Noisy fringe pattern demodulation by an iterative phase locked loop

Optics and Losers in Engineering 23 (1995) 355-365 Copyright 0 199s Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143-8166/95/$950 0143-8166(%)ooo41-0

ELSEVIER

Noisy Fringe Pattern Demodulation by an Iterative Phase Locked Loop

M. Servin, Centro de Investigaciones

R. Rodriguez-Vera

& D. Malacara

en Optica A.C., Apartado Mexico

Postal l-948, 37000 Leon, Gto.,

ABSTRACT The phase locked loop (PLL) technique applied to demodulate two-dimensional carrier-frequency fringe patterns has been developed recently. Here we present an extension to the basic PLL scheme to demodulate noisy fringe patterns. This modified technique estimates the phase in the fringe pattern iteratively; that is, the first wavefront estimation is done using a flat reference phase and the second iteration takes the demodulated phase found in the first iteration as the new reference. The third demodulating iteration uses the second phase estimation as the reference and so on, until further changes in the detected wavefront fall below a predefined threshold. During the iterative process the bandwidth of the iterative PLL system is gradually decreased to improve the signal-to-noise ratio of the detected phase as well as to resolve noise-generated phase inconsistencies.

INTRODUCTION Detecting the phase of interferograms may be carried out using several techniques. Among the best known phase detecting methods are the phase stepping technique,’ direct interferometry**” and the Fourier method.4,5 The phase stepping method is a temporal technique and requires typically three or more fringe patterns. The phase at each resolvable image pixel is found independently of the surrounding pixels. On the other hand, the direct and Fourier techniques require only one These methods are called spatial carrier-frequency interferogram. 355

M. Servin, R. Rodriguez-Vera,

356

D. Malacara

techniques because the phase at each pixel is found with reference to the surrounding ones. The techniques used for demodulating carrierfrequency interferograms were borrowed and adapted to interferometry from well-known demodulating techniques used in electronic communications engineering. The direct interferometric technique was adapted to fringe analysis from quadrature synchronous demodulation. The Fourier technique is the interferometric version of early FM radios. Recently, another well-established technique used in communication engineering for phase detection, named phase locked loop (PLL), was applied to demodulate carrier frequency interferograms.’ The PLL is also a spatial technique and its main feature is its faster phase detecting capabilities. The PLL technique also has the unique feature that the unwrapping process is implicit within this phase tracking loop. The PLL technique is extended in this paper to make it more accurate and robust. This of course requires more computing time but it may pay off, given that many noise-generated phase inconsistencies may be removed within the iterative process. The iterative PLL presented in this work starts with an estimation of the wavefront as reported by Servin & Rodriguez-Vera.’ Afterwards, this initial phase estimation is used as a reference to improve the detecting estimation by correcting discrepancies that the new estimation may have. In turn, the third estimation takes the second estimation as the new reference to improve, if possible, the detected wavefront. This process is continued until further changes in the detected wavefront fall below a predefined threshold. The first section describes the basics of the PLL in a somewhat more comprehensive way than described in Ref. 6, then the extension to an iterative PLL is described. Finally, experimental results are presented to show the advantages of using this new demodulating scheme.

BASIC PLL The basic components of a first-order PLL are shown in Fig. 1. The non-linear equation for a first-order PLL used to demodulate carrier frequency phase modulated signals is:

&x)=

-Tl

~0s

[oox+ 4(x)1 sinbhx

+ &)I

dx

-r

= -- 2K _.x (sin [24,x + 4(x) + +(x)1 + sin16(x)-

I

z

4(X)1)

dX (1)

357

Noisy fringe pattern demodulation

Fig. 1.

Basic phase locked loop (PLL) building blocks.

where r is the closed loop gain. We can see that eqn (1) is formed by two terms, a high frequency term and a low frequency one. The high frequency term is average out by the integral operator (low-pass filtered). As a consequence only the last term in eqn (1) remains. Now, assuming that the phase loop is operating in lock the phase error will be small so the following holds:

We can therefore write the linearised first-order phase locked loop as:

differential

-_x F=;[m(x) - c&r)]

equation

for a

(3)

As can be seen from eqn (3) the rate of change of the voltage controlled oscillator’s (VCO’s) phase is proportional to the phase error. In consequence the VCO phase will follow the input phase continuously whenever the input phase does not have large discontinuities. The linearised version of the PLL as stated in eqn (3) has been written to show how the estimated phase d(x) follows the form of the unknown phase 4(x). Another insight that can be obtained from writing the linear PLL approximation is to analyse its steady-state frequency response. Taking the Fourier transform of eqn (3) we may obtain the magnitude of its transfer function as:

Equation

(4) gives a cutoff frequency

equal to

0 cutoff= 5

(5)

In other words the parameter r controls the cutoff frequency and the stability of the PLL system. The actual value assigned to r will depend

358

M. Servin, R. Rodriguez-Vera, D. Malacara

mainly on the frequency spectrum of the modulating phase object. In other words, if the modulated fringe pattern is narrow band (as usually required by direct and Fourier interferometry), r is set to a small value to obtain good noise rejection. On the other hand, if the modulated fringe pattern is wide band, r should be substantially higher in order to track the wider bandwidth phase signal that modulates the interference fringes. In consequence, the PLL implicitly smooths the demodulated phase signal as well as filtering out the doubled carrier frequency obtained in eqn (1). A practical value for the closed loop gain r in the non-linear equation (eqn (1)) may be in the range 0401 to O-1. The linear analysis of the PLL was given to show in a simple manner interesting properties behind this phase detector; but the actual differential equation that must be wired or programmed in order to demodulate an interferogram is the non-linear PLL differential equation, eqn (1). Given that we are dealing with an interferogram, the fringes in a carrier frequency interferogram can be expressed as:

4(x, y) represents the desired phase information, a(_~,y) the background illumination and b(x, y) the amplitude modulation of the interferogram. In most cases a@, y), b(x, y) and 4(x, y) vary slowly compared with the variation introduced by the spatial carrier angular frequency oO. The background illumination a(x, y) can be filtered out using a high-pass filter. In practice, a high-pass filter as simple as the partial derivative of this irradiance with respect to the x coordinate will suffice to eliminate this unwanted signal almost entirely. High-pass filtering of a(x, y) is also convenient because the PLL’s low-pass filter will then have to reject only an unwanted signal with twice the interferogram’s carrier frequency, which is easier. Implementing this, the dynamic tracking system as given by eqn (1) modified to demodulate the phase of the interferogram is: a8(4 Y) =

ax

--z

%(x9 Y)

ax

sin

b”X + &x7Y )I

(7)

Assuming that the interferogram irradiance is digitised on an image of L X L pixels, we need a discretised version of eqn (7) which forms the updating rule for the interferogram’s phase determination. A discretising process of eqn (7) to deal with a digitised fringe pattern was proposed by Servin & Rodriguez-Vera.‘j

359

Noky fringe pattern demodulation

ITERATIVE

PLL

In the last section the main aspects of the PLL demodulating scheme were reviewed. In this section it is discussed how the basic PLL is modified to deal with noisy fringe patterns. The basic idea is already known. It consists of estimating the unknown phase which modulates the fringe pattern iteratively, i.e. eqn (7) is modified in order to demodulate the fringe pattern iteratively. For the first iteration the reference phase is a plane and the first phase estimation is achieved as presented in the basic PLL section. For the second iteration the outcome of the first iteration is used as the new phase reference and the PLL will therefore demodulate only the differences between the first estimation and the observed fringe pattern. For the third iteration the estimated phase found in the second iteration is used as the new reference and so on, until no further changes occur from one iteration to the next. During this iterative process the bandwidth of the PLL controlled by the parameter z is decreased continuously to increase the signal-to-noise ratio of the detected phase. Applying these ideas to the basic PLL (eqn (1)) the non-linear differential equation of the iterative PLL is:

&(x7 Y) dX

= -z cos [WG + 4(x, y)] sin [w,x + $F-‘(x, y) + $5(x, y)]

with the initial condition given by: I&

y) = 0

(9)

where k is the current iteration. As explained later on, function c$-‘(x, y) is the low-pass filtered version of the estimated phase $‘-‘(x, y) at time (k - 1); function 6:(x, y) is the estimated error between the unknown phase 4(x, y) and its estimation c$-‘(x, y). Assuming that the term at twice the carrier frequency has been filtered out, eqn (8) can be rewritten as G4X7Y)

ax

r

= 5 sin[4(x, Y> -

+‘(x,

Y>

- 8% ~41

the error between the interesting phase 4(x, y) and its estimation (k - 1) iteration is 44%

Y> = 4(x,

Y) - i@‘k

Y)

(10) at the (11)

Replacing eqn (11) in eqn (10) and assuming that the PLL is already

360

M. Servin, R. Rodriguez-Vera,

D. Malacara

working in lock we may approximate the sine function by its argument in order to obtain the linear model for the iterative PLL as:

d4~~y)=$#J.(x, y) - &(x, y)]

(12)

As seen from eqn (12) using the iterative PLL the current phase error estimation c$~(x,y) will demodulate the differences that may still exist between the observed phase 4(x, y) and the low-pass filtered estimated phase r$fk-l(x, y) at the last iteration. Finally the phase estimation at iteration k is:

where Q is the convergence rate which should be less than one to ensure stability of eqn (13). After each iteration the r constant in eqn (8) may be reduced to improve the signal-to-noise ratio in the estimated phase. If the estimated phase converges towards the true phase 4(x, y) the bandwidth of the remaining error eqn (11) will decrease. This bandwidth reduction of the detected phase error is valid whenever the estimated phase c$~(x,y) approaches monotonically the real phase 4(x, y), otherwise the algorithm risks settling down in a spurious minimum. Assuming that only smooth and continuous wavefronts 4(x, y) are expected, we may remove many noise-generated phase inconsistencies by passing the estimated phase at time k (eqn (13)) by a low-pass filter. In our case this low-pass filter was simply a 3 X 3 averaging window h(x, y), i.e. 4$(x, Y) = w,

y) * * @(x9

y)

(14)

where the symbol * * represents double convolution. This filtered estimation is in turn used by eqn (8) and the iterative process is continued until no further changes are observed in the estimated phase 8”(x9 Y 1. Unfortunately, using deterministic systems such as the one presented in this work, we are always at the risk of becoming trapped in a spurious solution. We may use a stochastic approach such as simulating annealing in order to reduce the risk of spurious solutions, but unfortunately these stochastic approaches are time consuming. In many cases, however, deterministic systems end up in a short period of time with solutions good enough to be considered as valuable estimations.

Noisy fringe pattern demodulation

361

The actual form used by the iterative PLL algorithm proposed herein to demodulate carrier-frequency fringe patterns is;

Function g(x, y) is the fringe pattern irradiance given by eqn (6). Equation (15), along with eqns (13) and (14) form the proposed iterative phase estimator. The discretized version of this equation also follows the one proposed in Ref. 6.

EXPERIMENTAL

AND COMPUTER

SIMULATION

RESULTS

Computer simulations were carried out to test quantitatively the iterative PLL scheme. The numerical experiment consists of demodulating the following computer-generated interferogram: g(x, y) = 128 + 127 cos [2O~rx+ 5.4~~’ + noise (-lr/2,7r/2)],

(p I

1) (16)

where p* = (x2 + y’), and noise (-n/2, n/2) is uniformly distributed white noise in between -n/2 and n/2. The interferogram is shown in Fig. 2. The demodulation was achieved using a discretised version of eqn (15) and using eqns (13) and (14). The first iteration is shown in Fig. 3. It can be seen how the noise-generated phase inconsistencies have severely corrupted the first estimation. This first estimation is the demodulation produced by the one-step PLL method described by Servin & Rodriguez-Vera.” After the 50 iterations which took about 3 min in a 33 MHz 486-PC, we obtained the phase estimation shown in Fig. 4. The iterative process was performed while reducing the r constant from O-02 to 0.005 to improve the signal-to-noise ratio of the estimated phase. We can now see how the noise-generated phase inconsistencies have been removed and the estimated phase now looks continuous. The root mean square (RMS) phase error between the current phase estimation and the real noise-free phase given by eqn (16) was calculated from l/2

RMS error =

fi i=l

- @m Y)12

MAY)

N-l

(17)

and Fig. 5 shows how this RMS error decreases from near 5 rad down to 0.42 rad.

362

M. Seruin, R. Rodriguez-Vera,

Fig. 2.

D. Malacara

Computer generated noisy interferogram.

Finally, Fig. 6 shows a Ronchi ruling projected over a dummy face using 120 X 100 pixel and 256 grey level resolution. In order to estimate its topography, the experimental set-up of the technique named moire profiioietiy &as used.? Figure 7 shows the estimated topography at the

Fig. 3.

First iteration of the iterative PLL.

Noisy fringe pattern demodulation

Fig. 4.

Smooth phase determination

363

given by the iterative PLL proposed.

first iteration. It can be observed that the shadows produced by the rough surface of the dummy face create phase inconsistencies that are not resolved by the first iteration. However, after 30 iterations these phase inconsistencies are resolved and we obtain the expected smooth face topography shown in Fig. 8. Of course the phase topography was not resolved very satisfactorily but this is due to the limited resolution given by our video frame grabber. This drawback may be reduced if a frame grabber with higher resolution is used to support a higher frequency of the projected Ronchi ruling.

CONCLUSION A new iterative method for phase determination of fringe patterns using a PLL has been presented. This iterative phase demodulating technique was applied to noisy fringe patterns phase modulated by RMS error in radians

Fig. 5.

RMS error between the noiseless phase given by the argument the iterative PLL estimation.

of eqn (16) and

M. Servin, R. Rodriguez-Vera,

Fig. 6.

D. Malacara

Ronchi ruling projected on a dummy face.

Fig. 7.

Fig. 8.

First estimation

of the iterative PLL.

Estimated phase after 30 iterations.

Noisy fringe pattern demodulation

365

smooth continuous functions. The main advantage of the method is that it can deal with noise-generated phase inconsistencies. In this manner the range of applications of the PLL is extended to cover highly noisy fringe patterns which can be demodulated in a direct way. Further research can be done in extending this iterative phase determination scheme to other carrier-frequency fringe-pattern analysis techniques, such as Fourier and direct interferometry. We believe that these traditionally one-step phase estimators may improve their performance by the use of some kind of iterative phase demodulation similar to the one presented in this work.

ACKNOWLEDGEMENT We would like to acknowledge the support of the Consejo National de Tecnologia (CONACYT) during the development of this work.

REFERENCES 1. Creath, K., Phase measurement interferometric techniques. In Progress in 2.

Optics, Vol. XXXVI, ed. E. Wolf. Elsevier, Amsterdam, 1988. Ichioka, Y. & Inuiya, M., Direct phase detecting system. Appl. Opt., 28

(1989) 3268-70. 3. Womack, K. H., Interferometric phase measurement using spatial synchrenous detection. Opt. Engng, 23 (1984) 391-5. 4. Takeda, M., Ina, H. & Kobayashi, S., Fourier transform methods of fringe pattern analysis for computer-based topography and interferometry. J. Opt. Sot. Am., 72 (1982) 156-60. 5. Roddier, C. & Roddier, F., Interferogram analysis using Fourier transform techniques. Appl. Opt., 26 (1987) 1668-73. 6. Servin, M. & Rodriguez-Vera, R., Two dimensional phase locked loop demodulation of interferograms. J. Mod. Opt., 40 (1993) 2087-94. 7. Gasvik, K. J., Optical Metrology. John Wiley, New York, 1987.