Determination of the local contrat of interference fringe pattern uing continuou wavelet tranform Jong Kwang Hyok, Kim Chol Su Intitute of Optic, Department of Phyic, Kim Il Sung Univerity, Pyongyang, DPR of Korea Abtract The contrat of fringe pattern cannot be uniform over the whole interferogram becaue of variou caue uch a non-uniformity of reflectance, air fluctuation or vibration, and ometime it i neceary to analyze it. In thi paper we propoe a method of determining the local contrat of interference fringe pattern uing continuou wavelet tranform. 1. Introduction In optical interferometry the phyical quantitie to be conidered modulate the intenity of the fringe pattern a follow. I( x, I [1 ( x, co ( x, ], (1) where I ( x, repreent the intenity of the pattern image, I i the background intenity, ( x, i the local contrat, and i the optical phae related to the phyical quantitie to be conidered. In interferometry phae retrieval from the fringe pattern i therefore eential if one want to recover the phyical quantitie to be conidered. To recover the phae information from a ingle interferogram require that it hould be modulated with a uitable patial carrier frequency. But a hown in Eq. (1), the fringe quality i trongly influenced by local contrat ( x, and it i important to determine it to enhance the accuracy of fringe analyi. For thee type of fringe pattern there are three well known method for phae recovery and fringe contrat, namely
Fourier tranform method [1], the windowed Fourier tranform [,3] and wavelet tranform [4]. The Fourier tranform method i baed on the filtering of Fourier pectrum of the fringe pattern. But in the cae of rapid variation of patial frequency or direction of fringe pattern, it i difficult to filter the effective pectrum accurately only by Fourier tranform. Furthermore, in Fourier tranform method it i impoible to verify the correponding relation between local intenitie and local fringe frequencie. Therefore it i difficult to eparate DC component and noie pectrum from effective pectrum and to analyze a local area of fringe pattern. In the windowed Fourier tranform, it i poible to analyze the local character of the fringe pattern. But in the cae of rapid variation of patial frequency, the accuracy of fringe analyi i not o high becaue of the fixed ize of window. Modulated fringe pattern can be bet decribed a non-tationary ignal with local intantaneou frequencie that are proportional to the gradient of the phae ditribution of the meaured field. The continuou wavelet tranformation, which i a correlation of a ignal with a mother wavelet with a certain cale and hift, can be ued to verify the correpondence between the local fringe intenity and contrat.. Determination of the local contrat of interference fringe pattern CWT i repreented by the correlation between the given ignal f and a wavelet function [5]. W 1 f a, f ( x) a x b ( dx, () a where W f ( a, i the coefficient matrix of CWT, f (x) i the ignal, and x b i the wavelet function with cale parameter a a and hift parameter b. The cale parameter decribe the feature of fringe, uch a fringe frequencie, and the hift parameter pecifie a local region where the ignal i analyzed. The CWT tranform the ignal in the patial domain into the pace-frequency domain. The local ignal analyi approach i a major advantage of CWT compared a Fourier tranform.
Becaue the CWT i eentially a correlation operation between a ignal and a wavelet, the analyi accuracy of the fringe pattern greatly depend on the choice of the mother wavelet function. A hown in Eq. (1), the interference fringe pattern i coine modulated, o a uitable choice of wavelet function i the complex Morlet wavelet. where exp( i x) 1 x ( x) exp exp( ix), (3) i the frequency of the Morlet wavelet. The term exp x / and repreent a fat-decaying Guaian envelope and a complex function with inuoidal characteritic repectively. Eq. (1) can be rewritten in term of the fringe frequencie a follow I( x, I [1 ( x, co( x)] (4) The wavelet tranformation of Eq. (4) uing the complex Morlet wavelet i a follow. W a a f ( a, I exp I exp exp( i a (5) Becaue the wavelet tranform coefficient i the correlation one, the higher coefficient value i, the more imilar the given ignal to the wavelet with a certain cale. Thu the maximal cale value during hifting the wavelet through the fringe pattern i the fringe period and the coefficient value i the contrat of the fringe pattern at thi poition. Now let the cale value with which the wavelet tranform coefficient i maximal be a r. Then
, (6) a r Fig.1 how a method of determining the local contrat of fringe pattern by CWT. Fig 1(a) how the imulated fringe pattern with a quadric phae and hence a linearly increaing frequency, but different contrat from place to place. Fig. 1( how local fringe contrat ditribution obtained by wavelet tranform. Becaue the wavelet function i the complex one, the wavelet coefficient are alo complex and therefore their moduli are the fringe contrat. We changed the cale parameter a of the wavelet tranform from 1 to 5 pixel and hifted the poition parameter b one by one pixel to get the correlation coefficient between the wavelet function and the given ignal. (a) ( Fig. 1. Fringe relative intenity (a) and contrat ditribution ( Fig. how the local contrat ditribution of the -D interference fringe obtained by the above-mentioned method. Fig. (a) how the computer generated -D fringe and Fig. ( how the local contrat ditribution of it.
(a) ( Fig. -D interference fringe (a) and local contrat ditribution ( 3. Concluion In thi paper we have propoed a method of determining the local contrat of interference fringe pattern uing continuou wavelet tranform. The CWT tranform the interference fringe pattern into the pace-frequency domain. The maximal cale value during hifting the wavelet through the fringe pattern i the fringe period and the coefficient value i the contrat of the fringe pattern at thi poition. Reference 1. Takeda M, et al, J. Opt. Soc. Am. 9, 156, 198. Kemao Q. Appl. Opt. 43, 695, 4 3. Kemao Q. Opt. Laer Eng. 45, 34, 7 4. Gdeiat M, et al, Opt. Laer Eng. 47, 1348, 9 5. H.G.Stark, Wavelet and Signal Proceing, Springer Verlag, 5