A combination correlation-based interrogation and tracking algorithm for digital PIV evaluation

A combination of the correlation-based interrogation algorithm and the correlation-based tracking algorithm is proposed for digital PIV evaluation. A zero-padding interrogation algorithm is adopted in which the interrogation windows differ in size and in which the number of pixels in the side length of the smaller window is not restricted to a power of 2. This greatly improves the algorithm's accuracy and measurement range. The correlation-based tracking algorithm is employed when the already-measured vector can serve as a good predictor of the next vector to be measured. In this case, only a very small searching scope is required and computation can be fast. Computational intensity analysis shows that, using the same-sized sampled window, the correlation-based tracking algorithm is more efficient than the conventional correlation-based interrogation algorithm if the searching scope is less than 4. Compared with some of the other correlation-based algorithms, the proposed combination method is faster, is more accurate, has a larger measurable range, and can utilize a sampled window of any size.     

A digital mask technique for reducing the bias error of the correlation-based PIV interrogation algorithm

 In this paper the bias phenomenon in the evaluation of PIV recordings by using the correlation-based interrogation algorithm is discussed, and a digital mask technique, that can effectively reduce the bias error, is introduced. The correlation-based interrogation algorithm, when masked with a Gaussian window function, can achieve a higher evaluation accuracy not only for PIV recordings of flows with small velocity gradients, but also for that of flows with large gradients. Received: 14 October 1998/Accepted: 20 July 1999     

A comparative study of five different PIV interrogation algorithms

Five different particle image velocimetry (PIV) interrogation algorithms are tested with numerically generated particle images and two real data sets measured in turbulent flows with relatively small particle images of size 1.0–2.5 pixels. The size distribution of the particle images is analyzed for both the synthetic and the real data in order to evaluate the tendency for peak-locking occurrence. First, the accuracy of the algorithms in terms of mean bias and rms error is compared to simulated data. Then, the algorithms ability to handle the peak-locking effect in an accelerating flow through a 2:1 contraction is compared, and their ability to estimate the rms and Reynolds shear stress profiles in a near-wall region of a turbulent boundary layer (TBL) at Re=510 is analyzed. The results of the latter case are compared to direct numerical simulation (DNS) data of a TBL. The algorithms are: standard fast Fourier transform cross-correlation (FFT-CC), direct normalized cross-correlation (DNCC), iterative FFT-CC with discrete window shift (DWS), iterative FFT-CC with continuous window shift (CWS), and iterative FFT-CC CWS with image deformation (CWD). Gaussian three-point peak fitting for sub-pixel estimation is used in all the algorithms. According to the tests with the non-deformation algorithms, DNCC seems to give the best rms estimation by the wall, and the CWS methods give slightly smaller peak-locking observations than the other methods. With the CWS methods, a bias error compensation method for the bilinear image interpolation, based on the particle image size analysis, is developed and tested, giving the same performance as the image interpolation based on the cardinal function. With the CWD algorithms, the effect of the spatial filter size between the iteration loops is analyzed, and it is found to have a strong effect on the results. In the near-wall region, the turbulence intensity varies by up to 4%, depending on the chosen interrogation algorithm. In addition, the algorithms computational performance is tested.     

A comparative study of the MQD method and several correlation-based PIV evaluation algorithms

 The Minimum Quadratic Difference (MQD) method is compared with methods conventionally used for the evaluation of PIV recordings, i.e. correlation-based evaluation with fixed interrogation windows (auto- or cross-correlation) and correlation-based tracking. The comparison is performed by studying the evaluation accuracy achieved when applying these methods to pairs of synthetic PIV recordings for which the true displacements are known. The influence of the magnitude of the particle image displacement, evaluation window size, density of particle image distribution, and particle image size on the accuracy are investigated. In all these cases the best results in terms of a statistical error are obtained with the MQD method. The superiority of the MQD method can be explained with its potential of accounting for non-uniformities in the particle image distribution and a non-uniform illumination. It is also shown that the conventional correlation-based methods may produce principal errors that are non-existent for the MQD method. The evaluation speed achievable for the MQD method by making use of the FFT is comparable to that common for the generally used auto- or cross-correlation algorithm. Finally, a quantitative explanation is given for the often observed phenomenon that PIV velocity results tend to be smaller than the true values. Received: 15 May 1998/Accepted: 24 April 1999     

Identification of a new source of peak locking, analysis and its removal in conventional and super-resolution PIV techniques

One of the key factors that limit accuracy of particle image velocimetry (PIV) is the peak-locking effect. In this paper, a previously uncharacterised source of peak locking is presented. This source is neither related to the sensor geometry nor the subpixel resolution peak-fitting algorithms. It is present even when the particles are well described in terms of sensor spatial resolution (i.e. for particle diameters larger than 2 pixels). If no specific actions to avoid it are taken, its effect is especially important in those super-resolution systems that are based on iteratively reducing the size of the interrogation window. In this work, the mentioned source and its effects are studied and modelled. Based on this study, the actions required to avoid this type of peak locking are described. This includes the most usual correlation-based PIV systems, as well as super-resolution ones. Once this source of inaccuracy is avoided, it is possible to discriminate the performance of different types of correlation algorithms. As a consequence, specific proposals for the algorithms in the last steps of multigrid super-resolution PIV systems are given. The performances of the proposed solutions are verified using both synthetic and real PIV images. Received: 31 January 2000/Accepted: 2 May 2000     

Biases of PIV measurement of turbulent flow and the masked correlation-based interrogation algorithm

Influences of evaluation bias of the correlation-based interrogation algorithm on particle image velocimetry (PIV) measurement of turbulent flow are investigated. Experimental tests in the Iowa Institute of Hydraulic Research towing tank with a towed PIV system and a surface-piercing flat plate and simulations demonstrate that the experimentally determined mean velocity and Reynolds stress components are affected by the evaluation bias and the gradient of the evaluation bias, respectively. The evaluation bias and gradient of the evaluation bias can both be minimized effectively by using Gaussian digital masks on the interrogation window, so that the measurement uncertainty can be reduced. Received: 16 September 1999/Accepted: 7 February 2000     

A new particle tracking algorithm based on deterministic annealing and alternative distance measures

We describe a new particle tracking algorithm for the interrogation of double frame single exposure data, which is obtained with particle image velocimetry. The new procedure is based on an algorithm which has recently been proposed by Gold et al. (Gold et al., 1998) for solving point matching problems in statistical pattern recognition. For a given interrogation window, the algorithm simultaneously extracts: (i) the correct correspondences between particles in both frames and (ii) an estimate of the local flow-field parameters. Contrary to previous methods, the algorithm determines not only the local velocity, but other local components of the flow field, for example rotation and shear. This makes the new interrogation method superior to standard methods in particular in regions with high velocity gradients (e.g. vortices or shear flows). We perform benchmarks with three standard particle image velocimetry (PIV) and particle tracking velocimetry (PTV) methods: cross-correlation, nearest neighbour search, and image relaxation. We show that the new algorithm requires less particles per interrogation window than cross-correlation and allows for much higher particle densities than the other PTV methods. Consequently, one may obtain the velocity field at high spatial resolution even in regions of very fast flows. Finally, we find that the new algorithm is more robust against out-of-plane noise than previously proposed methods. Received: 1 March 1999 / Accepted: 29 July 1999     

Simultaneous two-phase PIV by two-parameter phase discrimination

 A flexible and robust phase discrimination algorithm for two-phase PIV employs second-order intensity gradients to identify objects. Then, the objects are sorted into solids and tracers according to parametric combinations of size and brightness. Solids velocities are computed by tracking, gas velocities by cross-correlation. Tests in a fully-developed turbulent channel flow of air showed that the two phases do not contaminate or bias each other's velocity statistics. Error magnitude and valid data yield were quantified with artificial images for three particle sizes (25, 33, and 63 μm), two interrogation area sizes (32 and 64 pixels), and volumetric solids loads from 0.0022% to 0.014%. At the channel centerline, the gas valid data yield was above 98% and the RMS error in gas velocity was less than 0.1 pixels for all variations of these parameters. The solid-to-tracer signal ratio was found to be the major parameter affecting the magnitude of the RMS error. Received: 20 September 2000/Accepted: 2 July 2001 Published online: 29 November 2001     

PIV measurement of flow around an arbitrarily moving body

This paper presents a PIV (particle image velocimetry) image processing method for measuring flow velocities around an arbitrarily moving body. This image processing technique uses a contour-texture analysis based on user-defined textons to determine the arbitrarily moving interface in the particle images. After the interface tracking procedure is performed, the particle images near the interface are transformed into Cartesian coordinates that are related to the distance from the interface. This transformed image always has a straight interface, so the interrogation windows can easily be arranged at certain distances from the interface. Accurate measurements near the interface can then be achieved by applying the window deformation algorithm in concert with PIV/IG (interface gradiometry). The displacement of each window is evaluated by using the window deformation algorithm and was found to result in acceptable errors except for the border windows. Quantitative evaluations of this method were performed by applying it to computer-generated images and actual PIV measurements.     

On the resolution limit of digital particle image velocimetry

This work analyzes the spatial resolution that can be achieved by digital particle image velocimetry (DPIV) as a function of the tracer particles and the imaging and recording system. As the in-plane resolution for window-correlation evaluation is related by the interrogation window size, it was assumed in the past that single-pixel ensemble-correlation increases the spatial resolution up to the pixel limit. However, it is shown that the determining factor limiting the resolution of single-pixel ensemble-correlation are the size of the particle images, which is dependent on the size of the particles, the magnification, the f-number of the imaging system, and the optical aberrations. Furthermore, since the minimum detectable particle image size is determined by the pixel size of the camera sensor in DPIV, this quantity is also considered in this analysis. It is shown that the optimal magnification that results in the best possible spatial resolution can be estimated from the particle size, the lens properties, and the pixel size of the camera. Thus, the information provided in this paper allows for the optimization of the camera and objective lens choices as well as the working distance for a given setup. Furthermore, the possibility of increasing the spatial resolution by means of particle tracking velocimetry (PTV) is discussed in detail. It is shown that this technique allows to increase the spatial resolution to the subpixel limit for averaged flow fields. In addition, PTV evaluation methods do not show bias errors that are typical for correlation-based approaches. Therefore, this technique is best suited for the estimation of velocity profiles.     

Generating arbitrarily sized interrogation windows for correlation-based analysis of particle image velocimetry recordings

 We describe a technique that allows an arbitrary size of the interrogation window when using the traditional FFT algorithm in analysing PIV recordings by either cross- or auto-correlation methods. The length and width of the effective interrogation window are no longer required to be composed of a number of pixels making a power of 2 (16, 32, 64 etc). This gives a higher flexibility in selecting the appropriate window size. Received: 28 January 1997/Accepted: 11 August 1997     

Second-order accurate particle image velocimetry

 An adaptive, second-order accurate particle image velocimetry (PIV) technique is presented. The technique uses two singly exposed images that are interrogated using a modified cross-correlation algorithm. Consequently, any of the equipment commonly available for conventional PIV (such as dual head Nd: YAG lasers, interline transfer CCD cameras, etc.) can be used with this more accurate algorithm. At the heart of the algorithm is a central difference approximation to the flow velocity (accurate to order Δt ²) versus the forward difference approximation (accurate to order Δt) common in PIV. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms have been gaining popularity in recent years because they allow the spatial resolution of the PIV technique to be maximized. Adaptive shifting algorithms also have the virtue of helping to eliminate velocity bias errors. The second- order accuracy resulting from the central difference approximation can be obtained with relatively little additional computational effort compared to that required for a standard first-order accurate forward difference approximation. The adaptive central difference interrogation (CDI) algorithm has two main advantages over adaptive forward difference interrogation (FDI) algorithms: it is more accurate, especially at large time delays between camera exposures; and it provides a temporally symmetric view of the flow. By comparing measurements of flow around a single red blood cell made using both algorithms, the CDI technique is shown to perform better than conventional FDI-PIV interrogation algorithms near flow boundaries. Cylindrical Taylor–Couette flow images, both experimental and simulated, are used to demonstrate that the CDI algorithm is significantly more accurate than conventional PIV algorithms, especially as the time delay between exposures is increased. The results of the interrogations are shown to agree quite well with analytical predictions and confirm that the CDI algorithm is indeed second-order accurate while the conventional FDI algorithm is only first-order accurate. Received: 15 June 2000/Accepted: 2 February 2001     

Quantitative evaluation of PIV peak locking through a multiple Δ<Emphasis Type="Italic">t</Emphasis> 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.     

Performances of feature tracking in turbulent boundary layer investigation

In this paper, we describe the application of a feature tracking (FT) algorithm for the measurement of velocity statistics in a turbulent boundary layer over a flat plate at Re _θ ≃ 3,700. The feature tracking algorithm is based on an optical flow approach. Displacements are obtained by searching the parameters of the mapping between interrogation windows in the first and second image which minimize a correlation distance between them. The correlation distance is here defined as the minimum of the sum of squared differences of interrogation windows intensities. The linearized equation which governs the minimization problem is solved with an iterative procedure only where the solution is guaranteed to exist, thus maximizing the signal-to-noise ratio. In this process, the interrogation window first undergoes a pure translation, and then a complete affine deformation. Final mapping parameters provide the velocity and velocity gradients values in a lagrangian framework. The interpolation inherent to window-deforming algorithms represents a critical factor for the overall accuracy and particular attention must be devoted to this step. In this paper different schemes are tested, and their effects on algorithm accuracy are first discussed by looking at the distribution of systematic and random errors computed from synthetic images. The same analysis is then performed on the turbulent boundary layer data, where the effects associated with the use of a near-wall logical mask are also investigated. The comparison with single-point data gathered from the literature demonstrate the overall ability of the FT technique to correctly extract all relevant statistical quantities, including the spanwise vorticity distribution. Concerning the mean velocity profile, no evident influence of the interpolation scheme appears, while the near-wall accuracy is improved by the application of the logical mask. On the contrary, for the fluctuating components of the velocity, the interpolation based on B-Spline basis functions is found to perform better compared to the classical Bicubic scheme, particularly in the highly sheared region close to the wall.

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A correlation-based central difference image correction (CDIC) method and application in a four-roll mill flow PIV measurement

An experiment is conducted in a four-roll mill to verify a novel particle image velocimetry (PIV) recording evaluation method that combines the advantages of central difference interrogation and an image correction technique. Simulations and experiments in the four-roll mill geometry demonstrate that the central difference image correction method described in this paper can not only avoid the bias error resulting from the curvature and high-velocity-gradient flow but also effectively reduce the random error resulting from particle image distortion. Two image correction schemes and two base algorithms are discussed. A four-point image correction scheme is suggested on the basis of the traditional correlation-based interrogation algorithm to enable a fast, high-accuracy evaluation of PIV recordings in complex flows. In addition, the PIV experiment accurately determines the velocity field in the four-roll mill and confirms the linear distributions of the velocity components and the roller speed.     

An efficient anti-aliasing spectral continuous window shifting technique for PIV

A new sub-pixel correlation peak locating algorithm for PIV analysis is introduced. The method is theoretically consistent with the method of continuously shifting interrogation sub-windows by fractional displacements, which has proven to be an effective way to reduce the bias error associated with integer pixel aliasing, or peak-locking. However the proposed algorithm performs continuous window shifting in the spatial frequency domain using the shift property of the Fourier transform, thus it is equivalent to interpolating the original digital image with the Fourier transform reconstruction. Synthetic and real PIV images are used to test the new algorithms performance relative to that of traditional (non-iterative) peak-finding methods and other peak-locking reduction algorithms, such as the continuous window shifting technique. The resultant bias error of the proposed algorithm is smaller (by an order of magnitude in some cases), and importantly, the periodic nature of the bias error, the characteristic signature of peak-locking, is eliminated as long as the discrete particle images have been sampled at a rate greater than the Nyquist sampling frequency. Moreover, this new algorithm is shown to be computationally efficient and it converges faster than the competing algorithms.     

Modeling and correction of peak-locking in digital PIV

The systematic tendency of PIV evaluations to bias towards integral pixel values is known as peak-locking. These errors, although small, significantly affect the statistics extracted from such measurements. In this paper, the process by which such errors accrue is modeled, and a scheme for the removal of the same is suggested. Specifically, the modeling process considers FFT PIV with discrete window offset. The results are applied to actual situations and the results are found to be encouraging. The process is computationally inexpensive, and can be applied as a post processing technique to existing data to correct peak-locking.     

Particle tracer response across shocks measured by PIV

The experimental approach used for the evaluation of the particle response time across a stationary shock wave is assessed by means of PIV measurements. The study focuses on the experimental requirements for a reliable and unbiased measurement of the particle response time τ _p and length ξ _p based on a single-exponent decaying law. A numerical simulation of the particle response experiment returns the parameters governing the measurement: namely the normalized spatial and temporal resolution, shock strength, and digital resolution. Representing the velocity decay in logarithmic coordinates it is shown that measurements performed with laser pulse separation time up to τ _p and interrogation window up to ξ _p still yield unbiased results for the particle response. A set of experiments on the particle response across a planar oblique shock wave was conducted to verify the results from the numerical assessment. Liquid droplets of DEHS and solid tracer particles of silicon and titanium dioxide with different primary crystal size are compared. The resulting temporal response ranges from 2 to 3 μs, corresponding to values commonly reported in literature, to almost 0.3 μs when particles are properly dehydrated and a filter is applied before injection into the wind tunnel. It is the first experimental evidence of particle tracers with a measured response time lower than 0.4 μs. The same procedure is applied to attempt the measurement of individual particle tracers by particle tracking velocimetry to estimate the spread in the distribution of tracer time response. The latter analysis is limited by the particle image tracking precision error, which biases the results introducing a wider broadening of the particle velocity distribution.     

Effect of resolution on the speed and accuracy of particle image velocimetry interrogation

Particle image velocimetry incorporates a process by which an image of a flow field, bearing double images of seeding particles, is analyzed in small regions called “interrogation spots.” Each spot is imaged onto a photodetector array whose digitized output is evaluated computationally using the auto-correlation technique. This paper examines the effects of resolving the spot using arrays of various resolutions, motivated primarily by a gain in speed. For this purpose, two specially created test photographs representing (i) uniform flow and (ii) solid body rotation, were interrogated using array sizes ranging from 32 × 32 to 256 × 256. Each reduction in resolution by a factor of two gains a factor of four in interrogation speed, but this benefit is counteracted by a loss in accuracy. The particle image diameter strongly influences accuracy through two distinct error mechanisms. When the particle image is small compared to the pixel size, mean bias error becomes significant due to finite numerical resolution of the correlation function. Conversely, when the particle image is large, random error due to irregularities in the electronic images predominates. The optimum image size, therefore, lies not at either extreme but at an intermediate value such that the particle image is small in an absolute sense, and yet large relative to the pixel size. A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 September 1990     

On the uncertainty of digital PIV and PTV near walls

The reliable measurement of mean flow properties near walls and interfaces between different fluids or fluid and gas phases is a very important task, as well as a challenging problem, in many fields of science and technology. Due to the decreasing concentration of tracer particles and the strong flow gradients, these velocity measurements are usually biased. To investigate the reason and the effect of the bias errors systematically, a detailed theoretical analysis was performed using window-correlation, singe-pixel ensemble-correlation and particle tracking evaluation methods. The different findings were validated experimentally for microscopic, long-range microscopic and large field imaging conditions. It is shown that for constant flow gradients and homogeneous particle image density, the bias errors are usually averaged out. This legitimates the use of these techniques far away from walls or interfaces. However, for inhomogeneous seeding and/or nonconstant flow gradients, only PTV image analysis techniques give reliable results. This implies that for wall distances below half an interrogation window dimension, the singe-pixel ensemble-correlation or PTV evaluation should always be applied. For distances smaller than the particle image diameter, only PTV yields reliable results.