Straightforwardfiltering tophasedemodulationbyaFouriernormalized-fringe approach

R. Juarez-Salazar ∗, C. Robledo-Sanchez, C. Meneses-Fabian, G. Rodriguez-Zurita, W. Guerrero Sanchez, and J. Gonzalez-Garcia 

Benemérita Universidad Autónoma de Puebla - Facultad de Ciencias Físico-Matemáticas∗ rjuarezsalazar@gmail.com

Abstract

The fringe-pattern normalization method by parame-ter estimation is used to relieve the critical filter’s re-quirements in the Fourier transform method to phasedemodulation. By the normalization procedure, thezero order spectrum is suppressed. Thus, the simplehalf-plane filter is sufficient for the filtering stage.

Introduction

The Fourier transform method is an important tool forfringe pattern analysis [1]. In this approach the carrierfrequency is introduced in order to separates the desired±1 lobe from undesired zero and higher frequencies [2].However, the spectrum orders are in general not wellseparated. This leads to an inefficient filtering.In this work a fringe-pattern normalization approach[3] is presented to implement it for Fourier fringe anal-ysis. With this proposal, the filter design is no criticalbecause the normalization procedure deal to a zero-order spectrum suppression. Thus, the overlapping of the zero-order lobe with the ±1-order lobes is solvedand the leakage spectrum is avoided.

Conclusion

It was shown that the Fourier normalized-fringe anal-ysis is a more advanced technique to phase demodula-tion. The fringe-pattern normalization method by pa-rameter estimation was suggested. In this scheme, thespectrum leakage is avoided by the normalization pro-cedure because the zero order spectrum is effectivelyremoved. With this normalization approach, the fil-tering procedure is less critical; thus, even the simplehalf-plane filter is sufficient. Moreover, the phase de-modulation from a single fringe-pattern and real-timehigher sensitivity evaluations are kept.

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References[1] Malacara, D., Servin, M., and Malacara, Z., [ Interferogram 

analysis for optical testing ], Taylor & Francis Group, sec-ond ed. (2005).

[2] Takeda, M., Ina, H., and Kobayashi, S., “Fourier-transformmethod of fringe-pattern analysis for computer-based topog-raphy and interferometry,” J. Opt. Soc. Am. 72, 156–160(Jan 1982).

[3] Juarez-Salazar, R., Robledo-Sanchez, C., Meneses-Fabian,C., Guerrero-Sanchez, F., and Aguilar, L. A., “Generalizedphase-shifting interferometry by parameter estimation withthe least squares method,” Optics and Lasers in Engineer-

ing 51(5), 626 – 632 (2013).

Fourier normalized-fringe analysis

We consider a fringe-pattern modelled as

I ( p) = a( p) + b( p) cos[φ( p) + 2πf  · p]. (1)

The phase distribution φ( p) is recovered through twostage: fringe pattern normalization, and phase demodu-lation by the standard Fourier method.

Fringe-pattern normalization

We consider the finite degree polynomial representationof the parameters a and b. These polynomials are fittedto the fringe-pattern data. Thus, the fringe-pattern (1)is normalized as [3]:

I ( p) = sat

I − a

b

= cos[φ + 2πf  · p] (2)

= c exp[i2πf  · p] + c∗ exp[−i2πf  · p]. (3)

Phase demodulation

The Fourier transformation of  I  (3) leads to

 I (µ) = C (µ− f ) + C ∗(µ + f ), (4)

where the zero order (the spectrum A(µ) associated withthe background illumination a) is not present. The sim-ple half-spectrum filter is sufficient to carried out thefiltering procedure.After moving the selected spectrum lobe toward the ori-gin of µ-plane, the inverse Fourier transform is calculatedto obtain the complex function c. Finally, the wrappedphase distribution φw is obtained by

φw( p) = arg c( p). (5)

Simulation and optical experiments

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Figure 1: Simulation results. (a) Background light, (b) modulation light, (c) phase distribution, (d) wrapped phasedistribution. Phase demodulation using the half-plane filter in the standard Fourier transform (second row) and the Fourierfringe-normalized analysis (third row).

Figure 2: Experimental results. Phase demodulation using the half-plane filter in the standard Fourier transform (firstrow) and the Fourier normalized-fringe analysis (second row). (1st column) Fringe-pattern to b e processed, (2nd column)Fourier spectrum, (3rd column) filtering, and (4th column) recovered wrapped phase.