Fast frequency-guided sequential demodulationof a single fringe pattern

Li Kai1 and Qian Kemao2,*1Department of Mechanics, Shanghai University, Shanghai 200444, China

2School of Computer Engineering, Nanyang Technological University, Singapore 639798*Corresponding author: mkmqian@ntu.edu.sg

Received June 15, 2010; revised October 9, 2010; accepted October 9, 2010;posted October 13, 2010 (Doc. ID 130040); published November 1, 2010

Fast frequency-guided sequential demodulation (FFSD) for demodulating a single closed-fringe pattern is proposedas an improvement of frequency-guided sequential demodulation (FSD). Instead of using optimization to estimatethe local frequencies for determining the sign of the phase, the FFSD estimates the local frequencies by directlycalculating the gradient of the obtained phase with an undetermined sign. This improvement considerably reducesthe computational complexity of the FSD and leads to a faster and simpler method. Simulated and experimentalfringe patterns are used to test the proposed method and show that the demodulation speed of FFSD is about 150times faster than that of the FSD, while the robustness and accuracy remain almost the same. © 2010 OpticalSociety of AmericaOCIS codes: 120.2650, 100.2650, 100.5070, 120.3940.

In optical interferometry, phase demodulation from a sin-gle fringe pattern is of great interest in applications wherethe use of phase shifting or a carrier technique [1] is diffi-cult, for example, when transient phenomena are mea-sured or the environment is hostile. Various methodshave been proposed ([2,3] and references therein) to dealwith this problem. Of these, frequency-guided sequentialdemodulation (FSD) [2,3] is characterized by its simpleprinciple, easy implementation, and high robustness andaccuracy. In this Letter, we propose a fast frequency-guided sequential demodulation (FFSD) algorithm thathas much faster demodulation speed and similar robust-ness and accuracy. The details of the FFSD are describedbelow after a brief review of the FSD.A typical fringe pattern is modeled as

f ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½φðx; yÞ�; ð1Þ

where ðx; yÞ denotes a pixel coordinate; f ðx; yÞ is the ir-radiance; aðx; yÞ and bðx; yÞ are the background intensityand fringe amplitude, respectively; and φðx; yÞ is thephase to be estimated. Preprocessing of the fringe pat-tern is used to remove the background intensity andnormalize the fringe amplitude [2,3], which gives a nor-malized fringe pattern:

f ðx; yÞ ≈ cos½φðx; yÞ�: ð2Þ

The FSD directly calculates the phase as

φ0ðx; yÞ ¼ arccos½f ðx; yÞ�; ð3Þ

where φ0ðx; yÞ ∈ ½0; π�. The true phase, if wrapped, isφðx; yÞ ¼ �φ0ðx; yÞ with an undetermined sign. TheFSD then estimates the local frequencies ωxðx; yÞ andωyðx; yÞ from a neighborhood region Nx;y by minimizingthe following cost function:

Uðx; yÞ ¼X

ðε;ηÞ∈Nx;y

ff ðε; ηÞ − cos½φðx; y; ε; ηÞ�g2; ð4Þ

with

φðx; y; ε; ηÞ ¼ φ0ðx; yÞ þ ωxðx; yÞðε − xÞþ ωyðx; yÞðη − yÞ: ð5Þ

The sign of the phase then can be determined by for-cing the local frequencies to be spatially continuous anda frequency-guided scanning strategy is used to correctlyhandle the critical points. In the original FSD [2], Eq. (4)is optimized by an exhaustive search. Later, Levenberg–Marquardt optimization was adopted [3] to reduce theprocessing time. Recently, a more accurate FSD algo-rithm has been implemented [4] by replacing the locallylinear phase assumption of Eq. (5) with a locally quadra-tic phase assumption. However, all these methods sufferfrom the complicated optimization process introducedby Eq. (4), inevitably resulting in long processing time.

In this Letter, a more effective approach to determin-ing the sign of the phase is proposed. First, on both sidesof Eq. (2), performing a gradient operation, followed by adot product with ∇φðx; yÞ, gives [5]

∇f ðx; yÞ ·∇φðx; yÞ ¼ − sin½φðx; yÞ�j∇φðx; yÞj2; ð6Þ

where∇f ðx; yÞ ¼ ½f xðx; yÞ; f yðx; yÞ� is the intensity gradi-ent and ∇φðx; yÞ ¼ ½ωxðx; yÞ;ωyðx; yÞ� is the phase gradi-ent, i.e., the local frequencies. Since j∇φðx; yÞj2 ≥ 0 andsignfsin½φðx; yÞ�g ¼ sign½φðx; yÞ� for φðx; yÞ ∈ ð−π; π�,the following equation holds:

sign½φðx; yÞ� ¼ −sign½∇f ðx; yÞ ·∇φðx; yÞ�; ð7Þ

which indicates that the sign of phase can be determinedfrom ∇f ðx; yÞ and ∇φðx; yÞ.

Second, ∇f ðx; yÞ can be calculated by simply differen-tiating the normalized fringe pattern f ðx; yÞ. Similarly andinterestingly, ∇φðx; yÞ can be obtained by differentiatingthe phase obtained from Eq. (3) [6]. Note that ∇φðx; yÞalso has the phase ambiguity problem, i.e., ∇φðx; yÞ ¼�∇φ0ðx; yÞ. Thus solving the sign ambiguity for phase be-comes solving the sign ambiguity for phase gradient,which is easier.

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Third, isolated lines are observed in ∇φ0ðx; yÞ wherethe fringe intensity is �1. They are treated as noiseand eliminated by a smoothing process. To avoid thatthe vectors with opposite direction cancel themselvesduring the smoothing process [7], ∇φ0ðx; yÞ is convertedinto a complex form as

φ0xðx; yÞ þ jφ0yðx; yÞ ¼ ωðx; yÞe jθðx;yÞ; ð8Þ

where j ¼ ffiffiffiffiffiffi−1

pand ωðx; yÞ and θðx; yÞ denote the ampli-

tude and the angle of the complex number, respectively.In fact, ωðx; yÞ corresponds to the total local frequencydefined in the FSD [2,3]. The complex field is thensquared and averaged as

~ω2ðx; yÞe j2~θðx;yÞ ¼ 1N

Xðε;ηÞ∈Nx;y

ω2ðε; ηÞe j2θðε;ηÞ; ð9Þ

where Nx;y is a neighborhood region centered at ðx; yÞ, Nis the number of the pixels in this region, and ~ωðx; yÞ and~θðx; yÞ are the amplitude and angle of the smoothed com-plex field, respectively. Although other kernels, such asGaussian, can be used to weight the averaging in Eq. (9),the merit of using a rectangle window is that a fast algo-rithm can be implemented [8]. The smoothed version of∇φ0ðx; yÞ, ∇~φ0ðx; yÞ, can be calculated as

∇~φ0ðx; yÞ ¼ f~ωðx; yÞ cos½~θðx; yÞ�; ~ωðx; yÞ sin½~θðx; yÞ�g:ð10Þ

Finally, ∇φðx; yÞ without sign ambiguity is recoveredfrom∇~φ0ðx; yÞ by explicitly imposing a spatial continuitycondition as

∇φðxa; yaÞ

¼�∇~φ0ðxa; yaÞ if ∇φðxs; ysÞ ·∇~φ0ðxa; yaÞ ≥ 0

−∇~φ0ðxa; yaÞ if ∇φðxs; ysÞ ·∇~φ0ðxa; yaÞ < 0; ð11Þ

where ðxs; ysÞ denotes the selected pixel and ðxa; yaÞ de-notes one of the adjacent pixels. Subsequently, the signof the phase can be determined by Eq. (7).The above FFSD method can be implemented accord-

ing to the FSD [3]. The method is described in detail inthe following.

1. Calculate the phase φ0ðx; yÞ with an undeterminedsign by Eq. (3); calculate the gradients ∇f ðx; yÞ and∇φ0ðx; yÞ by a differential operator (Sobel operator [1]is used in this Letter); smooth ∇φ0ðx; yÞ by Eq. (9) andconstruct ∇~φ0ðx; yÞ by Eq. (10).2. Specify the pixel ðx0; y0Þ with the highest fre-

quency ~ωðx0; y0Þ as the seed pixel and set ∇φðx0; y0Þ ¼∇~φ0ðx0; y0Þ; determine the sign of φðx0; y0Þ according toEq. (7); push the seed pixel into a register R.3. Select the pixel ðxs; ysÞ with the highest local

frequency from R and process its four adjacent pixelsif they have not been processed. Estimate ∇φðxa; yaÞby Eq. (11), then determine the sign of φðxa; yaÞ byEq. (7); unwrap these pixels by comparing them withthe selected pixel; remove the selected pixel from Rand push the processed adjacent pixels into R.

4. Repeat step 3 until R is empty; the pixel-queuingstrategy [9] is used to improve the scanning speed.

To verify the effectiveness of the FFSD, a noisy fringepattern (280 × 280 pixels) is simulated and shown inFig. 1(a). The fringe pattern is denoised by windowedFourier filtering (WFF) [10] and then normalized [2,3],as shown in Fig. 1(b). The phase φ0ðx; yÞ with an unde-termined sign obtained from Fig. 1(b) using Eq. (3) isshown in Fig. 1(c). The angle of ∇φ0ðx; yÞ is shown inFig. 1(d). The angle of ∇~φ0ðx; yÞ calculated by Eq. (9)with Nx;y ¼ 9 × 9 is shown in Fig. 1(e). The smoothed fre-quency ~ωðx; yÞ is shown in Fig. 1(f), which clearly speci-fies the critical points with low values. The real phasegradient ∇φðx; yÞ is then estimated from ∇~φ0ðx; yÞ,and its angle is shown in Fig. 1(g). The sign of phaseis determined by Eq. (7), and the recovered phase isshown in Fig. 1(h). The demodulated result is satisfactoryexcept for some imperfect pixels. By performing postpro-cessing using the WFF again on Fig. 1(h), a refined resultis obtained, as shown in Fig. 1(i). All the phase resultsobtained in this Letter are continuous and unwrapped.These phases are rewrapped for comparison purposes.

Table 1 quantitatively compares some closely relatedmethods, including FSD [3], the frequency-guided regu-larized phase tracker (FGRPT) [3], quadratic phasematching and frequency-guided sequential demodulation(QFSD) [4], and the proposed FFSD, by demodulatingFig. 1(b). All calculations are performed on a personalPentium Dual E8400 computer with 3:0 GHz main fre-quency by C++ programming. The accuracy is measured

Fig. 1. Demodulation of a simulated fringe pattern: (a) noisysimulated fringe pattern, (b) normalized fringe pattern, (c)phase with undetermined sign, (d) angle distribution of thephase gradient∇φ0ðx; yÞ, (e) angle distribution of the smoothedphase gradient∇~φ0ðx; yÞ, (f) smoothed frequency, (g) angle dis-tribution of the recovered phase gradient ∇φðx; yÞ, (h) phasewith determined sign, (i) refined phase.

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as the mean absolute errors (MAEs) between the demo-dulated phases and the true phase. The consumed timedoes not include preprocessing and postprocessing. Thecalculation results show that all these methods can suc-cessfully demodulate the simulated fringe pattern andachieve similar accuracy. However, FFSD is about 150times faster than FSD and 340 times faster than theFGRPT. Although QFSD achieves the highest accuracy,its processing time is also the longest.An experimental fringe pattern (256 × 256 pixels) from

speckle shearography is shown in Fig. 2(a), which is nor-malized and shown in Fig. 2(b). The smoothed frequencyis shown in Fig. 2(c). The angle of the phase gradients∇φðx; yÞ is shown in Fig. 2(d). The demodulated phaseis shown in Fig. 2(e), and the refined result obtainedby using the WFF is shown in Fig. 2(f). It can be seenthat the demodulated result is satisfactory. The time usedfor this demodulation is 0:078 s.In conclusion, an FFSD method for phase demodula-

tion from a single fringe pattern is proposed. The method

obtains the phase with an undetermined sign from thenormalized fringe pattern, from which the phase gradientwith an undetermined direction is directly calculated.After a smoothing process, the real phase gradient is re-covered by imposing a spatial continuous condition witha frequency-guided strategy. The sign of the phase is thendetermined, and the demodulated phase is unwrappedsimultaneously. FFSD gives almost the same robustnessand accuracy as FSD but increases the demodulationspeed by about 150 times.

This work was partially supported by the NationalNatural Science Foundation of China (NSFC) (No.10902066).

References

1. D. Malacara, M. Servín, and Z. Malacara, InterferogramAnalysis for Optical Testing (Marcel Dekker, 1998).

2. Q. Kemao and S. H. Soon, Opt. Lett. 32, 127 (2007).3. H. Wang and Q. Kemao, Opt. Express 17, 15118 (2009).4. H. Wang, K. Li, and Q. Kemao, “Frequency guided methods

for demodulation of a single fringe pattern with quadraticphase matching,” (submitted to Opt. Express).

5. M. Servin, J. A. Quiroga, and J. L. Marroquin, J. Opt. Soc.Am. A 20, 925 (2003).

6. M. Servin, J. L. Marroquin, and J. A. Quiroga, J. Opt. Soc.Am. A 21, 411 (2004).

7. M. Kass and A. Witkin, Comput. Vis. Graph. Image Process.37, 362 (1987).

8. L. T. H. Nam and Q. Kemao, Proc. SPIE 7155, 71550T(2008).

9. J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, J.Opt. Soc. Am. A 22, 2766 (2005).

10. Q. Kemao, Opt. Lasers Eng. 45, 304 (2007).

Fig. 2. Demodulation of an experimental fringe pattern fromspeckle shearography: (a) original fringe pattern, (b) normal-ized fringe pattern, (c) smoothed frequency, (d) angle distribu-tion of the recovered phase gradient ∇φðx; yÞ, (e) phase withdetermined sign, (f) refined phase.

Table 1. Demodulation Results of the Simulated

Fringe Pattern

FGRPT FSD QFSD FFSD

MAEs 0.165 0.120 0.117 0.119Time 31:68 s 14:43 s 135:27 s 0:093 s

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