Peak-locking Error Research Articles

Comparative Analysis of Discrete Subtraction and Cross-Correlation for Subpixel Object Tracking

Many applications in physics and engineering require non-invasive, precise object tracking, which can be achieved with image processing methods at very good cost-efficiency ratios. The traditional method for measuring displacement with subpixel resolution involves cross-correlation between images and interpolation of the correlation peak. While this method enables target tracking with a resolution of thousandths of a pixel, it is computationally intensive and susceptible to peak-locking errors. Recently, a new method based on discrete subtraction between images has been presented as an alternative to cross-correlation to improve computational efficiency, which also results in being free of peak-locking errors. This manuscript presents an experimental evaluation of the performance of the discrete subtraction method (DSM) and compares it with the cross-correlation method in terms of subpixel accuracy and deviation errors. Four different targets were used with apparent displacements as small as 0.002 px, which approaches the theoretical digital resolution limit. The results show that the discrete subtraction method is more sensitive to noise but does not suffer from peak-locking error, thus being a reliable alternative to the correlation method, mainly for calibration processes.

Peak-locking full characterization: PIV error assessment and velocity ensemble measurement correction

Concluding a few previous approaches, the last work by the authors presented a complete analytic model for peak-locking error. That work was devoted to the mathematical foundation of the model. This paper separately focuses on the procedure for calibrating a given particle-image velocimetry measurement setup using the model. In addition to error assessment, it allows for the measurement correction of statistical ensembles such as: (a) the local velocity average and (b) the local velocity root mean square (RMS). This calibration is done by means of an inexpensive multiple Δt strategy, consisting of measuring the same flow field with different Δt subsets. This paper offers a comparison with previous empirical models produced by the authors, showing the advantages of the robust theoretical foundation established for this model. A real case corresponding to a hot high-speed coaxial jet, representative of the turbofan engine propulsion systems used for civil transport aircraft, is used for this comparison. The discrepancy plots between the different Δt subsets indicate that the peak-locking error description and correction are improved by 30% for the local velocity average. For the local velocity RMS, the results are even more striking, as the peak locking error can be almost completely corrected in situations where previous models only provided the order of magnitude of the error. Together with a detailed explanation of the calibration procedure, the paper offers guidelines to avoid getting into an ill-conditioned mathematical problem.

Full characterization of the peak-locking error by means of orthogonal functions and application to the flow behind a helicopter fuselage model

This work offers a 1D fully analytic complete modeling of the peak-locking error. Besides error assessment, this model allows for measurement correction. The paradigm of peak locking is revisited. The expressions of the functional dependence of the error are obtained for the two most relevant components of the measurement statistics: (1) the local velocity average, and (2) the local velocity rms. The prediction of the peak-locking error is related to complex interactions between the particle images size and shape and the measurement algorithms (peak fitting, image interpolation, etc.). The proposed model incorporates the different interactions by using a generalized approach that only requires the calibration of a few coefficients. This calibration is done by means of an inexpensive multiple Δt strategy, consisting in measuring the same flow field with different Δt subsets. Differently to previous empirical models from the authors, a robust theoretical foundation has been established. Besides providing rationale to the procedure, the results are significantly more accurate. A calibration using only three coefficients may correct up to 90% of the peak-locking error, while the previous empirical models would correct just in the order of 50%. This leaves a 10% of residual peak-locking error instead of 50%, reducing the final incidence by a factor of 5. The methodology is tested, in the context of turbulent flows, on real images corresponding to the measurement of the flow around the fuselage of a helicopter model. The test campaign was performed in the Italian Aerospace Research Centre (CIRA) CT-1 low speed wind tunnel. The proposed method, combined with the multiple Δt strategy, results in an error prediction accuracy that is good enough for correcting the measurements. Corrections to the average flow are in the order of 0.1 pixel (~ 3% of the convective velocity but in the order of 100% of the local flow variations). Corrections to the rms are in the same order; being the turbulence small, they cancel very large relative rms errors (~ 100%). The coherence between predicted error and actual error in real cases is further supported by the consistent matching that can be observed on their spatial patterns, associated with flow field statistics. An orthogonal expansion allows for full assessment of PIV peak-locking bias errors. The turbulent flow behind a helicopter fuselage model is used as test case. The expansion is calibrated and validated using a multiple Δt strategy. The resulting assessment is good enough to allow for correcting errors in the order of 0.1 pixel in the measured average as well as in the rms.

Multi-Δt approach for peak-locking error correction and uncertainty quantification in PIV

A novel approach is devised for the quantification of systematic uncertainty due to peak locking in particle image velocimetry (PIV), which also leads to correction of the peak-locking errors. The approach, applicable to statistical flow properties such as time-averaged velocity and Reynolds stresses, relies on image recordings with multiple time separations Δt and a least-squares regression of the measured quantities. In presence of peak locking, the measured particle image displacement is a non-linear function of Δt due to the presence of measurement errors which vary non-linearly with the sub-pixel particle image displacement. Additionally, the measured displacement fluctuations are a combination of the actual flow fluctuations and the measurement error. When the image recordings are acquired with multiple Δt’s, a least-squares regression among the statistical results yields a correction where systematic errors due to peak locking are significantly diminished. The methodology is assessed for planar PIV measurements of the flow over a NACA0012 airfoil at 10 degrees angle of attack. Reference measurements with much larger Δt than the Δt’s of the actual measurements, such that relative peak-locking errors are negligible for the former, are used to assess the validity of the proposed approach.

Influence of Neighborhood Size and Cross-Correlation Peak-Fitting Method on Location Accuracy

A known technique to obtain subpixel resolution by using object tracking through cross-correlation consists of interpolating the obtained correlation function and then refining peak location. Although the technique provides accurate results, peak location is usually biased toward the closest integer coordinate. This effect is known as the peak-locking error and it strongly limits this calculation technique’s experimental accuracy. This error may differ depending on the scene and algorithm used to fit and interpolate the correlation peak, but in general, it may be attributed to a sampling problem and the presence of aliasing. Many studies in the literature analyze this effect in the Fourier domain. Here, we propose an alternative analysis on the spatial domain. According to our interpretation, the peak-locking error may be produced by a non-symmetrical sample distribution, thus provoking a bias in the result. According to this, the peak interpolant function, the size of the local domain and low-pass filters play a relevant role in diminishing the error. Our study explores these effects on different samples taken from the DIC Challenge database, and the results show that, in general, peak fitting with a Gaussian function on a relatively large domain provides the most accurate results.

Peak-locking error reduction by birefringent optical diffusers

The use of optical diffusers for the reduction of peak-locking errors in particle image velocimetry is investigated. The working principle of the optical diffusers is based on the concept of birefringence, where the incoming rays are subject to different deflections depending on the light direction and polarization. The performances of the diffusers are assessed via wind tunnel measurements in uniform flow and wall-bounded turbulence. Comparison with best-practice image defocusing is also conducted. It is found that the optical diffusers yield an increase of the particle image diameter up to 10 µm in the sensor plane. Comparison with reference measurements showed a reduction of both random and systematic errors by a factor of 3, even at low imaging signal-to-noise ratio.

Peak-locking reduction for particle image velocimetry

A parametric study of the factors contributing to peak-locking, a known bias error source in particle image velocimetry (PIV), is conducted using synthetic data that are processed with a state-of-the-art PIV algorithm. The investigated parameters include: particle image diameter, image interpolation techniques, the effect of asymmetric versus symmetric window deformation, number of passes and the interrogation window size. Some of these parameters are found to have a profound effect on the magnitude of the peak-locking error. The effects for specific PIV cameras are also studied experimentally using a precision turntable to generate a known rotating velocity field. Image time series recorded using this experiment show a linear range of pixel and sub-pixel shifts ranging from 0 to ±4 pixels. Deviations in the constant vorticity field (ωz) reveal how peak-locking can be affected systematically both by varying parameters of the detection system such as the focal distance and f-number, and also by varying the settings of the PIV analysis.A new a priori technique for reducing the bias errors associated with peak-locking in PIV is introduced using an optical diffuser to avoid undersampled particle images during the recording of the raw images. This technique is evaluated against other a priori approaches using experimental data and is shown to perform favorably. Finally, a new a posteriori anti peak-locking filter (APLF) is developed and investigated, which shows promising results for both synthetic data and real measurements for very small particle image sizes.

Quantitative evaluation of PIV peak locking through a multiple Δt strategy: relevance of the rms component

The possibility of using different times between laser pulses (Δt) in a PIV (Particle Image Velocimetry) measurement of the same real flow field for error assessment has already been proposed by the authors in a recent paper Nogueira et al. (Meas Sci Technol 20, 2009). It is a simple procedure that is available with the usual PIV setup. In that work, peak locking was considered basically as a bias error. Later measurements indicated that, using appropriate processing algorithms, this error is not the main peak-locking effect. Scenarios with the rms (root mean square) error due to peak locking as the most relevant contribution are more common than initially expected and require a differentiated approach. This issue is relevant due to the impact of the rms error in the evaluation of flow quantities like turbulent kinetic energy. The first part of this work is centred on showing that peak-locking error in PIV is not always a measurement bias towards the closest pixel integer displacement. Insight in the subject indicates that this is the case only for algorithm-induced peak locking. The peak locking coming out of image acquisition limitations (i.e. resolution) is not ‘a priory’ biased. It is a random error with a peculiar probability density function. Discussion on the subject is offered, and a particular approach to use a simple multiple Δt strategy to asses this error is proposed. The results reveal that in real images where amplitude of the peak-locking bias error is assessed to be as small as 0.02 pixels, rms errors can be in the order of 0.1 pixels. As PIV approaches maturity, providing a quantitative confidence interval by estimating measurement error seems essential. The method developed is robust enough to quantify these values in the presence of turbulence with rms up to ~0.6 pixels. This proposal constitutes a relevant step forward from the traditional histogram-based considerations that only reveal whether strong peak-locking error is present or not, without any information on its magnitude or whether its origin is bias or rms.

Reduction of peak-locking errors produced by Gaussian sub-pixel interpolation in cross-correlation digital particle image velocimetry

Cross-correlation, whether by a direct or spectral calculation, remains the most common analysis method for digital particle image velocimetry. Gaussian sub-pixel interpolation likewise remains a common method for achieving measurement resolution at fractional pixel displacements. Specific to the use of Gaussian sub-pixel interpolation, the particular form of the cross-correlation function is shown to affect the peak-locking or mean-bias error that increases the probability of the measurement occurring at integer pixel displacements. The main effect is from nonzero mean image intensities that cause a shift in the amplitude of the cross-correlation function away from zero and introduce a bias error into the Gaussian interpolation formula. A simple, but not universal, method of reducing this error is to use the covariance function, often calculated in a single step after calculation of the cross-correlation function. This effectively shifts the amplitude back toward zero and reduces the bias that originates in the interpolation formula. Examples are shown using synthetic particle image simulations and experimental water jet PIV images from the 2003 PIV Challenge.

Multiple Δt strategy for particle image velocimetry (PIV) error correction, applied to a hot propulsive jet

PIV (particle image velocimetry) is a measurement technique with growing application to the study of complex flows with relevance to industry. This work is focused on the assessment of some significant PIV measurement errors. In particular, procedures are proposed for estimating, and sometimes correcting, errors coming from the sensor geometry and performance, namely peak-locking and contemporary CCD camera read-out errors. Although the procedures are of general application to PIV, they are applied to a particular real case, giving an example of the methodology steps and the improvement in results that can be obtained. This real case corresponds to an ensemble of hot high-speed coaxial jets, representative of the civil transport aircraft propulsion system using turbofan engines. Errors of ∼0.1 pixels displacements have been assessed. This means 10% of the measured magnitude at many points. These results allow the uncertainty interval associated with the measurement to be provided and, under some circumstances, the correction of some of the bias components of the errors. The detection of conditions where the peak-locking error has a period of 2 pixels instead of the classical 1 pixel has been made possible using these procedures. In addition to the increased worth of the measurement, the uncertainty assessment is of interest for the validation of CFD codes.

Elimination of peak-locking error in PIV analysis using the correlation mapping method

The peak-locking effect causes mean bias in most of the existing cross-correlation based algorithms for PIV data analysis. This phenomenon is inherent to the smooth curve-fitting through discrete correlation values, which is used to obtain the sub-pixel part of the displacement. Almost all of the existing effective methods to solve this problem require iterations. In this paper we introduce a new technique for obtaining sub-pixel accuracy, which bypasses the sub-pixel curve fitting, and eliminates the peak-locking effect, but does not require iterations. The principles of the ‘correlation mapping method’ (CMM) are based on the following logic: if one uses a bi-cubic interpolation to express the second image based on the first one and the unknown displacement, the correlation between them becomes a third-order polynomial of the displacement, whose coefficients depend on the first image. Matching this polynomial with the measured correlation provides an equation for the displacement for each point of the correlation map. A least-squares fit to the correlation values in the vicinity of the correlation peak (e.g. 5 × 5 points) provides an estimate for the particle displacement, including its sub-pixel part. We combine the new correlation mapping method with corrections for particle image distortion (PID) to further reduce the uncertainty in the velocity measurements. Three iterations typically achieve converged results. The CMM-PID method is tested using synthetic and experimental data. The peak-locking bias disappears in all cases. Even the ‘random’ error is substantially smaller than that obtained using a conventional sub-pixel curve fit. Issues related to streamline curvature and uncertainty in estimates of velocity gradients are also discussed.

A107 ピークロッキング現象の軽減によるPTVの計測精度向上

The spatio-temporal differential method(gradient method) can be one of the high precision algorithms for PIV analysis. But, when the method is applied to actual PIV pictures, measurement precision cannot be so good as is expected. It is found that inadequate numerical differentiations and peak-locking phenomenon degrade its intrinsic characteristics. The first issue is discussed in a separate paper. In this paper, the peak-locking phenomena for PTV with various picture conditions and influences of interpolation methods on the phenomenon are examined and discussed. Some methods to reduce the peak-locking error are recommended, and the precision of less than 0.05 pixel ambiguity could be realized.