Measurement of frequency response functions in noisy environments

The geometric mean is proposed as an alternative averaging technique for frequency response function (FRF) measurements of a linear system. It is shown that it produces almost unbiased measurements even if the input and output measurements are both disturbed with (normal distributed) noise. The properties of the arithmetic and geometric mean of expression are compared to those of the H₁ and H₂ methods. It is shown that if the S/N ratio on the input and the output measurements is higher than 3 dB, then it is possible to generate measurements without systematic errors     

High-precision shape measurement by white-light interferometry with real-time scanner error correction

White-light interferometric techniques allow high-precision shape measurement of objects with discontinuous structures by detecting the peak of the coherence envelope. These techniques assume a specific change in the optical path difference (OPD) between the interfering beams; however, the scanning device effecting that change often introduces OPD errors that are carried over to the measurements. We present a technique for measuring OPD changes from the collected interference fringes during each measurement. Information about the scan is directly fed into the algorithm, which compensates for the errors, resulting in improved measurement accuracy. The method corrects not only the scanner errors but also slowly varying vibrations. In addition, this technique can be easily adapted to any existing low-coherence interferometer because no large data storage or postprocessing is required.     

数字平均法的误差分析

文章针对以前提出的声时测量中的随机数字平均技术，给出了其理论依据，详细讨论了误差的大小，并利用概率统计理论进行了论证。同时，文中还提出均匀延时数字平均技术。在实际应用中，对该类仪器的测量误差的估计和测量次数的选取以及这类仪器的设计有着重要的指导作用。     

Multiple-target detection with use of a modified amplitude-modulated joint transform correlator

Multiple-target detection with a modified amplitude-modulated joint transform correlator is proposed. With this technique the joint power spectrum is first modified by the subtraction of the power spectrum of the input scene only and of the reference image from it; the resultant modified joint power spectrum is next multiplied by the amplitude-modulated filter function. The effect of noise in the input scene on the performance of the joint transform correlator is analyzed and quantified. This technique is found to deliver a correlation output and the capacity to accommodate noise in the input scene better than both the fringe-adjusted filter-based joint transform correlator and the modified fringe-adjusted filter-based joint transform correlator.     

Modified fringe-adjusted joint transform correlation to accommodate noise in the input scene

A modified fringe-adjusted joint transform correlator is proposed that is able to accommodate noise in the input scene. The effect of noise in the input scene on the performance of the joint transform correlator is analyzed and quantified. When the target is embeddedin aseverely noise-corrupted input scene, it is shown that the proposed modified fringe-adjusted filter joint transform correlator delivers a better correlation performance and the capacity to accommodate this noise than does the fringe-adjusted filter-based correlator. When the power spectra of the input image and the reference image are subtracted from the power spectrum of the joint-input image, it is found that the noise effect on the output plane is independent of the objects in the input scene and originates from the convolution of the reference image and noise in the input scene.     

Maximum likelihood for target location in the presence of substitutive noise

We consider the optimal likelihood algorithm for the estimation of a target location when the images are corrupted by substitutive noise. We show the relationship between the optimal algorithm and the sliced orthogonal nonlinear generalized (SONG) correlation. The SONG correlation is based on the application of a linear correlation to corresponding binary slices of both the input scene and the reference object with appropriate weight factors. For a particular case, we show that the optimal strategy is a function of only the number of pixels for which the gray values in the noisy image match the ones of the reference image when the substitutive noise is uniformly distributed. This is exactly what a particular definition of the SONG correlation does.     

Adaptive phase-input joint transform correlator

An adaptive phase-input joint transform correlator for pattern recognition is presented. The input of the system is two phase-only images: input scene and reference. The reference image is generated with a new iterative algorithm using phase-only synthetic discriminant functions. The algorithm takes into account calibration lookup tables of all optoelectronics elements used in optodigital experiments. The designed adaptive phase-input joint transform correlator is able to reliably detect a target and its distorted versions embedded into a cluttered background. Computer simulations are provided and compared with those of various existing joint transform correlators in terms of discrimination capability, tolerance to input additive noise, and to small geometric image distortions. Experimental optodigital results are also provided and discussed.     

Three-flat test solutions based on simple mirror symmetry

In interferometric surface and wavefront metrology, three-flat tests are the archetypes of measurement procedures to separate errors in the interferometer reference wavefront from errors due to the test part surface, so-called absolute tests. What is believed to be a new class of solutions of the three-flat problem for circular flats is described in terms of functions that are symmetric or antisymmetric with respect to reflections at a single line passing through the center of the flat surfaces. The new solutions are simpler and easier to calculate than the known solutions based on twofold mirror symmetry or rotation symmetry. Strategies for effective azimuthal averaging and a method for determining the averaging error are also discussed.     

Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements

Aspheric optical surfaces are often tested using diffractive optics as null elements. For precise measurements, the errors caused by the diffractive optical element must be calibrated. Recently, we reported first experimental results of a three position quasi-absolute test for rotationally invariant aspherics by using combined-diffractive optical elements (combo-DOEs). Here we investigate the effects of the DOE substrate errors on the proposed calibration procedure and present a set of criteria for designing an optimized combo-DOE. It is demonstrated that this optimized design enhances the overall consistency of the procedure. Furthermore, the rotationally varying part of the surface deviations is compared with the rotationally varying deviations obtained by an N-position averaging procedure and is found to be in good agreement.     

Simultaneous variable selection and outlier detection using a robust genetic algorithm

Given a dataset in which it is known that all spectra are representative, without error, and have matching accurate reference values, there are many tools which exist to determine the best set of variables to use for constructing an inverse model, such as partial least squares (PLS). Likewise, given that the best variables are known a priori, there are many tools that can be used to determine if any samples are outliers, either due to inaccurate reference values, or due to invalid spectra. However, in many real-world situations, the reference values contain error and the spectra are imperfect. In this situation, it is not always possible to determine either the best subset of samples or the best subset of variables. This paper presents a new technique for combining a robust outlier determination method with a genetic algorithm optimized for spectral variable selection. No assumptions are made as to the optimum set of variables or as to the amount and structure of the errors present in either the predictor (X) or predictand (Y) variables. The technique is best suited for datasets which contain redundant information, i.e., datasets from designed experiments with no replicates may not produce optimum results, as the experimental design implicitly assumes there are no outlier data.     

Improved Feature Extraction by use of a Joint Wavelet Transform Correlator

A new joint wavelet transform correlation-based technique is proposed for feature extraction such as the detection of edges in an unknown input scene. We exploited a modified version of the Roberts and the Sobel wavelet filters as reference images for extracting the edges of an unknown input scene. The performance of the proposed technique with the aforementioned wavelet filters is evaluated and compared by use of numerical simulations. For noise-free input scenes the Roberts wavelet filter was found to yield a superior output compared with that of the Sobel wavelet filter. However, for noisy input scenes the Sobel wavelet filter was found to yield a better output compared with the Roberts wavelet filter.     

Design and Implementation of a Fuzzy Area-Based Image-Scaling Technique

In this paper, we propose the design and implementation of an interpolation scheme for performing image scaling by utilizing a dynamic mask combined with a sophisticated neighborhood averaging fuzzy algorithm. The functions that contribute to the final interpolated image are the areas of the input pixels, overlapped by a dynamic mask, and the difference in intensity between the input pixels. Fuzzy if–then rules for these two functions are presented to carry out the interpolation task. Simulation results have shown a fine high-frequency response and a low interpolation error, in comparison with other widely used algorithms. The interpolation can be applied to both gray-scale and color images for any scaling factor. The proposed hardware structure is implemented in a field-programmable gate array (FPGA) chip and is based on a sequence of pipeline stages and parallel processing to minimize computation times. The fuzzy image interpolation implementation combines a fuzzy inference system and an image-interpolation technique in one hardware system. Its main features are the ability to accurately approximate the Gaussian membership functions used by the fuzzy inference system with very few memory requirements and its high-frequency performance of 65 MHz, making it appropriate for real-time imaging applications. The system can magnify gray-scale images of up to 10-bit resolution. The maximum input image size is 1024 $times$ 1024 pixels for a maximum of 800% magnification.      

Flexible error-reduction method for shape measurement by temporal phase unwrapping: phase averaging method

Temporal phase unwrapping is an important method for shape measurement in structured light projection. Its measurement errors mainly come from both the camera noise and nonlinearity. Analysis found that least-squares fitting cannot completely eliminate nonlinear errors, though it can significantly reduce the random errors. To further reduce the measurement errors of current temporal phase unwrapping algorithms, in this paper, we proposed a phase averaging method (PAM) in which an additional fringe sequence at the highest fringe density is employed in the process of data processing and the phase offset of each set of the four frames is carefully chosen according to the period of the phase nonlinear errors, based on fast classical temporal phase unwrapping algorithms. This method can decrease both the random errors and the systematic errors with statistical averaging. In addition, the length of the additional fringe sequence can be changed flexibly according to the precision of the measurement. Theoretical analysis and simulation experiment results showed the validity of the proposed method.     

Eddy current inversion for layered conductors

By making multifrequency eddy current measurements on a layered conductor, it is possible to acquire information on the depth dependence of the conductivity. We consider an inversion problem in which coil impedance data are used to determine either the layer thicknesses or layer conductivities. The algorithm is based on a well known forward model which gives the impedance of an air cored coil above a stratified conductor from a closed form expression. In the forward calculation, estimates of the unknown material parameters are used to get tentative predictions of the measurements. Differences between these predictions and measured impedances are expressed in terms of a global error that is minimized iteratively with the aid of a descent algorithm by varying the parameters of the structure. Examples of minimization searches for layer parameters are given.This article is dedicated to Professor Bertram A. Auld on the occasion of his 70th birthday and his retirement from Stanford University.     

Reconstruction of time-dependent boundary heat flux by a BEM-based inverse algorithm

The objective of this study is to reconstruct an unknown time-dependent heat flux distribution at a surface whose temperature history is provided by a broad-band thermochromic liquid crystal (TLC) thermographic technique. The information given for this inverse problem is the surface temperature history. Although this is not an inverse problem, it is solved as such in order to filter the errors in input temperatures which are reflected in errors in heat fluxes. We minimize a quadratic functional which measures the sum of the squares of the deviation of estimated (computed) temperatures relative to measured temperatures provided by the TLC thermography. The objective function is minimized using the Levenberg–Marquardt method, and we develop an explicit scheme to compute the required sensitivity coefficients. The unknown flux is allowed to vary in space and time. Results are presented for a simulation in which a spatially varying and time-dependent flux is reconstructed over an airfoil.     

Discrete-time method for signal-to-noise power ratio measurement

The measurement of signal-to-noise power ratio (SNR) is of fundamental importance in many areas of electrical engineering, such as communications, signal processing, tests and measurements, circuits and systems, etc. In this paper, we propose two algorithms for estimating the signal-to-noise ratio of a noisy sinewave from discrete-time data obtained by sampling the input signal. One algorithm is based on the estimation of the four parameters of the input sinewave. The second algorithm is based on estimating the average noise power by averaging the squared magnitude of the FFT bins attributed to the noise. Both methods show excellent performance. Simulation results indicate that the four-parameter method requires the input SNR to be at least 10 dB and the input signal frequency not exceeding one-third of the sampling frequency. On the other hand, the second approach, the spectrum averaging method, shows a remarkable robustness over a very wide range of normalized frequencies (with respect to the Nyquist frequency) and SNRs (well over 100 dB). This spectrum averaging method should prove to be very useful in a wide range of applications     

Identification of linear systems in the presence of nonlineardistortions

This paper treats the identification of linear systems in the presence of nonlinear distortions. It extends the theory developed for measurement setups where the input is exactly known and the output is observed with errors (output error framework) to measurement setups where both the input and output are observed with errors (errors-in-variables framework). An appropriate measurement strategy and identification algorithm are presented     

Determining the Uncertainty of X-Ray Absorption Measurements

X-ray absorption (or more properly, x-ray attenuation) techniques have been applied to study the moisture movement in and moisture content of materials like cement paste, mortar, and wood. An increase in the number of x-ray counts with time at a location in a specimen may indicate a decrease in moisture content. The uncertainty of measurements from an x-ray absorption system, which must be known to properly interpret the data, is often assumed to be the square root of the number of counts, as in a Poisson process. No detailed studies have heretofore been conducted to determine the uncertainty of x-ray absorption measurements or the effect of averaging data on the uncertainty. In this study, the Poisson estimate was found to adequately approximate normalized root mean square errors (a measure of uncertainty) of counts for point measurements and profile measurements of water specimens. The Poisson estimate, however, was not reliable in approximating the magnitude of the uncertainty when averaging data from paste and mortar specimens. Changes in uncertainty from differing averaging procedures were well-approximated by a Poisson process. The normalized root mean square errors decreased when the x-ray source intensity, integration time, collimator size, and number of scanning repetitions increased. Uncertainties in mean paste and mortar count profiles were kept below 2 % by averaging vertical profiles at horizontal spacings of 1 mm or larger with counts per point above 4000. Maximum normalized root mean square errors did not exceed 10 % in any of the tests conducted.     

Errors and uncertainties in the measurement of ultrasonic wave attenuation and phase velocity

This paper presents an analysis of the error generation mechanisms that affect the accuracy of measurements of ultrasonic wave attenuation coefficient and phase velocity as functions of frequency. In the first stage of the analysis we show that electronic system noise, expressed in the frequency domain, maps into errors in the attenuation and the phase velocity spectra in a highly nonlinear way; the condition for minimum error is when the total measured attenuation is around 1 Neper. The maximum measurable total attenuation has a practical limit of around 6 Nepers and the minimum measurable value is around 0.1 Neper. In the second part of the paper we consider electronic noise as the primary source of measurement error; errors in attenuation result from additive noise whereas errors in phase velocity result from both additive noise and system timing jitter. Quantization noise can be neglected if the amplitude of the additive noise is comparable with the quantization step, and coherent averaging is employed. Experimental results are presented which confirm the relationship between electronic noise and measurement errors. The analytical technique is applicable to the design of ultrasonic spectrometers, formal assessment of the accuracy of ultrasonic measurements, and the optimization of signal processing procedures to achieve a specified accuracy.