Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. 1: Computer simulations

A series of simulations were made for an ideal Twyman-Green interferogram of equally spaced straight fringes having tilt only about x. It was found that fitting polynomials to the interferometric data resulted in biased estimates of some of the fitting coefficients to the optical path difference. The acceptance of the Seidel aberrations grows with the noise level and diminishes when the number of fringes is increased.     

Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. II: Analytical study

We consider an ideal Twyman-Green interferogram with equally spaced straight fringes parallel to the x axis and fringe coordinates that are affected by Gaussian errors. We adjust the data points by polynomial fitting to the interferograms. We use a statistical analysis to obtain analytical formulas for the expected values of the aberration coefficients. The result of the analysis shows that the expected coefficients are zero, except for tilt about x and for the comatic term, and that such deviation increases with the noise level and decreases with the number of fringes. Formulas are also obtained for the expected values of the sum of squares of the residuals. We show that the problem of choosing the wrong polynomial order is a consequence of erroneous adjustment of the data points.     

Windowed Fourier transform for fringe pattern analysis

Fringe patterns in optical metrology systems need to be demodulated to yield the desired parameters. Time-frequency analysis is a useful concept for fringe demodulation, and a windowed Fourier transform is chosen for the determination of phase and phase derivative. Two approaches are developed: the first is based on the concept of filtering the fringe patterns, and the second is based on the best match between the fringe pattern and computer-generated windowed exponential elements. I focus on the extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single carrier fringe pattern. Principles as well as examples are given to show the effectiveness of the proposed methods.     

Fourier fringe analysis with improved spatial resolution

The spatial resolution of the phase image derived from the interferogram by Fourier fringe analysis is limited by the necessity to isolate a first order in the Fourier plane. By use of the two complementary outputs of the interferometer, it is possible to eliminate the zero order and thus to improve the spatial resolution by a factor of approximately 2. The theory of this improvement is presented and confirmed experimentally.     

High-precision fringe tracking invariant to systematic model errors

We present a new approach to processing of interferometric data, which dramatically improves operation on the smallest fraction of the fringe (approximately 1/1,000 of a wavelength and beyond). In particular, this approach, dubbed the variation-invariant subspace tracking approach (VISTA), makes it possible to operate a Michelson interferometer in a highly stable mode in which the estimation of optical path delay becomes invariant (insensitive) to a large class of systematic model errors. This previously unknown invariance property of interferometry follows from the fundamental odd-even symmetry of the channeled spectrum and its derivatives. VISTA offers a powerful algorithmic alternative for alleviating technological challenges in the design of high-precision long-baseline spaceborne interferometers.     

Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis

Based on the windowed Fourier transform, the windowed Fourier ridges (WFR) algorithm and the windowed Fourier filtering algorithm (WFF) have been developed and proven effective for fringe pattern analysis. The WFR algorithm is able to estimate local frequency and phase by assuming the phase distribution in a local area to be a quadratic polynomial. In this paper, a general and detailed statistical analysis is carried out for the WFR algorithm when an exponential phase field is disturbed by additive white Gaussian noise. Because of the bias introduced by the WFR algorithm for phase estimation, a phase compensation method is proposed for the WFR algorithm followed by statistical analysis. The mean squared errors are derived for both local frequency and phase estimates using a first-order perturbation technique. These mean square errors are compared with Cramer-Rao bounds, which shows that the WFR algorithm with phase compensation is a suboptimal estimator. The above theoretical analysis and comparison are verified by Monte Carlo simulations. Furthermore, the WFR algorithm is shown to be slightly better than the WFF algorithm for quadratic phase.     

Modified Fourier transform method for interferogram fringe pattern analysis

A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.     

Windowed Fourier transform for fringe pattern analysis: addendum

Novel approaches based on windowed Fourier transform for demodulation of fringe patterns were previously presented [Appl. Opt. 43, 2695-2702 (2004)], where extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single-carrier fringe pattern was the main focus. I show that the same methods can be applied to process a single closed-fringe pattern in either noise reduction or phase approximation, which adds to the versatility of the windowed Fourier-transform method for fringe pattern analysis.     

Windowed Fourier transform for fringe pattern analysis: theoretical analyses

A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c(20)=c(02)=0.005 rad/(pixel)(2)], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad. However, when using the WFR algorithm with a window size of sigma(x)=sigma(y)=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01 rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01 rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.     

Phase correction of interferograms using digital all-pass filters

A method for the phase correction of interferograms in Fourier transform infrared spectroscopy is presented. It is shown that phase error can be canceled to within an arbitrary angular precision by a low-order digital all-pass filter. Such a filter only modifies the phase of the Fourier transform of the interferogram and keeps the magnitude unchanged, like the Mertz method, for example. However, our method minimizes the asymmetric apodization that results in photometric errors when using the Mertz method alone. A practical example is provided in which phase correction over a frequency range of 800 cm(-1) to 4000 cm(-1) using a 9-pole all-pass filter resulted in a photometric error of <0.01%, much less than the 0.3% error of the Mertz method. An alternative and faster (approximately 100 ms) approach is to use an all-pass filter with lower angular precision followed by the Mertz method. Removing most of the phase error with the filter brings the interferogram to an optimal state so that the residual phase error can be completely removed with the Mertz procedure without introducing photometric error. The method can be used in most experiments, including emission spectroscopy, where conventional techniques are inadequate. A simple all-pass filter design algorithm is given.     

Minimizing the effect of extraneous spikes in open-path Fourier transform infrared interferograms

Intense spikes caused by extraneous factors are sometimes found in the interferograms of open-path Fourier transform infrared (OP/FT-IR) measurements. Those spikes result in dominant oscillations in the corresponding spectra and make the true spectral features indistinguishable from the noise. Three techniques were designed to remove the spikes: replacing the affected region by zeroes; grafting the data from the corresponding region on the other side of the centerburst ("homografting"); and grafting the data from the same region on the same side of the centerburst from a neighboring interferogram ("heterografting"). Results showed that all three techniques were effective to remove the spikes, but at the cost of introducing more noise into the corresponding spectrum. The performance of the heterograft technique was the best in terms of noise reduction. The factors affecting the noise level of spectra computed from spike-removed interferograms were also explored. A procedure was proposed to minimize the effect of spikes on OP/FT-IR spectra, and a Matlab program with a graphical user interface is available upon request to facilitate the implementation.     

Phase statistics of interferograms with applications to synthetic aperture radar

Interferometric methods are well established in optics and radio astronomy. In recent years, interferometric concepts have been applied successfully to synthetic aperture radar (SAR) and have opened up new possibilities in the area of earth remote sensing. However interferometric SAR applications require thorough phase control through the imaging process. The phase accuracy of SAR images is affected by decorrelation effects between the individual surveys. We analyze quantitatively the influence of decorrelation on the phase statistics of SAR interferograms. In particular, phase aberrations as they occur in typical SAR processors are studied in detail. The dependence of the resulting phase bias and variance on processor parameters is presented in several diagrams.     

Radiometric errors in complex Fourier transform spectrometry

A complex spectrum arises from the Fourier transform of an asymmetric interferogram. A rigorous derivation shows that the rms noise in the real part of that spectrum is indeed given by the commonly used relation sigmaR = 2X x NEP/(etaAomega square root(tauN)), where NEP is the delay-independent and uncorrelated detector noise-equivalent power per unit bandwidth, +/- X is the delay range measured with N samples averaging for a time tau per sample, eta is the system optical efficiency, and Aomega is the system throughput. A real spectrum produced by complex calibration with two complex reference spectra [Appl. Opt. 27, 3210 (1988)] has a variance sigmaL2 = sigmaR2 + sigma(c)2 (Lh - Ls)2/(Lh - Lc)2 + sigma(h)2 (Ls - Lc)2/(Lh - Lc)2, valid for sigmaR, sigma(c), and sigma(h) small compared with Lh - Lc, where Ls, Lh, and Lc are scene, hot reference, and cold reference spectra, respectively, and sigma(c) and sigma(h) are the respective combined uncertainties in knowledge and measurement of the hot and cold reference spectra.     

Digital filtering implementations for the detection of broad spectral features by direct analysis of passive Fourier transform infrared interferograms

Finite impulse response (FIR) filters and finite impulse response matrix (FIRM) filters are evaluated for use in the detection of volatile organic compounds with wide spectral bands by direct analysis of interferogram data obtained from passive Fourier transform infrared (FT-IR) measurements. Short segments of filtered interferogram points are classified by support vector machines (SVMs) to implement the automated detection of heated plumes of the target analyte, ethanol. The interferograms employed in this study were acquired with a downward-looking passive FT-IR spectrometer mounted on a fixed-wing aircraft. Classifiers are trained with data collected on the ground and subsequently used for the airborne detection. The success of the automated detection depends on the effective removal of background contributions from the interferogram segments. Removing the background signature is complicated when the analyte spectral bands are broad because there is significant overlap between the interferogram representations of the analyte and background. Methods to implement the FIR and FIRM filters while excluding background contributions are explored in this work. When properly optimized, both filtering procedures provide satisfactory classification results for the airborne data. Missed detection rates of 8% or smaller for ethanol and false positive rates of at most 0.8% are realized. The optimization of filter design parameters, the starting interferogram point for filtering, and the length of the interferogram segments used in the pattern recognition is discussed.     

Multiscale windowed Fourier transform for phase extraction of fringe patterns

A multiscale windowed Fourier transform for phase extraction of fringe patterns is presented. A local stationary length of signal is used to control the window width of a windowed Fourier transform automatically, which is measured by an instantaneous frequency gradient. The instantaneous frequency of the fringe pattern is obtained by detecting the ridge of the wavelet transform. The numerical simulation and experiment have proved the validity of this method. The combination of the windowed Fourier transform and the wavelet transform makes the extracted phase more precise than other methods.     

Vibration detection using focus analysis of interferograms

This paper proposed an automated technique for vibration detection using statistical focus measure to evaluate interferogram contrast. An interferogram sequence from a Mach-Zehnder interferometer setup is recorded (frame rate: 24 fps) and the gray-level variance (GLVA) is plotted versus time. Occurrence of induced vibration in the setup causes a decrease in the interferogram contrast which, in turn, manifests as an evident rapid drop in the variance plot. The technique is demonstrated experimentally using periodic microvibrations (frequency range, ≤6 Hz) and aperiodic disturbances.     

Phase reconstruction from three interferograms based on integral of phase gradient

A novel algorithm of phase reconstruction based on the integral of phase gradient is presented. The algorithm directly derives two real-valued partial derivatives from three phase-shifted interferograms. Through integrating the phase derivatives, the desired phase is reconstructed. During the phase reconstruction process, there is no need for an extra rewrapping manipulation to ensure values of the phase derivatives lie in the interval [?π, π] as before, thus this algorithm can prevent error or distortion brought about by the phase unwrapping operation. Additionally, this algorithm is fast and easy to implement, and insensitive to the nonuniformity of the intensity distribution of the interferogram. The feasibility of the algorithm is demonstrated by both computer simulation and experiment.