Method to eliminate the zero spectra in Fourier transform profilometry based on a cost function

Shi-lin Cui; Fei Tian; De-hua Li

doi:10.1364/AO.51.003194

Applied Optics
Vol. 51,
Issue 16,
pp. 3194-3204
(2012)
•https://doi.org/10.1364/AO.51.003194

Method to eliminate the zero spectra in Fourier transform profilometry based on a cost function

Shi-lin Cui, Fei Tian, and De-hua Li

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Shi-lin Cui, Fei Tian, and De-hua Li, "Method to eliminate the zero spectra in Fourier transform profilometry based on a cost function," Appl. Opt. 51, 3194-3204 (2012)

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Abstract

To increase the accuracy, speed, and robustness of 3D measurements in Fourier transform profilometry (FTP), this paper introduces a cost function according to the intrinsic features of the amplitude and frequency modulated (AF/M) signal and proposes two new algorithms to eliminate the background components of the fringe pattern based on the proposed cost function. Finally, the standard Fourier transform (FT) is used to calculate the phase of the pattern, which no longer contains background components. The two proposed methods are both data-driven and require no parameter adjustments in advance. Theoretical analysis and 80 experimental results show that the proposed cost function is valid. The results of more than 80 experiments with different types of fringe patterns, different carrier frequencies, and different noise variances with frequency overlap and sudden phase variation show that the proposed two methods are more accurate and robust than the 2D Gabor wavelet transform, the 2D Fan wavelet transform, and the 1D complex Morlet wavelet transform profilometry, and they are approximately 70 times faster than the 1D complex Morlet wavelet transform profilometry.

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Method/Noise variance	Proposed method 1	Proposed method 2	DOG wavelet using our cost function	1D Morlet with their cost function ridge	2D Gabor with direct modulus maximum ridge	2D Fan with direct modulus maximum ridge
0.0	0.0273	0.0045	0.0041	0.0241	0.0213	0.0171
0.2	0.0366	0.0265	0.0410	0.0289	0.0221	0.0211
0.5	0.0602	0.0582	0.0401	0.0468	0.0267	0.0349
0.9	0.0974	0.0727	0.0714	0.0764	0.0363	0.0574
1.2	0.1213	0.0969	0.0952	0.1000	0.0448	0.0752
1.5	0.1397	0.1213	0.1191	0.1242	0.0538	0.0934
1.8	0.1580	0.1459	0.1433	0.1492	0.0632	0.1118
2.0	0.1713	0.1625	0.1596	0.1663	0.0696	0.1240

Method/Noise variance	Proposed method 1	Proposed method 2	DOG wavelet using our cost function	1D Morlet with their cost function ridge	2D Gabor with direct modulus maximum ridge	2D Fan with direct modulus maximum ridge
0.0	0.0120	0.0116	0.0134	0.0913	0.0453	0.0699
0.2	0.0155	0.0183	0.0164	0.0921	0.0455	0.0701
0.5	0.0250	0.0259	0.0272	0.0957	0.0462	0.0709
0.9	0.0389	0.0395	0.0405	0.1046	0.0480	0.0732
1.2	0.0494	0.0507	0.0516	0.1137	0.0499	0.0757
1.5	0.0598	0.0621	0.0630	0.1244	0.0523	0.0789
1.8	0.0710	0.0738	0.0747	0.1365	0.0550	0.0826
2.0	0.0770	0.0816	0.0826	0.1451	0.0570	0.0854

Method/Noise variance	Proposed method 1	Proposed method 2	DOG wavelet using our cost function	1D Morlet with their cost function ridge	2D Gabor with direct modulus maximum ridge	2D Fan with direct modulus maximum ridge
0.0	0.0349	0.0465	0.0439	0.2410	0.2544	0.1998
0.2	0.0361	0.0468	0.0442	0.2413	0.2567	0.1995
0.5	0.0452	0.0479	0.0547	0.2420	0.2602	0.1991
0.9	0.0514	0.0503	0.0573	0.2440	0.2644	0.1987
1.2	0.0566	0.0525	0.0600	0.2461	0.2670	0.1984
1.5	0.0620	0.0549	0.0634	0.2492	0.2688	0.1982
1.8	0.0674	0.0643	0.0672	0.2556	0.2705	0.1981
2.0	0.0711	0.0658	0.0700	0.2597	0.2719	0.1980

Method/Noise variance	Proposed method 1	Proposed method 2	DOG wavelet using our cost function	1D Morlet with their cost function ridge	2D Gabor with direct modulus maximum ridge	2D Fan with direct modulus maximum ridge
0.0	0.0899	0.0282	0.0138	0.0959	0.0519	0.0729
0.2	0.0934	0.0332	0.0348	0.0971	0.0525	0.0737
0.5	0.1062	0.0523	0.0536	0.1031	0.0553	0.0776
0.9	0.1210	0.0842	0.0857	0.1188	0.0622	0.0871
1.2	0.1356	0.1098	0.1116	0.1349	0.0691	0.0972
1.5	0.1541	0.1360	0.1382	0.1542	0.0771	0.1095
1.8	0.1760	0.1628	0.1655	0.1762	0.0859	0.1238
2.0	0.1912	0.1810	0.1841	0.1914	0.0921	0.1347

Method/Noise variance	Proposed method 1	Proposed method 2	DOG wavelet using our cost function	1D Morlet with their cost function ridge	2D Gabor with direct modulus maximum ridge	2D Fan with direct modulus maximum ridge
0.0	0.0630	0.0804	0.0655	0.1297	0.4034	0.1641
0.2	0.0643	0.0812	0.0966	0.1499	2.8211	0.1649
0.5	0.0687	0.0855	0.1016	0.2176	2.9587	0.1662
0.9	0.0772	0.0961	0.1137	0.1734	2.8102	0.1688
1.2	0.0846	0.1068	0.1260	0.1806	2.9319	0.1734
1.5	0.0931	0.1192	0.1405	0.1857	2.6358	0.1776
1.8	0.1017	0.1330	0.1565	0.1985	2.6364	0.1845
2.0	0.1077	0.1428	0.1680	0.2058	2.6390	0.1907

Method	Proposed first method	Proposed second method	1D Complex Morlet with cost function ridge	2D fan WT	2D Gabor WT
Time-consuming (s)	6.91	3.89	482.48	51.86	9.22

	Simulated fringe pattern (1) with carrier frequency of $1 / 8$	Simulated fringe pattern (1) with carrier frequency of $1 / 16$	Simulated fringe pattern (1) with carrier frequency of $1 / 32$	Simulated fringe pattern (2) with carrier frequency of $1 / 8$	Simulated fringe pattern (2) with carrier frequency of $1 / 16$
Noise variance 0	5 (0.0196)	5 (0.0120)	6 (0.0349)	4 (0.0899)	5 (0.0630)
Noise variance 0	5 (0.0196)	5 (0.0120)	6 (0.0349)	4 (0.0899)	5 (0.0630)
Noise variance 0.2	5 (0.0315)	5 (0.0155)	6 (0.0361)	4 (0.0934)	5 (0.0643)
Noise variance 0.2	5 (0.0315)	5 (0.0155)	6 (0.0361)	4 (0.0934)	5 (0.0643)
Noise variance 0.5	5 (0.0579)	5 (0.0250)	5 (0.0452)	3 (0.1062)	5 (0.0687)
Noise variance 0.5	5 (0.0579)	5 (0.0250)	6 (0.0404)	4 (0.1012)	5 (0.0687)
Noise variance 0.9	5 (0.0966)	5 (0.0389)	5 (0.0514)	3 (0.1210)	5 (0.0772)
Noise variance 0.9	5 (0.0966)	5 (0.0389)	6 (0.0476)	4 (0.1172)	5 (0.0772)
Noise variance 1.2	5 (0.1211)	5 (0.0494)	5 (0.0566)	3 (0.1356)	5 (0.0846)
Noise variance 1.2	5 (0.1211)	5 (0.0494)	6 (0.0534)	4 (0.1328)	5 (0.0846)
Noise variance 1.5	5 (0.1405)	5 (0.0598)	5 (0.0620)	3 (0.1541)	5 (0.0931)
Noise variance 1.5	5 (0.1405)	5 (0.0598)	6 (0.0593)	4 (0.1523)	5 (0.0931)
Noise variance 1.8	5 (0.1597)	4 (0.0710)	5 (0.0674)	3 (0.1760)	5 (0.1017)
Noise variance 1.8	5 (0.1597)	5 (0.0697)	6 (0.0653)	4 (0.1750)	5 (0.1017)
Noise variance 2.0	5 (0.1736)	4 (0.0770)	5 (0.0711)	3 (0.1912)	5 (0.1077)
Noise variance 2.0	5 (0.1736)	5 (0.0760)	6 (0.0693)	4 (0.1907)	5 (0.1077)

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Equations (14)

Applied Optics