PosterRIAO2
2
Str aightfor wa rd filteri ng to phasedemodulation by a Fourier norma lized- fring e appro ach R. Juar ez-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 ∗ [email protected] 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 phase demodulation. By the normaliz atio n procedure, the zero order spectrum is suppressed. Thu s, the simp le half-plane filter is sufficient for the filtering stage. Introduction The Fourier transform method is an important tool for fringe pattern analysis [1]. In this approach the carrier frequency 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 well separated. 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 . Wit h this propos al, the filter design is no critica l because the normaliz atio n proced ure deal to a zero - order spectrum suppressio n. Thus, the ove rlapping of the zero-order lobe with the ±1-order lobes is solved and 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, the spectrum leakage is avoided by the normalization pro- cedure because the zero order spectrum is effectively removed. With this normali zati on approac h, the fil- tering procedure is less critical; thus, even the simple half- plane filter is suffic ien t. More ov er, the phase de- modulation from a single fringe-pattern and real-time higher sensitivity evaluations are kept. Do you wish to learn more? Get the full document , suppl emen tary materia ls and related papers. Capture the following figure with a QR Code Reader from your mobile device (smart phone, tablet, etc.) and enjoy. References [1] Malac ara, D., Servin, M., and Malacar a, Z., [ Interferogram analysis for optical testing ], Ta ylor & Francis Group, sec- ond ed. (2005). [2] Takeda, M., Ina, H., and Kobayashi, S., “Fourier-transform method of fringe-pattern analysis for computer-based topog- raphy and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (Jan 1982). [3] Juare z-Sala zar, R., Roble do-San chez , C., Meneses-F abian, C., Guerr ero-Sa nche z, F., and Aguil ar, L. A., “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Enginee r- 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 two stage: fringe pattern normalization, and phase demodu- lation by the standard Fourier method. Fringe-pattern normalization We consider the finite degree polynomial representation of the parameters a and b. These polynomials are fitted to 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 with the background illumination a) is not present. The sim- ple half-spectrum filter is sufficient to carried out the filtering procedure. After moving the selected spectrum lobe toward the ori- gin of µ-plane, the inver se Fourier transform is calculated to obtain the complex function c. Finally , the wrapped phase distribution φ w is obtained by φ w ( p) = arg c( p). (5) Simulation and optical experiments −1 0 1 −1 0 1 100 150 (a) −1 0 1 −1 0 1 0 50 (b) −1 0 1 −1 0 1 −20 0 20 (c) (d) 50 100 150 2 00 50 100 150 200 (e) 50 1 00 1 50 200 50 100 150 200 −0.4 0 0.4 −0.4 0 0.4 6 8 10 12 (f) (g) −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (h) 50 100 150 2 00 50 100 150 200 (i) 50 100 150 200 50 100 150 200 −0.4 0 0.4 −0.4 0 0.4 2 4 6 8 10 12 (j) (k) −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (l) 50 100 150 200 50 100 150 200 Figure 1: Simulation resu lts. (a) Backgro und light, (b) modulatio n light, (c) phase distr ibut ion, (d) wrapped phase distribution. Phase demodulation using the half-plane filter in the standard Fourier transform (second row) and the Fourier fringe-normalized analysis (third row). Figure 2: Experimental results. Phase demodulation using the half-plane filter in the standard F ourier transform (first row) and the Fourier normalized-fringe analysis (second row). (1st column) Fringe-pa ttern to b e processed, (2nd column) Fourier spectrum, (3rd column) filtering, and (4th column) recovered wrapped phase.
-
Upload
rigoberto-juarez-salazar -
Category
Documents
-
view
214 -
download
0
Transcript of PosterRIAO2
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∗ [email protected]
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.
Do you wish to learn more?
Get the full document, supplementary materials andrelated papers. Capture the following figure with a QRCode Reader from your mobile device (smart phone,tablet, etc.) and enjoy.
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
−10
1
−10
1
100
150
(a)
−10
1
−10
10
50
(b)
−10
1
−10
1−20
0
20
(c) (d)
50 100 150 200
50
100
150
200
(e)
50 100 150 200
50
100
150
200−0.4
00.4
−0.40
0.4
6
8
10
12
(f) (g)
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
(h)
50 100 150 200
50
100
150
200
(i)
50 100 150 200
50
100
150
200−0.4
00.4
−0.40
0.42468
1012
(j) (k)
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
(l)
50 100 150 200
50
100
150
200
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.
