I5 September 1996
OPTICS COMMUNICATI~~K ELSEVIER
Optics communications 130 ( 1996) 7- 12
PLL interferometry on non-continuous domains Noe Alcal Ochoa, Manuel Servin, Ram& Rodriguez-Vera, Fernando Mendoza Santoyo Centro de Inoestigaciones
en Opticu, A.C., Apartado Postal I-948, Leh.
Gto., CP 37000, Mexico
Received 15 November 1995; revised version received 8 February 1996; accepted 27 Febroary 1996
Abstract The phase-locked loop technique (PLL) is a fast demodulating scheme recently adapted to carrier fringe pattern analysis. The main advantage of the phase-locked loop is that the phase extraction and the unwrapping process are performed simultaneously. One drawback of the PLL is its path dependent phase demodulation. This means that even noise free fringe patterns containing holes of invalid data cannot be successfully detected. The work presented here extends the validity of the phase locked loop, allowing its use with connected regions of valid fringe data having any shape. The extended method was tested and implemented numerically and experimentally.
1. Introduction The phase extraction from a fringe pattern containing a linear carrier may be realized in several manners. Two of these use Fourier transform [2] and spatial synchronous detection [3] methods. With the Fourier and synchronous methods, the borders of the fringe pattern are not well reconstructed, and a wrapped phase map is obtained that has to be later unwrapped. Recently Servin and Rodriguez-Vera [ 11 applied the phase-locked loop (PLL) method to demodulate linear carrier frequency fringe patterns. The PLL method has the advantage that the process of phase detecting and phase unwrapping are both achieved simultaneously. Furthermore, the system may be used with fringe patterns bounded by a finite pupil with minimum distortion of the detected phase at the boundary. However, the PLL algorithm, in its actual form, only demodulates fringe patterns that have continous domains and smooth frontiers. By masking the fringe pattern domain and modifying the
bound~y conditions of the PLL method, a modified algorithm is obtained, overcoming the mentioned drawbacks. The new and former PLL algorithms give the same results for closed regions bounded by smooth boundaries with no inside holes. In the following section the basic theory of the PLL technique is presented for reference, a detailed discussion may be found in Ref. [l], and in Section 3 the implemented modifications will be shown. Finally, the last two sections of this work will show the application of the extended PLL method to simulated and ex~rimental fringe patterns respectively.
2. Basic theory Let a fringe pattern be given by the equation 1(x,
y) =A(x,
~3~-~18/96/$12.~ Copyright 0 1996 Elsevier Science B.V. All rights reserved. Plf SOO30-4018(96)00185-X
Y) +B(x,
x cos[(6,( x,
Y> Y> + 4 x3
Y,l
’
(1)
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where A(x, y) and B(x, y) are smooth functions related to the dc-level and the fringe contrast, respectively. 4,(x, y) is a phase function related with parameters to be measured, and c(x, y) a carrier function. c may have any linear form: like the one used in this paper c(x, y) = wc,x, where w0 would be the carrier frequency, or c( x, y) = w0 x + o, y, with w0 and w, the carrier frequency along the x, and y axis respectively. In what follows, consider I( x, y) inside an area of size N X M, with x and y contained in the intervals 0 to N - 1 and 0 to M - 1, respectively (N and M are integer numbers). The PLL method establishes [ll that the phase 4,(x, y) may be calculated iteratively, performing: (i) a forward calculation along the x axis, 4,(x+
1, Y)
= 4,(x,
Y) -
X sin[ +,(
T[ I( x + 1, Y) - I( XT Y)]
x,
Y)
+
qx],
x=O,...,N-2,
(2)
where T is defined as the closed-loop gain, and is chosen to be small. The initial condition is taken as zero, that is 4,(0,
Y) = 07
(3)
and (ii) a backward axis, 40(x,
calculation
along
the same
x
Y)
=&(x+1, Xsin[+,(x+l,
y)-7[Z(x+l,
Y)-1(x,
Y)]
130 (1996) 7-12
process may be started from another row instead of the top one, and the forward-backward process may be reversed (backward-forward).
3. Modifications
to the PLL algorithm
In order to give a proper explanation to the modifications introduced in the PLL method, some useful concepts will be defined. A binary domain for a fringe pattern I( x, y) may be defined as F(x7
Y)
= 1 =
0
if(x,
y) ED(Z(x,
Y)),
otherwise,
(6)
where D(Z(x, y)) is the domain of valid fringe pattern data. The binary domain function @x, y) can be calculated using several methods, depending on the fringe nature. For the case presented here, it was determined by measuring the standard deviation inside a 5 X 5 pixel window. When this value was lower than a background threshold value (given by the user), the points were not considered as part of @x, y). This method may not be useful for carrier fringe data with strong noisy background because the standard deviation of the noise may be mistakenly detected as good modulation fringe data. A binary domain 8 is defined as connected if any two points p,( x, y> and p2( x, y> in 57(x, y> can be joined by a continuous line lying entirely within SZ. This includes, closed regions with holes. Fig. la
y)+wOx],
x=N-2,...,0
(4)
but now taking as an initial condition the last calculated value $,( N - 1, y> given by Eq. (2). Finally, to obtain the phase for all the fringe pattern the object is assumed smooth and continuous in both coordinates, and the starting phase value used for the next ( y + 1) line to be demodulated, is the last value calculated using Eq. (2) for the ( y) line, $,(O,
y+l)=$J,(O,
Y),
y=O,...,M-2. (5)
Note that the process described is applied from top to bottom and from left to right (forward) and then from right to left (backwards). However, the
a) connected
b) unconnected
Fig. 1. (a) Connected domain, because any two points can be joined by a line lying entirely within it. (b) Unconnected domain because it contains at least two points P, and P, that cannot be joined with a continuous line.
N.A. Ochou et al./ Optics Communicutions
shows an example of a connected domain. An unconnected domain is shown in Fig. lb. It is not connected because there are no continuous lines within g7( x, y) that join p, and pz. Due to the demodulation nature of the PLL, it is convenient to brake down the g domain into n horizontally connected domains. A domain is said to be horizontally n-connected if there are n disjoint subsets Z,, of 57(x, y) that, for any two horizontal points xn,, x,~ in Z’,,, can be joined by a horizontal line lying entirely within Z,,. Similarly, a vertical m-connected domain is defined if there are m disjoint subsets 7, of E’(x, y> that, for any two vertical points ym,, ym2 in V,, can be joined by a vertical line lying entirely within y;. Clearly the set G?(x, y> is the union of all the horizontal n-connected subsets Z,,, or all the vertical m-connected 2;T,, this is @(x,
y) = u Zn = u y;. n m
The frontiers of each horizontal n-connected subset are horizontal lines and those for the vertical m-connected are vertical. Consider for example the connected domain E’(x) y) of Fig. 2a. It could be separated in two subsets, 3, and Z1 (Fig. 2b), where each one is a l-connected horizontal domain, making Hx, y> a 2connected horizontal domain. The PLL method as described in Section 2 demodulates only l-connected horizontal domains with smooth frontiers. With the former definitions the PLL method may be extended to l-connected horizontal domains
130 (1996) 7-12
Fig. 3. I-connected horizontal domain that cannot be demodulated correctly with the original PLL version [ll (for demonstrating purposes this domain is considered that of a fringe pattern). The arrows show the demodulation trajectories, and the dotted line is the reference line for the top to bottom demodulation.
Z?‘,( x, y) without smooth frontiers and later generalized to n-connected horizontal domains. In order to show the feasibility of the extended PLL method, assume the domain of the fringe pattern to be that of Fig. 3, whose binary domain %x, y) = 2,(x, y) may be calculated using EQ. (6). It is necessary to select a vertical line x = u,(y) within 2, to use it to start demodulating the fringe pattern under analysis. This line must contain only one point x for each row of the 2,(x, y> region. To begin the phase extraction process select a point on the u,(y) line, as the starting position. Forward demodulate the y-row using the equation 4,(x+
1, Y)
= $“(X,
Y) -#(x+
1, Y) -1(x,
Xsin[ 4,( x, Y) + wo.x]z,( x, Y), x = u,( y), . . . , N - 2, with the initial condition 4,(x = u,(y), then backward demodulate using, 4”(X,
Y)--7[qX+L
Xsin[&,(x+l, x=N-2,...,0
c IX, Yi
(a)
(W
Fig. 2. (a) The domain S is a l-vertical connected domain. (b) It is separated in two l-horizontal connected domains X’,( x, y) and .Tz(x,
y).
(7)
y) = 0, and
Y)
=4Jx+l,
~
Y)l
Y)-l(x,
y)+w,x]Z,(x+1,
Y)] y),
(8) Fig. 3 explains this graphically: from the reference curve the demodulation starts towards the right of the figure, continues all the way to the left and then to the right. This process is continued for the next row y + 1, using as a starting the position u ,( y + 11, and the initial condition +,( u,( y + 11, y + 1) =
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130 (1996) 7-12
qbo(u,( y), y), where +,(u,( y), y) is the phase value calculated for the y-row. This process is continued until 4”(x, y) is obtained over all the image. The demodulation process is similar to the one described in Section 2, except that here explicit use is made of the horizontal boundaries, and the starting position was choosen on the curve x = u,(y). As an example, Fig. 3 shows a l-connected horizontal domain that cannot be demodulated correctly using the PLL algorithm as described in Ref. [l]. However, using the vertical dotted line x = u,(y) as reference for Eqs. (7) and (8), with the modified method as described here, it is possible to do it. Now, to demodulate an n-connected horizontally binary domain @x, y), the PLL technique described above is applied independently to each Xn domain. Due to the fact that each subset is demodulated independently, an offset constant phase 0, is present among them. With this in mind, the continuous phase 4,(x, y) is obtained as, 4”(XT Y) = c [k&x9 n
Y> - OnI.
(9)
Some fringe patterns are easier to demodulate if a horizontal carrier is used (vertical demodulation), this is because the number of connected subsets in the vertical case may be lower than in the horizontal case.
4. Numerical
simulations
In order to show the behaviour of the modified algorithm under controlled conditions, numerical simulations were made. Images of 256 X 240 pixels and 256 gray levels were used, using a 486 PC-computer, at 33 MHz. The domain is constructed to cover the following aspects: (i) It has 4 unconnected domains. (ii) One domain is 3-connected horizontally. (iii) In some areas there are abrupt changes from one line to the next (non-smooth boundary). The fringe pattern is computer generated using Eq. (l), over the above domain. A( x, y) and B( x, y) where chosen as constants, w0 = 2~(7/2.56), and 4,(x,
Y)=qX2+Y2)+qX2+3Y2),
(10)
Fig. 4. (a) Computer generated non-continuous fringe pattern. The phase function introduced was I$,,(x, y)= D(x* + y*)+ E(x’ + 3 y’), with D = - 2, E = I. (b) Demodulated phase obtained by applying the modified PLL to (a). (c) Phase difference in radians between the phase obtained with the modified PLL method and the ideal phase given by Eq (IO). The X-Y axis coordinates are normalized.
N.A. Ochoa et al./ Optics Communicrrtions
130 (1996) 7-12
with D= -2, E= 1, and x, y normalized to a [ - 1, I] interval. The fringe pattern is shown in Fig. 4a, and Fig. 4b shows the demodulated phase +“(x, y> obtained. As seen from Fig. 4b, the isolated areas are not connected with the main domain, having then a different constant phase value 0,. However, inside the connected area the result is independent of the line used for the initial demodulation. In order to know the accuracy in ideal conditions of the modified PLL method, the phase obtained for the larger connected domain of Fig. 4b was subtracted point to point from the one given by Eq. (10). Peak to valley and RMS values of n-/ 16 and rr/ 100 rad were obtained respectivelly. The mesh graph of the subtraction is shown in Fig. 4c. This figure shows that the greater errors are found at the edges of the fringe domain.
5. Experimental
results
Fig. 5 shows the experimental arrangement. A conventional slide projector is used to project an 80 line/in on a 3-connected horizontal 10 X 20 cm2 object (Fig. 6a). Line fringes are deformed according to the topography of the object surface. This deformed pattern, actually a phase modulated one, is captured by a 256 X 240 (N X A4) pixels CCD camera and processed by a 33 MHz PC computer. The
-0.99
-0.73
-0.46
-3.19
0.07
0.34
0.60
0.87
Fig. 6. (a) Experimental fringes obtained by projecting an 80 lines/in Ronchi ruling on an irregular metallic surface. (b) Demodulated phase obtained by applying the modified PLL to (a).
Fig. 5. The experimental arrangement used. projector, a CCD camera and a PC-computer.
It shows
a slide
processing time to get the phase values was about 6 seconds, with this type of hardware. The frequency w,, was measured and the process described in Section 2 applied to the fringe image. Finally, the demodulated phase was calculated, and shown in Fig. 6b. In this image, the measured frequency w0 was 2~(5/256).
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The binary domain 5F( x, y) of this image determined by using a 5 X 5 pixel window, measuring the variance inside it.
was and
Communications
130 (1996) 7-12
required to find the binary image domain E(x, y). It is clear that each of the n-connected subsets should have enough fringes to be demodulated correctly.
6. Conclusions
Acknowledgements
The phase-locked loop is generalized in order that it may be applied on fringe patterns with irregular frontiers and with holes inside it. When the fringe pattern has smooth frontiers and is continuous, the generalized and the original algorithm give the same results. Limitations and accuracy of the modified algorithm are similar to the original method, and the computing time required to calculate the phase is practically the same [4]. The amount of additional time to apply the proposed algorithm is the time
Noe Alcala would like to thank the financial support through a scholarship from CONACYT, Mexico.
References [l] M. Servin and R. Rodriguez-Vera, J. Mod. Optics 40 (1993) 2087. [2] M. Takeda and K. Mutoh, Appl. Optics 22 (1983) 3977. [3] K.H. Womack, Opt. Eng. 23 (1984) 391. [4] N. Alcal et al., Optics Comm. 117 (1995) 213.