting light axis so that the imaging plane of the grating scans over the entire extent of the object.
The grating images were processed by second- order PLL, first-order PLL and FTP-based algo- rithms to calculate the corresponding phase planes. The theory of second- and first-order PLL
grating
reference plane l /
ð21Þ Fig. 2. Set-up for 3-D measurement (crossed axis geometry).
D. Ganotra et al. / Optics Communications 217 (2003) 85-96
has already been given in Section 2. In Fourier transform profilometry [10,11] the intensity pat- terns projected over an object and over a reference plane are Fourier transformed numerically on a computer. Their first-order spectra are filtered and multiplied element by element. The imaginary part of the log of the above product gives the phase change between the object and reference plane. If PLLs are used to calculate the phase, then it is also necessary to know the spatial period (in terms of pixels) of the projected fringes. We have used Fourier transform of the image of the grating to calculate this period.
Once the phases of the three reference planes are known, the neural network is trained using these phases, and the corresponding spatial coor- dinates as the input. The position of the reference plane is taken as the desired output of the network.
Analysis of the second- and first-order PLL was done at this stage. Parameters used such as closed loop gain and spatial period were varied to observe the changes in the corresponding phase planes.
They were also compared with the phase planes calculated using FTP. A raster scan of this phase map was fed to the neural network to calculate the corresponding height.
The set of images of the projected fringes cap- tured by the CCD camera were of the size 256 x 340 pixels with 256 grey levels. The frame grabber card was adjusted such that the entire image had almost uniform contrast. This helped to make the procedure image illumination invariant.
The 256 grey levels were rescaled to range )1 to + 1. The adjacent pixels were subtracted from each other (Eq. (11)) to filter the variation caused by background illumination. This is similar to differ- entiation of the fringe pattern with respect to the x coordinate.
The equations of PLL require the spatial pe- riod of the projected grating to be known. One method to obtain the spatial period is to count the number of pixels between the successive maxima of the intensity pattern observed. An- other way is to count the number of pixels be- tween the peaks and the center in the plot of modulus of 1-D Fourier transform of the pro- jected fringes. This can be done by using the following relation:
P = (N/2)
-1 + r
N ð23Þwhere N is the number of x-pixels in the projected fringes image, I the number of pixels from the centre to the peak of the Fourier transform, and p is the calculated spatial period.
Different values of s; s1, and s2 were chosen and the corresponding results are discussed in Section 5. Cuevas et al. [3] used closed loop gain s = 0:1 for the first-order PLL. Servin et al. [2] stated that a practical value for the closed loop gain s may be in the range 0.001-0.1. Servin and Rodriguez-Vera [1] stated this range as 0.001-0.2, but the choice of the loop gain also depends upon the range of the captured irradiance matrix. Gdeisat et al. [4] used values of s1 as 0.07, 0.15, 0.2 and s2 = Xj/2. For the first-order PLL, Eqs. (9) and (10) are itera- tively used to extract the phase information. The scanning sequence used is same as mentioned by Cuevas et al. [3]. For the second-order PLL Eqs.
(19)-(22) are iteratively used to extract the phase information. Forward and backward scanning technique is used in this algorithm. Here the same line is scanned twice [4]. When the scanning di- rection of the fringe pattern was changed during iterations for phase calculations, the signs of sðxÞ and /ðxÞ were changed while giving them as initial values for the next iteration. This improves the tracking ability of the second-order PLL and helps the loop to remain locked when the scanning di- rection is reversed.
For FTP, the images of the projected fringes which are in the form of 256 grey level matrix are normalised to the range 0-1. The image of the projected fringes on the reference plane at 0 cm is used as the base plane here.
4. Use of neural network
Neural networks have been used for profilom- etry by Cuevas et al. [12] who proposed a cali- bration technique using multilayer NN that can calculate directly the local height gradient from the local irradiance of the fringe pattern. The geometrical equations for phase to height conver- sion can be applied where a pinhole camera is
Fig. 3. Phases recovered for the three reference planes and the object using: (a) first-order PLL, (b) second-order PLL, and (c) Fourier transform profilometry, respectively. These phases are recovered from same set of projected fringes images. Note that for clarity only one row (x-pixel) passing through the object is shown (spatial period of the projected fringes 10.0).
(C)4 3 2 1
| o
0 50 100 150 200 250 300 x-pixel
Fig. 4. Phases recovered for the three reference planes and the object using: (a) first-order PLL, (b) second-order PLL, and (c) Fourier transform profilometry, respectively. These phases are recovered from same set of projected fringes images. Note that for clarity only one row (x-pixel) passing through the object is shown (spatial period of the projected fringes 11.3).
^ ^ 5 cm ^ |
\ 2.5 cm ^^ Z I ^><
object /
D. Ganotra et al. / Optics Communications 217 (2003) 85-96 91
considered and the test object is located far away from the projector-camera system. Ratnam et al.
[13] have proposed a technique of integrating the moire method and neural networks to recognise and classify shapes of continuous surfaces that vary only slightly in geometry. They introduced neural network because in their shape classifica- tion problem, the measurement becomes repetitive and tedious, and therefore becomes impractical in most cases. Pham and Aslantas [14] used multi- layer perceptron (MLP) neural network trained by the backpropagation algorithm to compute the depth of objects from their blurred images. The derivative images of blurred edges were used for training the MLP to determine the distance of the edge from the camera lens.
To accomplish the calibration process, i.e.
conversion of the phase (in radians) to physical measurements, it is required to know the pa- rameters of the optical experimental set-up and this is a difficult task. In profilometric case, where a projected fringe pattern is analysed, the conversion process should use the optical set-up parameters such as focal distance, reference plane location, frequency of projected fringe pattern, camera-fringe projector angle, among others. In addition, the geometrical aberrations such as spherical, coma, astigmatism and radial distor- tion of the optical components should be con- sidered. These have non-linear effects on the recovered depth. Thus the phase-to-depth con- version becomes a problem if formulae from
^ ^ J M A M J H T ^
^^JJy/J^Jlt^J^jAjAl^^M^
. .0 50 100 150 200 250 300 , ..0 50 (c) xpw (d)
Fig. 5. Effect of change of the closed loop gain (S) for first-order PLL: (a) S = 0:01, (b) S = 0:2, (c) S = 0:5, and (d) S = 0:65.
Fig. 6. Effect of change of the closed loop gain ðs1) for second-order PLL: (a) s1 = 0:01, (b) s1 = 0:2, (c) s1 = 0:5, and (d) s1 = 0:65.
geometrical optics are used directly. Neural net- work-based phase-to-depth conversion seems to provide promising solution to this problem. Once the phases of the three reference planes are known, the neural network is trained using phase and corresponding spatial x, y coordinates as the input, and the position of the reference plane as the desired output of the network. Seventy-five points were randomly sampled during training.
We used a feedforward backpropagation neural network with three input, two hidden, and one output neurons. The learning rate was kept 0.05, and 1500 epochs were used for training. Both the input and output were rescaled to the range )1 to +1.
5. Results
The images of the projected grating were cap- tured with the CCD camera. The distance and the angles (Fig. 2) were varied to change the spatial period of the captured fringes. The PLL algorithm to recover the phase was applied to these images.
The iterations were applied for different closed loop gains, and spatial periods, etc. The object used was in the shape of a pyramid with a square base of side 9 cm and height 5 cm.
It was observed that the second-order PLL re- covers larger phase changes correctly compared to the first-order PLL. Figs. 3(a)-(c) show the phases recovered for the three reference planes and the
D. Ganotra et al. / Optics Communications 217 (2003) 85-96
Fig. 7. Effect of change of spatial period. (a) Spatial period used is lower than that calculated by Fourier transform, (b) spatial period used is equal to the one calculated by Fourier transform, (c) spatial period used is more than that calculated by Fourier transform.
object using first-order PLL, second-order PLL, and FTP, respectively. These phases are recovered from the same set of projected fringe images. Note that for clarity, the plot of only one row (x-pixel) passing through the object in the shape of a pyr- amid is shown. The three reference planes were placed at 0, 2.5, and 5.0 cm. Note also that the phase change due to the object did not even reach the phase plane due to the 2.5 cm plane in the case of first-order PLL (Fig. 3(a)) whereas it did cross the phase plane due to the 2.5 cm plane in the second-order PLL (Fig. 3(b)). Thus, we can con- clude that the second-order PLL recovers larger phase changes correctly compared to the first-or- der PLL. Changes in the recovered phase can also be observed with changes in closed loop gain.
It can also be observed (Figs. 3(a) and (b)) that the values of the phases of the reference planes calculated by the first-order PLL and the second- order PLL are same. The values recovered with the FTP method (Fig. 3(c)) leads by a phase of P = 2 over the above-calculated phases. Also note that the phase calculated using FTP method wraps in the range — p to p.
phase ðFTPÞ = phase ðPLLÞ þ p : ð24Þ For wrapped phases, the above relation still holds true if integral multiples of 2P are added/sub- tracted logically. Similar results as discussed above are shown in Fig. 4. The spatial period of the projected fringes is 10 pixels in Fig. 3 and 11.33 pixels in Fig. 4. Comparing Figs. 3(c) and 4(c) we find that the position of the point of wrapping shifts with the change in spatial period.
With the increase in the value of the closed loop gain Y, the noise in phases recovered in the first- order PLL also increases (Figs. 5(a)-(c)). With smaller loop gain value, large phase changes are calculated as relatively smaller phase changes.
Thus a higher loop gain recovers the correct phase values, though corrupted with high frequency noise. Comparing Fig. 5(d) with Fig. 5(a)-(c), we notice that large values of the closed loop gain do not affect the positions of the phases of the refer- ence plane surfaces but add a constant phase to the surfaces in which there is some change of the phase due to the presence of the object. Also the phase
becomes noisy with increase in the closed loop gain. The cause of this high frequency noise can be
• the finite pixelation of the assumed sinusoidal projected waveform which manifests in terms of digital artefacts;
• the forced contrast saturation of the captured fringes with the CCD to make them illumina- tion invariant which makes them square wave fringes instead of sinusoidal;
• and PLL algorithm in digital form which as- sumes the derivative limit as one pixel.
With the increase in the value of 'TI' the high frequency noise in the phases recovered using second-order PLL also increases (Figs. 6(a)-(c)).
We have used 'x2' equal to s21=2 as done by Gdeisat et al. [4]. The problem of constant phase addition as in first-order PLL did not occur in second-order PLL even for large value of closed loop gain (Figs.
5(d) and 6(d)).
It was observed that if the spatial period (1/f) used in Eqs. (10), (21) and (22) is smaller than the calculated spatial period, then the slopes of the phase plane are negative. Whereas, if its value is more, then the slopes of the phase planes are po- sitive (Fig. 7). The spatial period calculated using Fourier transform of image of grating is 9.84
pixels. The corresponding plot of the phase plane is shown in Fig. 7(b). The plots using spatial pe- riod as 9 and 10 pixels for the same set of projected fringe images are shown, respectively, in Figs. 7(a) and (c). These plots are for the second-order PLL iterations, but the effect was also observed in first- order PLL. We may conclude that the period where the plot is flat is the best spatial period that fits into the calculation.
The reconstructed object is shown in Fig. 8.
Here Fig. 8(b) shows the reconstructed pyramid using first-order PLL. Fig. 8(c) shows the recon- structed object in the shape of a pyramid by using second-order PLL phase planes and Fig. 8(d) using Fourier transform profilometry. Fig. 8(a) shows the approximate profile of pyramid measured mechanically. It can be observed that the sharp- ness of the edges very much degraded in the case of results obtained by FTP. The PLL-based phase recovery method definitely performs better than the FTP. The object reconstructed using second- order PLL does not seem to be better than that reconstructed by first-order PLL. The mean square error between Figs. 8(a) and (b) is 0.22 and be- tween 8(a) and (c) is 0.16. Also the regions of larger phase change show saturation, i.e. the neu-
Fig. 8. Reconstructed object. (a) Profile of the mechanically measured object, (b) object reconstructed using first-order PLL and neural network, (c) object reconstructed using second-order PLL and neural network, and (d) object reconstructed using FTP and neural network.
D. Ganotra et al. / Optics Communications 217 (2003) 85-96 95
300
Fig. 9. Side view of one row of the object reconstructed using:
(a) first-order PLL and neural network, (b) second-order PLL and neural network, (c) FTP and neural network.
ral networks have transformed phases corre- sponding to upper regions of the pyramid as al- most of the same height. Side views of one of the rows of the reconstructed matrix for each methods are shown in Fig. 9. Note that with the first-order PLL, the position of the object appears to be shifted. This shift depends upon the scanning di- rection during iteration.
Servin et al. [1] used image of size 120 x 100 pixels with 256 grey levels. Their algorithm took 1.5 s on a 33.0 MHz 486 IBM PC. To calculate phase object Servin et al. [2] iterated PLL loop to obtain phase estimation for 50 iterations, which took about 3 min on a 33 MHz 486 PC. Their images were of size 120 x 100 pixels with 256 grey levels. Gdeisat et al. [4] demonstrated that their algorithm has the ability to operate at 25 images/s.
They used image processing card to grab the fringe pattern and to demodulate them, each image being 256 x 256 pixels. They mentioned that the use of lookup table can reduce the time to calculate the sinusoidal function and reduce the execution time of the algorithm by ^400%. In our simulations the time required was about 12 s to calculate the phases of a 256 x 340 pixel image using second order PLL on Matlab software on IBM PC with Pentium III processor. For first-order PLL the time required was about 9.6 s. The phase calcula- tion of the similar matrix using Fourier Transform profilometry took 8.9 s. Training the NN required 17.92 s.
6. Conclusion
Fringe pattern demodulation with phase-locked loops is extended from first-order to second-order and is analysed using neural networks. The algo- rithms and neural network were tested experi- mentally with fringe patterns. Such a test provides a better understanding of the methods and the underlying processes. Also it provides an under- standing of the effect of change of the closed loop gains, and spatial periods. The reconstruction of the objects using neural networks shows the ob- jects in correct perspective without the knowledge of the optical parameters. The training images should be selected to fit the requirement of a spe-
cific problem to which the neural network is to be applied, and the projected grating should have appropriate fringe carrier frequency. Servin et al.
[2] reported that the phase topography was not resolved very satisfactorily due the limited resolu- tion given by video frame grabber. They men- tioned that this drawback may be reduced if a frame grabber with higher resolution is used to support a higher frequency of the projected Ron- chi ruling. The resolution of measurement of the object height is dependent on the projected grating pitch and the direction of projection. The mea- surement range can be flexibly adjusted to fit the size of the specimen if laser diode light source and the lateral shearing interferometer is used [15]. A computer controlled grating can offer advantage [16] of altering at will the grating pitch. The problem can be overcome to some extent by pro- jecting a smaller spatial period grating. Projection of a dual spatial period grating can also be a possibility [17]. The position change in fringe processing can be substantially avoided by the use of second-order PLLs instead of first order PLLs (Fig. 9). Grating intensity variation of any form can be generated to match the object surface, thus optimising the dynamic range of measurement. If neural network is trained using the grating pitch and grating intensity variation as additional in- puts, the system can be used for a much wider range and robust measurements.
Acknowledgements
One of the authors D. Ganotra acknowledges financial support from the Council of Scientific and Industrial Research (CSIR), Government of India.
References
[1] M. Servin, R. Rodriguez-Vera, J. Mod. Opt. 40 (1993) 2087.
[2] M. Servin, R. Rodriguez-Vera, D. Malacara, Opt. Lasers Eng. 23 (1995) 355.
[3] F.J. Cuevas, M. Servin, R. Rodriguez-Vera, Opt. Com- mun. 163 (1999) 270.
[4] M.A. Gdeisat, D.R. Burton, M.J. Lalor, Appl. Opt. 39 (2000) 5326.
[5] N.A. Ochoa, R. Rodriguez-Vera, M. Servin, F.M. San- toyo, Opt. Commun. 117 (1995) 213.
[6] S.M. Pandit, D.P. Chan, Appl. Opt. 38 (1999) 6540.
[7] R.A. Jarvis, IEEE Trans. PAMI-5 (1983) 122.
[8] S. Haykin, Neural Networks; A Comprehensive Founda- tion, Macmillan, NJ, 1994.
[9] D.W. Patterson, Artificial Neural Networks, Theory and Applications, Prentice-Hall, Singapore, 1996.
[10] M. Takeda, K. Mutoh, Appl. Opt. 22 (1983) 3977.
[11] X. Su, W. Chen, Opt. Lasers Eng. 35 (2000) 263.
[12] F.J. Cuevas, M. Servin, O.N. Stavroudis, R. Rodriguez- Vera, Opt. Commun. 181 (2000) 239.
[13] M.M. Ratnam, C.P. Lim, A.N. Khor, Opt. Eng. 40 (2001) 2036.
[14] D.T. Pham, V. Aslantas, Pattern Recogn. 32 (1999) 715.
[15] M. Chang, K.-H. Lin, Opt. Eng. 40 (2001) 2057.
[16] Y.Y. Hung, L. Lin, H.M. Shang, E.G. Park, Opt. Eng. 39 (2000) 143.
[17] Y. Hao, Y. Zhao, D. Li, Appl. Opt. 38 (1999) 4106.
