Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm Munther A. Gdeisat, David R. Burton, and Michael J. Lalor
A novel technique called a two-dimensional digital phase-locked loop 共DPLL兲 for fringe pattern demodulation is presented. This algorithm is more suitable for demodulation of fringe patterns with varying phase in two directions than the existing DPLL techniques that assume that the phase of the fringe patterns varies only in one direction. The two-dimensional DPLL technique assumes that the phase of a fringe pattern is continuous in both directions and takes advantage of the phase continuity; consequently, the algorithm has better noise performance than the existing DPLL schemes. The twodimensional DPLL algorithm is also suitable for demodulation of fringe patterns with low sampling rates, and it outperforms the Fourier fringe analysis technique in this aspect. © 2002 Optical Society of America OCIS codes: 120.2650, 100.5070, 100.0100.
1. Introduction
There are a number of fringe pattern demodulation algorithms such as Fourier fringe analysis,1 phase stepping,2 and direct phase detection.3 These algorithms use an arctangent function to calculate the phase components of a fringe pattern. The output of the arctangent function is limited between ⫺ and ; consequently the resultant phase map may contain 2 discontinuities that should be removed. An unwrapping algorithm is required to remove the 2 steps. The unwrapping is often the most challenging and time-consuming step in the above techniques. Recently, Servin and Rodriguez-Vera proposed use of a first-order conventional digital phase-locked loop 共DPLL兲 for fringe pattern demodulation.4 Phase maps produced by this algorithm do not contain 2 discontinuities; consequently the need for an unwrapping algorithm is eliminated. In a previous paper, we suggested use of a second-order conventional DPLL for fringe pattern demodulation, and we implemented the algorithm in real time.5 The second-
The authors are with the School of Engineering, Coherent and Electro-Optics Research Group, Liverpool John Moores University, James Parsons Building, Room 114, Byrom Street, Liverpool L3 3AF, United Kingdom. The e-mail address for M. A. Gdeisat is
[email protected]. Received 21 November 2001; revised manuscript received 23 May 2002. 0003-6935兾02兾265479-09$15.00兾0 © 2002 Optical Society of America
order conventional DPLL has better tracking ability and more noise immunity than the first-order conventional DPLL. Phase maps produced by both algorithms suffer from high-frequency disturbances added by the multiplier phase detector 共PD兲 of the conventional DPLL.4,5 Kozlowski and Serra have modified the conventional DPLL to improve its tracking features.6 This modified DPLL produces phase maps with a better signal-to-noise ratio than the conventional DPLL because the PD of the modified DPLL does not generate high-frequency disturbances that corrupt the demodulated phase maps. The linear DPLL has better tracking ability than the conventional DPLL or its modified version.7 This algorithm has been used for fringe pattern demodulation, and it has been implemented in real time.8 Phase maps produced by this algorithm can be considered better than phase maps generated by the conventional and the modified DPLL algorithms in terms of accuracy and signal-to-noise ratio. Consider a fringe pattern whose phase is continuous in two directions that is analyzed with the abovementioned basic DPLL algorithms 共e.g., conventional, modified, and linear DPLLs兲. The basic DPLL techniques demodulate the fringe pattern row by row and assume that the phase of the fringe pattern is continuous in only one direction. These algorithms do not make any use of the continuity of the phase of the fringe pattern in the two directions. Servin et al. proposed a nonrecursive twodimensional DPLL to demodulate fringe patterns 10 September 2002 兾 Vol. 41, No. 26 兾 APPLIED OPTICS
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that are continuous in two directions.9 In this paper we extend the recursive one-dimensional DPLL to a recursive two-dimensional DPLL and discuss use of the recursive two-dimensional DPLL for fringe pattern demodulation. In this paper we refer to the recursive two-dimensional DPLL as a twodimensional DPLL. Servin et al. proposed an iterative DPLL algorithm that can be used in conjunction with any basic DPLL technique.10 The iterative DPLL algorithm demodulates a fringe pattern as follows. In the first iteration, a flat reference is used to estimate the phase of the fringe pattern. The estimated phase map resulting from the first iteration is smoothed by use of an m ⫻ n window, and then it is used as a reference to estimate the phase of the fringe pattern in the second iteration. In the third iteration, the phase map resulting from the second iteration is smoothed and used as a reference to estimate the phase of the fringe pattern in the fourth iteration and so on until the demodulated phase map has a good signal-to-noise ratio. The smoothing of an estimated phase map in an iteration prior to its use as a reference to estimate the phase of the fringe pattern in the next iteration makes this algorithm have better noise immunity than the basic DPLL algorithms because the algorithm assumes that the phase of the fringe pattern is continuous in both directions. The improvement in the noise immunity of the iterative DPLL algorithm is slight in comparison with the basic DPLL techniques because the smoothing is carried out after the demodulation process. Also, the interative DPLL algorithm is computationally intensive. In this paper we present a new method that we call a two-dimensional DPLL for fringe pattern demodulation. Similar to the iterative DPLL algorithm, the two-dimensional DPLL technique takes advantage of the continuity of the phase of a fringe pattern in two directions, and it can be used in conjunction with any basic DPLL technique. The two-dimensional DPLL technique performs the smoothing during the demodulation process; consequently it is more noise immune than the iterative DPLL algorithm. Also, the two-dimensional DPLL technique is generally less computationally intensive than the iterative DPLL algorithm. In Section 2 an overview of the operation of a DPLL is presented. In Section 3 we present a theoretical background of the two-dimensional DPLL algorithm and its use for fringe pattern demodulation. Some of our results of applying the proposed technique to demodulate real fringe patterns are presented and discussed in Section 4. 2. Digital Phase-Locked Loop
A block diagram of a DPLL is shown in Fig. 1 and consists of the following three basic elements: 1. a PD, 2. a digital filter 共DF兲, and 3. a digital controlled oscillator 共DCO兲. 5480
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Fig. 1. Block diagram of a DPLL.
The PD is a device whose output level depends on the phase difference between its two inputs. The DF is an infinite-impulse-response DF. The DCO is an oscillator whose output frequency depends directly on its input level. When the control signal applied to the DCO is zero, its output signal will have a constant frequency. This frequency is called the free-running frequency of the DCO 共 fr兲. A DPLL’s operation can be explained as follows: The PD compares the phase of a discrete input signal against the phase of the DCO output; the output of the PD is a measure of the phase difference between these two signals. The DF filters this output, and then the filtered output is applied to the DCO. The filtered output gradually changes the frequency of the DCO in a direction that reduces the phase difference between the input signal and the DCO output. A DPLL can be used to phase demodulate its input. Suppose that a phase-modulated signal is applied to the DPLL. Also, consider that the free-running frequency of the DCO is equal to the carrier frequency of the modulated signal. The instantaneous frequency of the modulated signal depends on the derivative of the modulating signal. When the loop is in the tracking state, it is necessary that the frequency of the DCO be close to the instantaneous frequency of the input signal, and the frequency of the DCO is proportional to the control signal. Consequently, the control signal should be a close replica of the derivative of the modulating signal. The modulating signal could be recovered if we integrate the control signal using an accumulator. A.
Conventional Digital Phase-Locked Loop
A block diagram of a conventional DPLL is shown in Fig. 2. The PD of the conventional DPLL is a multiplier. The DF is an infinite-impulse-response DF. A proportional path DF 共zero-order DF兲 yields a firstorder DPLL, whereas a first-order infinite-impulseresponse DF, normally a proportional plus accumulator paths, gives a second-order DPLL.5 If a phase-modulated signal is applied to the conventional DPLL, a closed replica of the modulating signal can be extracted from the output of the DCO accumulator, which accumulates the control signal as shown in Fig. 2.
Fig. 2. Block diagram of a conventional DPLL.
B.
Linear Digital Phase-Locked Loop
A block diagram of a linear DPLL is shown in Fig. 3.7,8 The operation of the linear DPLL can be explained as follows. The input of the linear DPLL c共 x兲 is split, and one is phase shifted by 兾2. This phase shift can be achieved by the application of a Hilbert transform.11 The phase components of the input can be calculated when we substitute the input c共x兲 and its 兾2 phase-shifted version q共x兲 into ⫺1
共 x兲 ⫽ tan
冋 册
q共 x兲 , c共 x兲
(1)
where x is the sample index. The phase components of the input 共x兲 contain 2 steps because we use the arctangent function. The wrapped phase 共x兲 is applied to the loop’s PD that subtracts it from the DCO output. The output of the subtractor PD is applied to the modular circuit that limits its output between ⫺ and . The output of the modular circuit is applied to the DF whose output controls the phase of the DCO so as to decrease the phase difference between the input and the DCO output. The linear DPLL unwraps and demodulates the wrapped phase 共 x兲兲 simultaneously. If the input to the linear DPLL is a phase-modulated signal, the demodulated signal can
be extracted from the output of the DCO accumulator as shown in Fig. 3. 3. Theoretical Background
Fringe intensity in a fringe pattern can be expressed as1 g共 x, y兲 ⫽ a共 x, y兲 ⫹ b共 x, y兲cos关2f 0 x ⫹ 共 x, y兲兴, (2) where a共x, y兲 represents the background illumination, b共x, y兲 is the amplitude modulation of the fringes, f0 is the spatial carrier frequency, x and y are the sample indices, and 共x, y兲 is the desired phase information. The background illumination of a fringe pattern must be removed prior to the application of a DPLL algorithm. Servin and RodriguezVera have suggested differentiating the fringe pattern with respect to the x coordinate to remove its background illumination.4 The direction of fringes is considered to be perpendicular to the x coordinate. c共 x, y兲 ⫽ g共 x ⫹ 1, y兲 ⫺ g共 x, y兲.
(3)
Basic DPLL algorithms demodulate the differentiated fringe pattern c共x, y兲 pixel by pixel. The phase of the DCO is compared with the phase of a pixel in the fringe pattern. The output of the PD is filtered 10 September 2002 兾 Vol. 41, No. 26 兾 APPLIED OPTICS
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Fig. 3. Block diagram of a linear DPLL.
with the DF whose output is used to control the phase of the DCO so as to track the phase of the fringe pattern. Operation of the proposed two-dimensional DPLL algorithm can be explained as follows. The PD compares the phase of the DCO with the phase of a number of pixels within a window 共e.g., an m ⫻ n window兲 in the fringe pattern. The phase differences between the DCO and the pixels in the window are averaged, and their average is applied to the DF. The output of the DF is used to control the phase of the DCO to enable the loop to track and demodulate its input. For simplicity, suppose that the fringe pattern consists of L ⫻ L pixels and the size of the analyzing window is 3 ⫻ 3 as shown in Fig. 4. Also, consider pixel 共2,2兲 as a starting point. The PD compares the phase of the DCO output with the phase of the center pixel 共2,2兲 and its surrounding pixels according to e共x, y兲 ⫽
1 1 1 ᏼ兵关2f0共x ⫹ i兲 ⫹ 共x ⫹ i, y ⫹ j兲兴 9 i⫽⫺1 j⫽⫺1 (4) ⫺ 关2f0共x ⫹ i兲 ⫹ ˆ 共x ⫺ 1 ⫹ i, y ⫹ j兲兴其,
兺兺
where the term 关2f0共x ⫹ i兲 ⫹ 共x ⫹ i, y ⫹ j兲兴 represents the phase of the fringe pattern, the term 关2f0共x ⫹ i兲 ⫹ ˆ 共x ⫺ 1 ⫹ i, y ⫹ j兲兴 represents the phase of the DCO output 共an estimation to the phase of the fringe pattern where ˆ is the demodulated phase of the fringe pattern兲, and ᏼ共.兲 is the function of the PD that is determined by the phase detection characteristics of the basic DPLL algorithm that will 5482
be used in conjunction with the two-dimensional DPLL technique to demodulate the fringe pattern. The demodulated phase ˆ is considered to be constant within the analyzing window, whereas the phase of the DCO output is not constant because of the inclusion of the phase of the free-running frequency of the DCO within the window. The averaged phase error is then applied to the DF whose
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Fig. 4. Example of the two-dimensional DPLL algorithm in operation.
Fig. 5. Dual scanning algorithm.
output is used to control the phase of the DCO to enable the loop to track and demodulate its input. Herein the free-running frequency of the DCO is considered to be equal to the spatial carrier frequency of the fringe pattern. We carry out the analysis of the next pixel 共2,3兲 by comparing the phase of this center pixel and its surrounding pixels within the window with the phase of the DCO using Eq. 共4兲. The same procedures are repeated for pixel 共2,4兲 and the following pixels in the same row until pixel 共2,L ⫺ 1兲 is reached and analyzed. The procedures described above are equivalent to a forward pass for the dual scanning algorithm shown in Fig. 5.5 The two-dimensional DPLL technique is a dynamic system and requires a settling time. Initially the two-dimensional DPLL algorithm starts out of lock but eventually reaches a tracking state. Consequently the demodulated phase values resulting from our scanning and analyzing the fringe pattern in the forward direction are incorrect and must be discarded. The desired demodulated phase values can be recovered when the same row is scanned in the opposite direction.5 After pixel 共2,L ⫺ 1兲 has been demodulated in the forward pass, the same row should be scanned and analyzed in the backward direction as indicated in Fig. 5. To avoid a newly transient response for the loop when it processes the first row in the backward direction, the phase of the DCO should be preserved when the scanning direction is changed. The pixel 共2,L ⫺ 1兲 is analyzed by Eq. 共4兲 as explained above, and the algorithm proceeds to analyze the pixels in the second row until pixel 共2,2兲 is reached. As stated above, it is assumed that the object is continuous in both directions. After the second row in the fringe pattern has been analyzed, the algorithm commences analyzing the third row starting from pixel 共3,2兲 and proceeds from the left to the right until it reaches pixel 共3,L ⫺ 1兲. The content of the DCO accumulator should be maintained when the scanning direction is reversed. The third row is then scanned and processed in the backward direc-
tion. The above procedures are repeated until all the rows of the fringe pattern are processed. The phase values resulting from the forward passes are ignored, and the phase values resulting from the backward passes are considered to be the desired demodulated phase map. The basic DPLL algorithm can be considered as a special case of the twodimensional DPLL technique, where the analyzing window size is 1 ⫻ 1. We can retrieve the phase components of the borders of the fringe pattern by starting the analysis process from pixel 共1,1兲. The PD compares the phase of the DCO with the phase of pixel 共1,1兲 and its surrounding pixels 共three pixels in the case of a 3 ⫻ 3 window兲. The phase of the next pixel 共1,2兲 and its surrounding pixels 共five pixels in the case of a 3 ⫻ 3 window兲 are compared with the phase of the DCO and so on until the first row of the fringe pattern is processed and scanned forward and backward. The same procedures can be carried out to analyze the fringe pattern including its borders. The advantage of this algorithm is that the smoothing of the phase map is carried out during the tracking process, so the demodulated phase map is smoothed, not the modulated fringes as we show in Section 4. The analyzing window can be any size 共e.g., m ⫻ n兲. Use of large window sizes 共e.g., 20 ⫻ 20兲 makes the algorithm computationally intensive and removes the fine details of the object. The main disadvantage of the two-dimensional DPLL algorithm is its suitability for analysis of narrow-bandwidth fringe patterns rather than widebandwidth fringe patterns. This stems from the assumption that the demodulated phase of a fringe pattern ˆ within the analyzing window is constant. The two-dimensional DPLL algorithm can be used in conjunction with any basic DPLL technique. In this paper we use the algorithm in conjunction with the conventional and linear DPLL algorithms. In the linear DPLL case, the phase of each pixel within the window is compared with the phase of the DCO, and the resultant phase errors are applied to the modular circuit individually. The averaging process is performed at the output of the modular circuit. 4. Experimental Results
Figure 6共a兲 shows a real fringe pattern that was actually taken from the thorax of a female mannequin. The fringe pattern was demodulated by use of the basic and the two-dimensional second-order conventional DPLL algorithms. The spatial carrier frequency of this fringe pattern is approximately 1兾15 共i.e., 15 pixels兾fringe兲. The fringe pattern was differentiated with respect to the x coordinate prior to the demodulation process. First, the fringe pattern was demodulated with the basic second-order conventional DPLL algorithm, and the demodulated phase map is shown in Fig. 6共b兲. In the second case, the fringe pattern was demodulated with the twodimensional second-order conventional DPLL algorithm employing 7 ⫻ 7 and 15 ⫻ 15 window sizes. The demodulated phase maps are shown in Figs. 6共c兲 10 September 2002 兾 Vol. 41, No. 26 兾 APPLIED OPTICS
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Fig. 6. 共a兲 Fringe pattern demodulated with the two-dimensional second-order conventional DPLL employing 共b兲 1 ⫻ 1, 共c兲, 7 ⫻ 7, and 共d兲 15 ⫻ 15 window sizes.
and 6共d兲, from which we can conclude that the phase maps produced with the two-dimensional conventional DPLL algorithm have a better signal-to-noise ratio than the phase map generated by the basic conventional DPLL algorithm. As mentioned above, Fig. 6共d兲 shows the phase map that we produced by demodulating the fringe pattern with a 15 ⫻ 15 window. If the smoothing with the two-dimensional DPLL algorithm was carried out for fringes, the smoothing window would remove the fringes, and the two-dimensional DPLL algorithm would fail in demodulating the fringe pattern. Conversely, this phase map has better signal-to-noise ratio characteristics than the phase maps shown in Figs. 6共b兲 and 6共c兲 because the smoothing process was carried out for the demodulated phase map during the demodulation process and not for the fringes themselves. Also, the high-frequency components generated by the multiplier PD were attenuated by the smoothing process. This demonstrates that the smoothing was carried out for the demodulated phase map, not for the fringes. This feature enables the two-dimensional DPLL algorithm to demodulate fringe patterns with sampling rates close to the Nyquist rate12 where the fringe patterns cannot be smoothed prior to the demodulation process. Figure 7共a兲 shows a fringe pattern whose spatial carrier frequency is approximately 1兾2.5. The fringe pattern cannot be smoothed prior to the demodulation process because the smoothing may remove the modulated fringes. The fringe pattern was differentiated and demodulated with the basic second-order conventional DPLL algorithm, and the resultant phase map is shown in Fig. 7共b兲. The basic DPLL algorithm failed to demodulate the fringe pattern. The two-dimensional second-order conventional DPLL algorithm was used to demodulate the 5484
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Fig. 7. 共a兲 Fringe pattern demodulated with the two-dimensional second-order conventional DPLL algorithm employing 共b兲 1 ⫻ 1, 共c兲 3 ⫻ 3, and 共d兲 9 ⫻ 9 window sizes.
differentiated fringe pattern, and the demodulated phase maps produced with 3 ⫻ 3 and 9 ⫻ 9 windows are shown in Figs. 7共c兲 and 7共d兲. The twodimensional DPLL algorithm succeeded in demodulating the fringe pattern as indicated in both Figs. 7共c兲 and 7共d兲. The two-dimensional DPLL algorithm can be used in conjunction with the linear DPLL technique to demodulate the fringe patterns. Figure 6共a兲 shows a fringe pattern, which was differentiated to remove its background illumination. The differentiated fringe pattern was then phase shifted by 兾2 by use of a seven-element Hilbert transformer.11 The fringe pattern was applied to the Hilbert transformer row by row. We calculated the phase components of a row in the fringe pattern by substituting the row and its 兾2 phase-shifted version in Eq. 共1兲. This operation was repeated until the phase components of all the rows in the fringe pattern were calculated. The resultant phase map contains 2 discontinuities that are due to use of the arctangent function, this is shown in Fig. 8共a兲. The wrapped phase map shown in Fig. 8共a兲 was demodulated and unwrapped with the basic linear DPLL algorithm. The demodulated phase map is shown in Fig. 8共b兲. The basic linear DPLL algorithm failed to unwrap and demodulate the wrapped phase map. The two-dimensional linear DPLL algorithm was used to unwrap and demodulate the phase map shown in Fig. 8共a兲 by use of a 3 ⫻ 3 window. The demodulated phase map is shown in Fig. 8共c兲. The two-dimensional algorithm succeeded in unwrapping and demodulating the wrapped phase map.
Fig. 8. 共a兲 Wrapped phase map for the fringe pattern shown in Fig. 6共a兲. The demodulated phase maps were generated with the two-dimensional linear DPLL with 共b兲 1 ⫻ 1 and 共c兲 3 ⫻ 3 window sizes. 共d兲 The demodulated phase map produced with the iterative linear DPLL algorithm after 50 iterations.
The iterative linear DPLL algorithm described in Ref. 10 was used to unwrap and demodulate the wrapped phase map, and the demodulated phase map resulting after 50 iterations is shown in Fig. 8共d兲. The iterative linear DPLL technique failed to demodulate the fringe pattern. From Figs. 8共c兲 and 8共d兲, we can conclude that the two-dimensional linear DPLL algorithm is more noise immune than the iterative linear DPLL technique. Figure 9共a兲 shows a fringe pattern whose spatial carrier frequency is approximately 1兾2.3. The fringe pattern was differentiated, and the wrapped phase map was calculated as explained in the above example. The basic linear DPLL algorithm was used to unwrap and demodulate the wrapped phase map, and the demodulated phase map is shown in Fig. 9共b兲. As can be seen from Fig. 9共b兲, the basic linear DPLL algorithm failed to unwrap and demodulate the wrapped phase map. The two-dimensional linear DPLL technique was used to unwrap and demodulate the wrapped phase map, and the demodulated phase maps produced by use of 3 ⫻ 3 and 7 ⫻ 7 windows are shown in Figs. 9共c兲 and 9共d兲. The twodimensional linear DPLL algorithm succeeded in unwrapping and demodulating the wrapped phase map as can be seen in both Figs. 9共c兲 and 9共d兲. The fringe pattern shown in Fig. 9共a兲 was demodulated with the two-dimensional second-order conventional DPLL algorithm with a 9 ⫻ 9 window size. The demodulated phase map is shown in Fig. 10共a兲. This algorithm succeeded in demodulating the fringe pattern by use of a 9 ⫻ 9 window size, but the technique failed to demodulate this fringe pattern by use of 3 ⫻ 3, 5 ⫻ 5, and 7 ⫻ 7 window sizes. How-
Fig. 9. 共a兲 Fringe pattern demodulated with the two-dimensional linear DPLL algorithm that employs 共b兲 1 ⫻ 1, 共c兲 3 ⫻ 3, and 共d兲 7 ⫻ 7 window sizes.
ever, Figs. 9共c兲 and 9共d兲 clearly show that the twodimensional linear DPLL algorithm succeeded in demodulating this fringe pattern with 3 ⫻ 3 and 7 ⫻ 7 window sizes. This clarifies that the twodimensional linear DPLL algorithm has better performance in demodulating fringe patterns with low
Fig. 10. 共a兲 Fringe pattern shown in Fig. 9共a兲 demodulated with the two-dimensional conventional DPLL algorithm by use of a 9 ⫻ 9 window. 共b兲 The demodulated phase map for the fringe pattern produced with the Fourier fringe analysis technique. 10 September 2002 兾 Vol. 41, No. 26 兾 APPLIED OPTICS
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Fig. 11. 共a兲 Fringe pattern and its demodulated phase maps produced with a two-dimensional linear DPLL by use of 共b兲 and 共e兲 1 ⫻ 1, 共c兲 3 ⫻ 3, and 共f 兲 5 ⫻ 5 window sizes. 共d兲 The fringe pattern in 共a兲 smoothed by use of a 5 ⫻ 5 moving average window.
sampling rates than the two-dimensional conventional DPLL technique. The fringe pattern shown in Fig. 9共a兲 was demodulated with the Fourier fringe analysis algorithm,13 and the demodulated phase map is shown in Fig. 10共b兲. From Fig. 10共b兲, it is clear that the Fourier fringe analysis technique that employs Schafer and Oppenheim’s unwrapper14 failed to analyze the fringe pattern. The Fourier fringe analysis algorithm is not quite suitable for analysis of fringe patterns with low sampling rates 共i.e., less than four pixels per fringe兲, and this can be justified as follows. The frequency spectrum of a fringe pattern modulated by an object becomes wider, and its amplitude relative to the background noise amplitude becomes smaller as the number of pixels per fringe decreases. It can be concluded from Figs. 9共c兲, 9共d兲, 10共a兲, and 10共b兲 that the two-dimensional DPLL algorithm has better performance in demodulating fringe patterns with sampling rates close to the Nyquist rate than the Fourier fringe analysis technique. Figure 11共a兲 shows a fringe pattern whose spatial carrier frequency is approximately 1兾5.4. The fringe pattern was smoothed by use of a 3 ⫻ 3 moving average window. The smoothed fringe pattern was 5486
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demodulated with the one-dimensional linear DPLL, and the resultant phase map is shown in Fig. 11共b兲, where the one-dimensional linear DPLL was incapable of demodulating the smoothed fringe pattern. The fringe pattern shown in Fig. 11共a兲 was demodulated with the two-dimensional linear DPLL by use of a 3 ⫻ 3 window size, and the demodulated phase map is shown in Fig. 11共c兲. The fringe pattern shown in Fig. 11共a兲 was smoothed by use of a 5 ⫻ 5 moving average window. The smoothed fringe pattern, shown in Fig. 11共d兲, was demodulated with the one-dimensional linear DPLL, and the demodulated phase map is shown in Fig. 11共e兲. The smoothing process attenuates the fringes to a high degree; consequently the onedimensional linear DPLL was incapable of demodulating the fringe pattern. The fringe pattern shown in Fig. 11共a兲 was demodulated with the twodimensional linear DPLL by use of a 5 ⫻ 5 window size. The two-dimensional linear DPLL succeeded in demodulating the fringe pattern, and the demodulated phase map is shown in Fig. 11共f 兲. A comparison of Figs. 11共b兲 and 11共e兲 with Figs. 11共c兲 and 11共f 兲 shows that use of the two-dimensional DPLL with a specific window size to demodulate fringe patterns produces better phase maps than when the fringe patterns are smoothed with a moving average window with the same window size and the smoothed fringe patterns are demodulated with the one-dimensional DPLL. 5. Conclusions
A novel technique called a two-dimensional DPLL technique was presented. This technique is shown to be more suitable for demodulation of fringe patterns with continuous phase than the basic DPLL techniques. The two-dimensional DPLL technique assumes that the phase of a fringe pattern is continuous in two directions, and it makes use of this phase continuity. The two-dimensional DPLL technique smoothes the demodulated phase map of a fringe pattern during the demodulation process. This makes the algorithm suitable for demodulation of fringe patterns with low sampling rates, where the smoothing of such fringe patterns prior to the demodulation process is difficult to accomplish. The two-dimensional DPLL algorithm outperforms the Fourier fringe analysis algorithm in demodulating fringe patterns with low sampling rates. References 1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156 –160 共1982兲. 2. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson and G. T. Reid, eds. 共Institute of Physics, Bristol, UK, 1993兲, pp. 94 –140. 3. Y. Ichioka and M. Inuiya, “Direct phase detection system,” Appl. Opt. 11, 1507–1514 共1972兲.
4. M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087–2094 共1993兲. 5. M. Gdeisat, D. Burton, and M. Lalor, “Real-time fringe pattern demodulation with a second-order digital phase-locked loop,” Appl. Opt. 39, 5326 –5336 共2000兲. 6. J. Kozlowski and G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025– 2030 共1997兲. 7. M. Hagiwara and M. Nakagawa, “DSP-type first order digital phase locked loop using linear phase detector,” Electron. Commun. Jpn. 共Part I: Communications兲. 69, 99 –107 共1986兲. 8. M. Gdeisat, D. Burton, and M. Lalor, “Real-time hybrid fringe pattern analysis using a linear digital phase locked loop for demodulation and unwrapping,” Meas. Sci. Technol. 11, 1480 – 1492 共2000兲.
9. M. Servin, J. Quiroga, and F. Cuevas, “Demodulation of carrier fringe patterns by the use of non-recursive digital phase locked loop,” Opt. Commun. 200, 87–97 共2001兲. 10. M. Servin, R. Rodriguez-Vera, and D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 共1995兲. 11. L. Rabiner and R. Schafer, “On the behaviour of minimax FIR digital Hilbert transformers,” Bell Syst. Tech. J. 53, 363–390 共1974兲. 12. F. Stremler, Introduction to Communication Systems 共Addison-Wesley, Reading, Mass., 1989兲. 13. C. Gorecki, “Interferogram analysis using a Fourier transform method for automatic 3D surface measurement,” Pure Appl. Opt. 1, 103–110 共1992兲. 14. R. Schafer and A. Oppenheim, Digital Signal Processing 共Prentice-Hall, Englewood Cliffs, N.J., 1975兲.
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