Time efficient Chinese remainder theorem algorithm ... - OSA Publishing

Time efficient Chinese remainder theorem algorithm for full-field fringe phase analysis in multi-wavelength interferometry Catherine E. Towers, David P. Towers, and Julian D.C. Jones School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK [email protected]

Abstract: We present a computationally efficient method for solving the method of excess fractions used in multi-frequency interferometry for absolute phase measurement. The Chinese remainder theorem, an algorithm from number theory is used to provide a unique solution for absolute distance via a set of congruence’s based on modulo arithmetic. We describe a modified version of this theorem to overcome its sensitivity to phase measurement noise. A comparison with the method of excess fractions has been performed to assess the performance of the algorithm and processing speed achieved. Experimental data has been obtained via a full-field fringe projection system for three projected fringe frequencies and processed using the modified Chinese remainder theorem algorithm. ©2004 Optical Society of America OCIS codes: (120.3180) Interferometry; (100.2650) Fringe Analysis

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(C) 2004 OSA

Received 19 February 2004; revised 13 March 2004; accepted 15 March 2004

22 March 2004 / Vol. 12, No. 6 / OPTICS EXPRESS 1136

1. Introduction Interferometry has become a powerful tool in metrology owing to the sensitivity of optical phase to a range of relevant measurands [1, 2]. In single frequency interferometry the fractional fringe value can be calculated by either phase stepping or Fourier transform techniques [3], but the intrinsic periodic nature of the fringe function means that fringe order information is lost. This fundamental limitation must be resolved in order to obtain absolute phase measurement. Multi-frequency interferometry has provided a solution for fringe order identification with classical analysis performed using the method of excess fractions [4, 5, 6, 7]. In this technique the fractional fringe values measured at each frequency are used to form a set of simultaneous equations that may be solved for the integral fringe order. The measurements are absolute within an unambiguous measurement range (UMR) that is maximized by ensuring that none of the measurement wavelengths share common factors, i.e. they are pair-wise relatively prime. This gives a single coincidence of wavelength at all measurement frequencies over the UMR [7]. Alternative approaches for absolute phase measurement have been proposed, based on multi-wavelength interferometry with beat frequency processing to generate a synthetic wavelength from a coincidence of phase [8, 9, 10]. It was found that the UMR of these techniques was limited by phase measurement noise. Techniques to further extend the UMR beyond the beat wavelength have also been reported and are relevant when the optical measurement wavelengths are well separated [10]. Recently, we introduced a robust optimization process for frequency selection in multi-wavelength interferometry where a geometric series of synthetic wavelengths is defined to maximize the overall process reliability [11]. This imposes a different frequency selection criterion compared to the method of excess fractions and hence both approaches will find particular applications depending on specific experimental constraints and the available optical sources. The processing time to find the absolute phase using the method of excess fractions is linearly dependent on the UMR and becomes computationally prohibitive for data from fullfield interferometers. Important applications include transient events and real time control of dynamic forming [12]. Processing times may run to several days for typical data sets [7], hence faster processing algorithms are desired. A solution to this computational problem has been proposed based on the Chinese Remainder Theorem (CRT) from number theory [13, 14, 15, 16] where wavelength selection is based on pair-wise relatively prime wavelengths. Unlike the excess fractions method a direct calculation of the measurand is produced using modulo arithmetic. To apply the CRT method, wrapped phase values are scaled to integers and multiplied by a large integer wavelength product, hence the approach is very susceptible to phase measurement noise [14, 15]. Agurok [14] demonstrated that considering a reduced UMR could accommodate small errors in the wrapped phase measurements. Alternatively, Takeda et al. [15] proposed a method where phase offsets were used to move data away from integer boundaries and it was recognized that fringe order errors form a structured distribution. Spatial comparisons between neighboring pixels were checked against this distribution and if a match was found, the values were corrected. This error correction procedure assumed relatively clean data, i.e., where the integer wrapped phase errors have a maximum magnitude of 1. Zhong et al. also reported a modified form of this approach based on an irrational ratio of wavelengths [16]. In all cases data was presented for a two-wavelength system over a limited number,