Precise phase demodulation of single carrier-frequency interferogram by pixel-level Lissajous figure and ellipse fitting

Abstract

Phase demodulation from a single carrier-frequency fringe pattern is becoming increasingly important particularly in areas of optical metrology such as dynamic interferometry, deflectometry and profilometry. The Fourier transform (FT) method and the spatial-carrier phase-shifting technique (SCPS) are two popular and well-established approaches to demodulation. However FT has the drawback of significant edge errors because of the Gibbs effect, whilst detuning errors for the local phase shift occur when SCPS is applied. A novel demodulation method based on pixel-level Lissajous figure and ellipse fitting (PLEF) is presented in this paper. Local demodulation in the spatial domain makes PLEF more flexible than the FT method, without spectral leakage. Based on a more adaptable approach, account is taken of variations in illumination and phase distribution over a few neighboring pixels. The mathematic demodulation model is of interest and has been demonstrated via simulation. Theoretical phase extraction error is as low as 10 −4 rad. Experiments further corroborate the effectiveness of the proposed method. In conclusion, various influencing factors, e.g. variations of background/modulation, phase amplitude, carrier frequency, additive noise that may affect the precision of PLEF are discussed in detail.Interferometry, deflectometry and profilometry are diagnostic tools frequently used in optical metrology to achieve highly accurate, non-contact and full-field measurements. Since the relevant physical information is usually encoded in the fringe phase, phase demodulation always plays a very critical role in fringe analysis. Temporal phase shifting (TPS) is a phase demodulation method, which has been used extensively in optical metrology since its introduction 1 .Phase shifting interferometry (PSI), for example, has already become a standard configuration of a modern high accuracy interferometer 2 . A well-calibrated phase shifter and stable environment are crucial factors to TPS due to equal steps conventionally used in phase-shifting algorithms (PSA) 3,4 . Though various kinds of generalized PSI have been put forward in recent years as a means of coping with random phase shift error caused by vibration 5-9 , it remains difficult to achieve real time phase demodulation in dynamic measurement operations since at least three interferograms are usually required in order to record and process data. Dynamic measurement is an attractive option not only in optical shop testing, e.g. high-aperture and long-distance primary mirror testing 10 but also in many other phase-demodulation operations, e.g. visual monitoring of cell culture in the biomedical domain 11 . With the exception of simultaneous phase shifting interferometry (SPSI) 12 , which is not our concern in this contribution, demodulation of a single shot fringe pattern is an attractive option. The spatial carrier-frequency (SCF) method is a good choice since implementation of spatial carrier-frequency interferometry (SCFI) is relatively easily accomplished on most advan...

Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method

“…(RMS) for X and Y fringe patterns, respectively. These phase residual errors are almost similar to well-known SCPS methods in the literature 23 . By increasing the carrier frequency and reducing the noise in the fringe pattern, we can obtain even higher phase accuracy.…”

“…We can increase this number to enhance the stability of ellipse construction robust to noise. The 2D mode of creating Lissajous figure can be used for relatively lower carrier frequencies and higher noise levels 23 . We only present 1D mode here for simplicity.…”

Phase retrieval from single interferogram without carrier using Lissajous ellipse fitting technology

“…We use Lissajous Figure and Ellipse Fitting technique 24 – 27 to calculate the parameters of Eq. ( 4 ), i.e., .…”

Snapshot 3D reconstruction of liquid surfaces

“…Limitations of the method are the requirement of parameter initialisation and in some cases pre-normalisation of the fringe pattern [30]. A different single image phase demodulation method shows that a fringe pattern describes Lissajous Ellipses [32]. By estimating the parameters of these ellipses the phase is demodulated.…”

“…FTP uses a filtering in the Fourier domain of the image and after inverse Fourier transform the parameters of the fringe pattern, A, B and ϕ, can be estimated. The method is computationally fast but its greatest drawback is connected to that it is a global method that has trouble to adapt to the non-stationary fringe patterns often found in phase demodulation problems [33][34][35]. To solve this problem either the Windowed Fourier Transform (WFT) or CWT method can be used which gains localisation in the spatial domain [33,35,36].…”