White-light interferometry using a channeled spectrum. 2. Calibration methods, numerical and experimental results

In the companion paper, [Appl. Opt. 46, 5853 (2007)] a highly accurate white light interference model was developed from just a few key parameters characterized in terms of various moments of the source and instrument transmission function. We develop and implement the end-to-end process of calibrating these moment parameters together with the differential dispersion of the instrument and applying them to the algorithms developed in the companion paper. The calibration procedure developed herein is based on first obtaining the standard monochromatic parameters at the pixel level: wavenumber, phase, intensity, and visibility parameters via a nonlinear least-squares procedure that exploits the structure of the model. The pixel level parameters are then combined to obtain the required "global" moment and dispersion parameters. The process is applied to both simulated scenarios of astrometric observations and to data from the microarcsecond metrology testbed (MAM), an interferometer testbed that has played a prominent role in the development of this technology.     

Complete fringe order determination in scanning white-light interferometry using a Fourier-based technique

White-light interferometry uses a white-light source with a short coherent length that provides a narrowly localized interferogram that is used to measure three-dimensional surface profiles with possible large step heights without 2π-ambiguity. Combining coherence and phase information improves the vertical resolution. But, inconsistencies between phase and coherence occur at highly curved surfaces such as spherical and tilted surfaces, and these inconsistencies often cause what are termed ghost steps in the measurement result. In this paper, we describe a modified version of white-light interferometry for eliminating these ghost steps and improving the accuracy of white-light interferometry. Our proposed technique is verified by measuring several test samples.     

Multiperiod fringe projection interferometry using a backpropagation method for surface profile measurement

Interference fringes with different periods are projected on an object surface. There is a constant phase point where the phase of the fringe is kept at a constant value while the period is scanning. Multiple optical fields with different periods on the object surface are made from detected phases of the fringes. The multiple optical fields are backpropagated to the constant phase point of the phase where all of the phases of the multiple backpropagated fields become the same value and the amplitude of the sum of the multiple backpropagated fields becomes maximum. The distance of the backpropagation provides the position of the object surface. Some experiments show that this method can measure an object surface with discontinuities of several millimeters with high accuracy of several micrometers.     

Error sources and algorithms for white-light fringe estimation at low light levels

We address the problem of highly accurate phase estimation at low light levels, as required by the Space Interferometry Mission (SIM). The most stringent SIM requirement in this regard is that the average phase error over a 30-s integration time correspond to a path-length error of approximately 30 pm. Most conventional phase-estimation algorithms exhibit significant enough bias at the signal levels at which the SIM will be operating so that some correction is necessary. Several algorithms are analyzed, and methods of compensating for their bias are developed. Another source of error in phase estimation occurs because the phase is not constant over the integration period. Errors that are due to spacecraft motion, the motion of compensating optical elements, and modulation errors are analyzed and simulated. A Kalman smoothing approach for compensating for these errors is introduced.     

Model-based estimation of ultrasonic echoes. Part I: analysis and algorithms

The patterns of ultrasonic backscattered echoes represent valuable information pertaining to the geometric shape, size, and orientation of the reflectors as well as the microstructure of the propagation path. Accurate estimation of the ultrasonic echo pattern is essential in determining the object/propagation path properties. In this study, we model ultrasonic backscattered echoes in terms of superimposed Gaussian echoes corrupted by noise. Each Gaussian echo in the model is a nonlinear function of a set of parameters: echo bandwidth, arrival time, center frequency, amplitude, and phase. These parameters are sensitive to the echo shape and can be linked to the physical properties of reflectors and frequency characteristics of the propagation path. We address the estimation of these parameters using the maximum likelihood estimation (MLE) principle, assuming that all of the parameters describing the shape of the echo are unknown but deterministic. In cases for which noise is characterized as white Gaussian, the MLE problem simplifies to a least squares (LS) estimation problem. The iterative LS optimization algorithms when applied to superimposed echoes suffer from the problem of convergence and exponential growth in computation as the number of echoes increases. In this investigation, we have developed expectation maximization (EM)-based algorithms to estimate ultrasonic signals in terms of Gaussian echoes. The EM algorithms translate the complicated superimposed echoes estimation into isolated echo estimations, providing computational versatility. The algorithm outperforms the LS methods in terms of independence to the initial guess and convergence to the optimal solution, and it resolves closely spaced overlapping echoes     

Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window

I propose a systematic way to derive efficient, error-compensating algorithms for phase-shifting interferometry by integer approximation of well-known data-sampling windows. The theoretical basi of the approach is the observation that many of the common sources of phase-estimation error can be related to the frequency-domain characteristics of the sampling window. Improving these characteristics can therefore improve the overall performance of the algorithm. Analysis of a seven-frame example algorithm demonstrates an exceptionally good resistance to first- and second-order distortions in the phase shift and a much reduced sensitivity to low-frequency mechanical vibration.     

Multitaper scan-free spectrum estimation using a rotational shear interferometer

Multitaper methods for a scan-free spectrum estimation that uses a rotational shear interferometer are investigated. Before source spectra can be estimated the sources must be detected. A source detection algorithm based upon the multitaper F-test is proposed. The algorithm is simulated, with additive, white Gaussian detector noise. A source with a signal-to-noise ratio (SNR) of 0.71 is detected 2.9 degrees from a source with a SNR of 70.1, with a significance level of 10(-4), approximately 4 orders of magnitude more significant than the source detection obtained with a standard detection algorithm. Interpolation and the use of prewhitening filters are investigated in the context of rotational shear interferometer (RSI) source spectra estimation. Finally, a multitaper spectrum estimator is proposed, simulated, and compared with untapered estimates. The multitaper estimate is found via simulation to distinguish a spectral feature with a SNR of 1.6 near a large spectral feature. The SNR of 1.6 spectral feature is not distinguished by the untapered spectrum estimate. The findings are consistent with the strong capability of the multitaper estimate to reduce out-of-band spectral leakage.     

Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform

It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.     

Discrete cosine transform-based shift estimation for fringe pattern profilometry using a generalized analysis model

What is believed to be a new analysis algorithm to carry out profile measurement with low computational complexity and less noise sensitivity is presented. First, a discrete cosine transform (DCT)-based representation method is introduced to express the height distribution of a 3D surface. Then a novel shift estimation algorithm, called the DCT-based shift estimation (DCT-SE), is presented to reconstruct 3D object surfaces by using the proposed expression and the generalized analysis model. The advantage of DCT-SE is that without loss of measurement precision it provides lower computational complexity to implement 3D reconstruction from nonlinearly distorted fringe patterns and, at the same time, survives the random noise. Simulations and experiments show that the proposed DCT-SE is a fast, accurate, and efficient reconstruction algorithm for digital projection- based fringe pattern profilometry techniques.     

Robust fringe density and direction estimation in noisy phase maps.

A method for automated fringe analysis is presented. It robustly estimates local fringe density and direction in noisy wrapped phase maps. Such information can be used to improve the performance of two-dimensional phase unwrapping methods, to construct phase-jump-preserving filtering strategies, and also to perform robust segmentation of phase data. The method, which is highly insensitive to noise, is model based and performs the estimation in the Fourier domain.     

Towards simplified and optimized a posteriori error estimation using PGD reduced models

The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the finite element method. More specifically, it focuses on the constitutive relation error concept, which has been widely used over the last 40 years for FEM verification of computational mechanics models. A technical key‐point when using constitutive relation error is the construction of admissible fields, and we propose here to use the proper generalized decomposition to facilitate this task. In addition to making the implementation into commercial FE software easier, it is shown that the use of proper generalized decomposition enables to optimize the verification procedure and to get both accurate and reasonably expensive upper bounds on the discretization error. Numerical illustrations are presented to assess the performance of the proposed approach.     

Signal spectrum analysis and period estimation by using delayedsignal sampling

The paper describes a signal power spectrum analyzer and a signal period estimator whose bandwidth is not limited by the mean sampling time. The procedure relies on the evaluation of the input signal autocorrelation function in different delayed time instants, located at either equispaced or random time instants. To do this, a recursive random sampling process in the time domain was used in order to avoid any bandwidth limitation due to the sampling strategy in the evaluation of each autocorrelation function. The signal power spectrum as well as its period, provided that an approximate value of the fundamental frequency is known, can finally be evaluated. Some theoretical background and experimental work are reported in the paper for validating the performance of the method     

Large deformation mechanical testing of biological membranes using speckle interferometry in transmission. I: Experimental apparatus

This paper describes an apparatus designed to study large mechanical deformations in biological membranes. The task of mechanically characterizing biological membranes is challenging because of the anisotropic and nonlinear nature of their material properties. The apparatus described here is well suited to the task because it uses speckle interferometry to measure in-plane displacements in a distributed fashion and has multiple degrees of freedom in the applied stress mechanism. In this way few a priori assumptions or restrictions are imposed on the applied stress and strain fields. The interferometer operates in transmission mode to increase the light efficiency of the system since the sample biological membranes are translucent and reflect little light. The experimental results confirm that the strain fields in the biological membranes that are generated in the experiments are highly nonuniform and cannot be properly estimated from a small number of point measurements.     

Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications

A local frequency estimation approach for the fringe pattern with a spatial carrier by which the 2D spatial frequencies at a certain pixel are estimated from its neighborhood is presented. The applications of this approach in the fringe pattern analyses are also introduced. First, a 2D spatial carrier phase-shifting algorithm is derived. With it the detuning errors induced by frequency mismatching are avoided, and the stronger phase deformations can be successfully coped with. Second, a novel aperture extrapolation method is developed by which the phase accuracies of the Fourier-transform method at the aperture boundaries are effectively improved.     

Directional velocity estimation using focusing along the flow direction. I: Theory and simulation

A new method for directional velocity estimation is presented. The method uses beam formation along the flow direction to generate data in which the correct velocity magnitude can be directly estimated from the shift in position of the received consecutive signals. The shift is found by cross-correlating the beamformed lines. The approach can find the velocity in any direction, including transverse to the traditionally emitted ultrasound beam. The velocity estimation is studied through extensive simulations using Field II. A 128-element, 7-MHz linear array is used. A parabolic velocity profile with a peak velocity of 0.5 m/s is simulated for different beam-to-flow angles and for different emit foci. At 45/spl deg/ the relative standard deviation over the profile is 1.6% for a transmit focus at 40 mm. At 90/spl deg/ the approach gave a relative standard deviation of 6.6% with a transmit focus of 80 mm, when using 8 pulse-echo lines and stationary echo canceling. Pulsatile flow in the femoral artery was also simulated using Womersley's flow model. A purely transverse flow profile could be obtained with a relative standard deviation of less than 10% over the whole cardiac cycle using 8 pulse emissions for each imaging direction, which is sufficient to show clinically relevant transverse color flow images.     

Thermal stresses in a coating layer. I. General theoretical scheme

The evolution of thermal stresses generated by a thermal perturbation of cylindrical symmetry in a flat infinite isotropic thermoelastic layer of constant (before perturbation) thickness on a half-space rigid substrate is theoretically investigated. The problem is considered in the framework of uncoupled linear thermoelasticity, in the quasi-stationary displacement field approximation. Thermal insulation of the layer is considered in two versions: (a) both its surfaces are adiabatically insulated, and (b) the free surface is adiabatically insulated, whereas the contact surface is kept at constant temperature. The free surface of the layer is assumed to be stress free, and the contact surface is kept on a rigid substrate. The proper equations are solved using suitable Hankel and Fourier transformations. Part I of the paper presents a general theoretical scheme of the problem in a general case (without specification of the perturbation), and it is illustrated by a simple, detailed example (time evolution of stresses at the contact surface of the layer, generated by a surface point instantaneous heat pulse in case a). The analysis of stresses in more realistic cases, modeling realistic situations will be presented in Part II.     

Spectral analysis of R-lines and vibronic sidebands in the emission spectrum of ruby using genetic algorithms

The advancement in spectral analysis methods for the emission spectrum of ruby has been driven by the characterization of R-line peak shifts with stress in order to establish piezospectroscopic relationships. These relationships form the basis for the development of photo-stimulated luminescence spectroscopy (PSLS) as a nondestructive method to determine the integrity of the thermally grown oxide (TGO) layer on jet engine turbine blades. Besides the measurement technique, the accuracy of PSLS in stress measurements is influenced by the spectral analysis methodology, which is the focus of this paper. Gradient-based algorithms have been used widely in the methods developed thus far. The approach of using genetic algorithms in the spectral analysis of R-lines and vibronic bands is presented here for the first time and validated with the wellknown piezospectroscopic coefficients of the R-lines. The implementation of this method has led to significant new results in the quantification of peak shifts with uniaxial stress in the vibronic bands of the spectrum. The use of genetic algorithms is instrumental in the deconvolution and fitting of the numerous peaks in these bands. Fitting statistics, such as the fitness function and number of function evaluations, were used to assess the effectiveness of the procedures used in this method.     

Numerical calculation of a converging vector electromagnetic wave diffracted by an aperture by using Borgnis potentials. I. General theory

A method is proposed, on the basis of the vector electromagnetic theory, for the numerical calculation of the diffraction of a converging electromagnetic wave by a circular aperture by using Borgnis potentials as auxiliary functions. The diffraction problem of vector electromagnetic fields is simplified greatly by solving the scalar Borgnis potentials. The diffractive field is calculated on the basis of the boundary integral equation, taking into consideration the contribution of the field variables on the diffraction screen surface, which is ignored in the Kirchhoff assumption. An example is given to show the effectiveness and suitability of this method and the distinctiveness of the diffractive fields caused by the vector characteristics of the electromagnetic fields.