Bias and precision errors of digital particle image velocimetry ...

Experiments in Fluids 28 (2000) 436±447 Ó Springer-Verlag 2000

Bias and precision errors of digital particle image velocimetry D. J. Forliti, P. J. Strykowski, K. Debatin

436 Abstract The bias and precision errors of digital particle image velocimetry are quanti®ed. Uniform displacement images are used to evaluate the uncertainty attributed to various sub-pixel peak ®nding algorithms. Bias errors are found to exist for all algorithms, and the presence of bias error tends to affect the precision error. The ability to ``calibrate'' out the bias error is explored using a rectangular free jet experiment. The calibration was effective in removing the bias error in the potential core and less effective in the shear layer. The bias error is found to functionally depend on the displacement gradients present in the interrogation region. The study stresses the need for in situ quanti®cation of DPIV uncertainty.

1 Introduction Particle image velocimetry (PIV) has developed into an important and widely used instrument in the study of ¯uid mechanics. Review articles by Adrian (1991) and Grant (1997) report the historical development of PIV over the past several years, including the subject of digital particle image velocimetry (DPIV). In PIV, the ¯ow of interest is seeded with tiny ¯ow-following particles. A laser beam transformed into a thin light sheet (using optics) passes through the ¯ow where it is scattered by the seed particles. A camera positioned along the axis perpendicular to the light sheet captures the images of the illuminated particles. The laser light is pulsed (typically with a pulse width on the order of nanoseconds) and the camera captures the

Received: 3 November 1998 / Accepted: 26 June 1999

D. J. Forliti, P. J. Strykowski Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455, USA K. Debatin Institute for Fluid Mechanics University of Karlsruhe 76128 Karlsruhe, Germany Correspondence to: P. J. Strykowski We would like to acknowledge Dr. Wing Lai of TSI Incorporated for his assistance with regards to the Insight software package. This work was supported by the ONR under technical monitor Dr. Gabriel Roy.

particle images at that instant. Traditionally, an individual photographic ®lm would be exposed for two or more pulses of laser light with a known time separation between pulses, thus capturing the sequential locations of the particles as they are convected by the ¯ow. The particle displacements are extracted through the interrogation of the photographic ®lm using a laser beam. This method of PIV has been described by Adrian and Yao (1984), Lourenco et al. (1989), Adrian (1988), and Keane and Adrian (1990). Digital particle image velocimetry incorporates the use of digital imaging into the measurement technique. Digital images, either taken with a digital camera or digitized from photographic ®lm, are processed with computer assistance to obtain displacement ®elds. Early experiments used digital cameras in an autocorrelation mode, with each image containing two or more laser pulses. In recent years, digital cameras capable of recording separate images with short time separations have lead to the use of cross-correlation techniques in DPIV. The uncertainty issues of interest in the current study pertain to both autocorrelation and cross-correlation modes of DPIV. The displacement peak maximum in the digital correlation function can be easily located to a precision of ‹0.5 pixels. However, there are various methods that can be used to estimate the displacement peak location to a more accurate sub-pixel level. Lourenco and Krothapalli (1995) studied the accuracy of four different peak ®nding algorithms which included: Parabolic and Gaussian curve-®ts using a 3 ´ 3 pixel domain centered on the displacement peak maximum; the Centroid method which consisted of calculating the centroid of the correlation function in a region near the displacement peak; and an interpolation scheme known as Whittaker's reconstruction which re®nes the resolution of the correlation to 1/64th of a pixel. The Parabolic and Gaussian curve-®t algorithms are used because they have shapes which are similar to the expected displacement peaks. The Centroid method which uses the ``center-of-mass'' as the peak location, is intuitively appropriate for this application and has been a common method used in previous studies (e.g. Keane and Adrian, 1990; Prasad et al., 1992). The Whittaker's reconstruction algorithm is a technique used to interpolate between discrete data samples (Stearns and Hush, 1990). Since the correlation function is sampled once per pixel, this technique can be used to calculate the correlation function at fractional pixel locations in the correlation plane, which allows for the location of the displacement peak maximum

to be determined to a higher sub-pixel accuracy. All of the above mentioned methods are legitimate peak ®nding algorithms. The study by Lourenco and Krothapalli (1995) established the accuracy attributed to the four peak ®nding algorithms using an autocorrelation mode setup where the correlation function is generated from the digitization and subsequent processing of the optically generated Young's fringe pattern. Whittaker's reconstruction interpolation scheme was found to produce the most accurate results. The two curve-®t methods had very different results; the Parabolic ®t tended to be biased toward integer pixel values while the Gaussian method was found to have good sub-pixel accuracy, which was attributed to the fact that the Gaussian algorithm was a better approximation to the displacement peak than the Parabolic algorithm. The centroid method was found to have errors for small displacements. This is due to the fact that the displacement peak is too close to the origin, thus there is an overlap between portions of the displacement and self correlations, which is a complication of using autocorrelation mode DPIV. This error can be avoided by forcing the displacement peak to be suf®ciently displaced from the self correlation peak, or by using cross-correlation mode DPIV. Prasad et al. (1992) studied the effect of image resolution on the accuracy of autocorrelation mode DPIV. It was found that a Centroid peak ®nding method had a bias error which was a function of the particle image diameter and sub-pixel displacement. With the technological growth and performance improvement of digital cameras, the use of cross-correlation mode DPIV has become widespread (cross-correlation DPIV is covered by Willert and Gharib, 1991). Crosscorrelation removes the self correlation peak and directional ambiguity (i.e. there is only one peak) which are present in autocorrelation mode DPIV. Recently, Huang et al. (1997) described two methods which reduced the bias error of cross-correlation mode DPIV when using a Gaussian estimator (independent estimates for the x and y components of displacement) to locate the peak maximum. The study found that the bias error can be signi®cantly reduced if the correlation function is not in¯uenced by the changing overlapping area during the convolution of the two interrogation regions. It should be stressed that the uncertainty characteristics of a sub-pixel peak ®nding method is not only dependent on the mathematical model employed, but also on the domain used for the peak location calculation (for a ®xed optical con®guration).

correlation plane. Another example of an error linked to the correlation process is the effect caused by the changing overlapping image area as the images are shifted with respect to one another, this tends to bias the displacement peak toward the correlation origin (zero displacement), which we refer to as window bias; this topic has been described by Westerweel (1993). The use of a Parabolic curve-®t to estimate the maximum of the displacement peak is an example of a potentially inappropriate method for peak ®nding when using the Young's fringe method (Lourenco and Krothapalli, 1995). The goal of the present work is to quantify the bias and precision errors of various peak ®nding algorithms, and attempt a correction of systematic errors which are present in the mean displacements. An experimental approach will be employed to study the uncertainty of DPIV using an autocorrelation mode system. Although cross-correlation has become the preferred mode of DPIV, the differences between the two modes in the correlation plane in the vicinity of the displacement peak are small (assuming that for autocorrelation mode the displacement peak is suf®ciently displaced from the self correlation peak). Thus the results will be generally representative of both cross and autocorrelation modes of DPIV. Uniform displacement ®elds will be initially used to describe the bias and precision errors of different sub-pixel peaking ®nding schemes. A rectangular free jet will also be used to explore some of the aspects of DPIV uncertainty for a real experimental scenario.

3 DPIV System The DPIV setup is shown in Fig. 1. Two aligned Continuum Surelite I-10 lasers (each laser is capable of 200 mJ/ pulse at a wavelength of 532 nm) are used to illuminate the test section. The laser beams are transformed into a thin light sheet using a cylindrical and spherical lens of focal lengths of )50 mm and 1000 mm, respectively. The lasers and optics were con®gured such that the beam waist was located below the image capture region (i.e. test section), such that the light sheet in the test section was thin enough to generate intense particle images, yet thick enough to reduce the loss of particle image pairs due to displacements normal to the plane of the light sheet. This was particularly important for the free jet experiments, where

2 Uncertainty of DPIV There are a variety of sources of error in velocity measurements using DPIV. The present study is only concerned with errors in the particle displacement portion of the velocity calculation. Generally, errors in the displacement calculation result from correlation phenomena (i.e. distortions of the correlation function) or errors (or inappropriateness) in the method used to extract the displacement from the peak in the correlation function. For example, velocity gradients (Keane and Adrian, 1990) may cause errors by distortion of the displacement peak in the Fig. 1. Experimental setup

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three-dimensional turbulence produces displacements normal to the light sheet, causing a reduction in the number of correlated particle image pairs. The light sheet in the test section was approximately 0.8 mm thick. A spinning mirror image shifting system (TSI Model 610055) was used to generate spatially ``uniform'' particle displacement images. The shifter is also used in experiments to resolve the directional ambiguity, which is a consequence of the autocorrelation mode. The digital images were recorded with a KODAK MEGAPLUS Image Capture System (TSI model 630042), with each image containing two laser pulses; images are stored on a PC using a frame grabber. The camera has a CCD array made up of 1320 horizontal and 1035 vertical square pixels. The synchronization between the two lasers, image shifter, and camera is controlled by a TSI Model 610030 LaserPulse synchronizer. The digital images were captured and post processed using TSI's InsightÓ software (version 1.22). Olive oil droplets generated with a Laskin nozzle aerosol generator are used as the light scattering seed particles. The details of the Laskin nozzle are given in Gillgrist (1999). Particle size measurement studies (Gerbig and Keady, 1985; Crosswy, 1985) have shown that this type of aerosol generator tends to generate polydisperse particle distributions predominately in the sub-micron diameter range. The outlet port of the Laskin nozzle is open to the room for the uniform displacement studies, while some of the seed is passed into the jet facility for the free jet experiment. As described in Westerweel (1997), it is important that PIV images do not contain gradients of seed concentration. For the free jet experiments, the seed ¯ow rates to the room (ambient) and the jet were balanced such that no seed gradients were observed (an example digital image is shown in Fig. 12). Because of the homogenous seeding, the image does not present any qualitative information about the ¯ow, i.e. no visualization of ¯ow structure is present. The particle image diameter which accounts for the diffraction limitations of the lens is calculated from

heat added to the air ¯ow by the compressor and return the jet to the same temperature as the surroundings. The temperature of the jet is measured with a thermocouple upstream of the nozzle contraction and ¯ow conditioning. A bleed-off line from the compressor outlet allows for variation of the jet exit velocity. In the present experiments, the jet exit velocity was set to a value of 65 m/s. See Van der Veer (1995), Van der Veer and Strykowski (1997) and Gillgrist (1999) for more detailed information on the jet facility. As shown in Fig. 1, the laser sheet is con®gured such that the center plane of the short dimension of the rectangular nozzle is contained in the light sheet. The camera and spinning mirror were positioned to capture the jet ¯ow from approximately 3H to 5H downstream of the nozzle exit, where H is the short dimension of the rectangular nozzle in the exit plane. The streamwise domain was selected because the shear layers are suf®ciently thick to allow for reasonably accurate spatial resolution of the instantaneous velocity ®eld, and this domain contained a wide range of ¯ow gradients.

4 Measurement of DPIV error A peak ®nding algorithm is characterized by the type of mathematical model employed, and the domain which is used by the model to compute the peak location. For instance, a Gaussian curve-®t of a 3 ´ 3 pixel region would have different uncertainty characteristics than a Gaussian ®t of a 6 ´ 6 pixel region. For the present study, the peak ®nding algorithms studied are those found on the Insight software package. In the current software, there are four different peak ®nding algorithms available. Two of the methods involve least-squared curve-®tting of a portion of the digital correlation function. The Insight software package offers both Parabolic and Gaussian curve-®tting methods, which use a 3 ´ 3 pixel domain centered on the displacement peak maximum. Additional algorithms include the Centroid method which calculates the centroid of q the correlation function in the region near the displacede ˆ M2 dp2 ‡ …2:44…1 ‡ M†f # k†2 …1† ment peak, and Whittaker's reconstruction, which interpolates in the vicinity of the displacement maximum to where de is the particle image diameter, M is the magni- improve the resolution of the correlation function. All ®cation, dp is the size of the particles, f # is the f-number of results presented for these four algorithms must be kept in the lens, and k is the wavelength of the laser light. The the context of being traits of the peak ®nding algorithms optical setup used for this study included; M ˆ 0:31, available with the Insight software package. k ˆ 532 nm, and f # ˆ 5:6. Assuming a particle size of 1 lm, the particle image size is 9.5 lm, which is equivalent 4.1 to 1:4dr , where dr is the length of one side of a pixel in the Procedure camera array. The particle image size is dominated by the The uncertainty of the algorithms will be studied using diffraction limited component of Eq. (1). digital images containing displacements induced with a spinning mirror. The displacement generated from a 3.1 spinning mirror has been described by Adrian (1986). It is Jet facility common to assume that the shift is spatially invariant. In A rectangular free jet is used to further explore the disactuality, the shift induced by a spinning mirror does placement measurement uncertainties. The rectangular depend on the position in the image plane. The distance nozzle, having an aspect ratio of 4.0, has an exit plane area from the mirror to the light sheet is not the same for all of 4 cm2 and a large contraction ratio which produces a locations in the light sheet, which causes variable shift top-hat velocity pro®le at the nozzle exit. The jet ¯ow is displacements within the image domain. The spatial varidriven continuously by a 10 hp Fuji ring compressor. The ation of the shift produced by a spinning mirror has been facility has an air-to-water heat exchanger to remove the modeled by Zhang and Eisele (1995). The model suggests

that the shift variation can be minimized by having the spinning mirror located close to the camera relative to the distance from the mirror to the laser light sheet. Using the Zhang and Eisele (1995) model and the optical con®guration used in this study, the shift variation is expected to be less than 0.01 pixel in both the horizontal and vertical directions. The measured variation of the shift will be addressed in a later section. The uncertainty associated with different peak ®nding algorithms can be evaluated from images generated with the spinning mirror. For the con®guration shown in Fig. 1, the spinning mirror is oriented for shifting in the x direction. If digital images are taken with the jet turned off but with the mirror spinning, the resulting images will contain nearly spatially invariant displacement ®elds. The room is ®lled with olive oil droplets generated with the Laskin nozzle, and doors/windows/vents are closed off in the room to minimize drafts. It will be taken for granted at this time that the effects of room drafts and mirror shift variation are negligible compared to the bias and precision errors, but this will be addressed subsequently. Other approaches for estimating PIV accuracy have implemented the use of arti®cial particle images (e.g. a piece of paper with tiny dots) which have been translated and/or rotated a known amount. Since the shape of the particle image intensity pro®le determines the shape of the displacement peak, it is expected that the performance of a peak ®nding algorithm will also be dependent on the particle image intensity pro®le. This is the motivation for using actual particle images for the calibration employed in this study. The bias and precision errors of the peak ®nding algorithms can be determined for a particular experimental setup using the uniform displacement images generated with the spinning mirror under no-¯ow conditions. Gillgrist (1999) has shown that the mirror speed is stable and constant for the shifting hardware used in this study. The spinning mirror was set at a ®xed rotational frequency, and digital images were captured at a series of different pulse separations. As anticipated, the set of images have particle displacements which linearly depend on pulse separation. The images are interrogated on a spatial grid of 30 ´ 10 displacement measurements over the image domain. The images were processed with square interrogation regions of 128 pixels on each side; employing large interrogation regions minimized the window bias (Keane and Adrian, 1990). The mean displacement and standard deviation for each image is then calculated from the 300 measured values. The 300 measured displacements did not suffer from erroneous vectors because the interrogation regions were so large, and the displacement peak was always larger than the noise peaks. Two images at each pulse separation were taken. An example set of data using the Parabolic ®t algorithm with a mirror rotation frequency of 28 Hz is shown in Fig. 2. In order to measure the bias error, the actual displacement must be known and compared to the measured mean displacements at each pulse separation. Since the displacement is generated from a spinning mirror, the actual displacement is not known a priori to high accuracy, but must be extracted from the measured values.

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Fig. 2. Mean displacement vs. pulse separation for a mirror rotation frequency of 28 Hz. (Parabolic ®t-128 ´ 128 interrogation region)

The measured mean displacements are expected to be most accurate near integer and half-integer pixel values (Prasad et al., 1992; Fincham and Spedding, 1997); these points can be used to determine the actual displacement induced with the spinning mirror. The solid symbols in Fig. 2 are points which are located symmetrically on each side of an integer pixel displacement value. The data points represented by the solid symbols are interpolated to ®nd the pulse separation at the integer pixel displacement located between the two points. Using the integer pixel values and the interpolated pulse separation, a linear function can be constructed which represents the actual displacement as a function of pulse separation. The line included in Fig. 2 is the actual displacement calculated in the manner described above. The linear function can then be used to calculate the actual displacement at each pulse separation. It is seen that the measured displacement data points systematically oscillate about the actual displacement line. The wavy nature of the measured displacements has a large wavelength which can be visualized by viewing Fig. 2 at a shallow angle along the line (i.e. ``looking'' along the line). The bias error is expected to have a period of one pixel, but the pulse separation cannot be incremented in small enough steps to resolve the bias error between neighboring pixels. The pulse separation increments generally resulted in displacement steps on the order of 1 pixel, thus the wavelength of the bias observed in Fig. 2 is caused by aliasing. The actual displacement as a function of pulse separation could have been calculated from the least-squared ®t of the whole set of measured values, as long as the data had an integer number of wavelengths of the systematic error. It was simpler to use the method described previously to construct the actual displacement relationship with pulse separation. The difference between the measured and actual displacements (at each pulse separation) is the error in the mean displacement measurement, i.e. the bias error. This was done for mirror speeds of 28, 29, 30, 34, and 35 Hz, and pulse separations of 1.2 to 4.0 lsec in 0.2 lsec time steps (for each mirror speed). The same set of images were pro-

cessed with each of the four peak ®nding algorithms available on the Insight software package.

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as seen in Fig. 3b. Of the two curve-®t algorithms studied, the Gaussian appears to be the most appropriate ®t to the correlation peak for these uniform displacement images. This is not a surprising result, since real particle images were used, which tend to have nearly Gaussian intensity pro®les (Keane and Adrian, 1990). These ®ndings are in qualitative agreement with Lourenco and Krothapalli (1995). Figure 4 shows the bias error for the Centroid peak ®nding algorithm. As done for the curve-®t algorithms, the data is presented in different displacement ranges. It appears that the bias error is less systematic than the bias error of the curve-®t algorithms. As mentioned above,

4.2 Bias and precision errors From Prasad et al. (1992), the bias error (for a ®xed particle image size) is expected to be primarily a function of the sub-pixel component of the displacement. The bias errors for the Parabolic and Gaussian curve-®t peak ®nding algorithms are shown in Fig. 3. Each point presented is the mean displacement of two images (600 total samples) at each mirror speed/pulse separation combination. The bias data is displayed using different symbols for different total displacement ranges (Dx is the total displacement in the x direction). The lines presented in Fig. 3 are polynomial curve-®ts of the complete set of bias error data. These curve-®ts can be used to predict the bias error as a function of sub-pixel displacement. The bias error is calculated for total particle displacements ranging from approximately 5 to 22 pixels, where the sub-pixel displacement is the non-integer portion of the displacement (e.g. the sub-pixel displacement of 14.27 is 0.27). It appears that the bias error does collapse over the full range of particle displacements. This suggests that the total displacement does not play a signi®cant role in the bias error, at least for the 128 ´ 128 pixel interrogation region size. Additionally, the digital images had mean image intensities ranging from 20±35 on the gray scale out of a total range of 0 to 255 (8 bit images). This is caused by ¯uctuations in the particle density from image to image. Thus the bias error is highly systematic over a wide range of total displacements and particle densities, which is a desirable trait if the bias is to be calibrated out. The Parabolic algorithm seen in Fig. 3a, suffers from signi®cant bias errors, as large as ‹0.17 pixels. The bias error for the Fig. 4. Bias error for the Centroid algorithm using 128 ´ 128 Gaussian has a similar trend with a much lower magnitude pixel interrogation region

Fig. 3. Bias error for the curve-®t subpixel algorithms using 128 ´ 128 pixel interrogation region: a Parabolic ®t, b Gaussian ®t

there is a variation in the particle seed density from image to image. The solid symbols are the bias error of the set of images which have a narrowed mean image intensity range between 30 and 35 gray levels. This set of images which have similar seed concentrations, are seen to have a more systematic error. This suggests that the Centroid method has a bias error which is more sensitive to the image parameters. The ®nding suggests that the Centroid algorithm has a bias error which would be less appropriate for calibration. The general trend of the Centroid algorithm does follow that of the Parabolic and Gaussian curve-®t algorithms, except for the apparent discontinuity at a sub-pixel displacement of 0.5, which is a trend predicted by Westerweel (1993) for a Centroid estimator. The magnitude of the bias error is much less than the Centroid estimator model presented by Westerweel (1993). The Westerweel Centroid estimator uses independent centroid calculations for the horizontal and vertical directions using 3 co-linear points. The difference in bias magnitude between that of the current study and the Westerweel Centroid estimator is most likely due to the different domains used to calculate the centroid. The bias error of the Whittaker algorithm is shown in Fig. 5a, and indicates a trend similar to that of the parabolic curve-®t algorithm, with large bias error magnitude (‹0.13 pixels) relative to the Gaussian and Centroid algorithms. This is a much different ®nding than that of Lourenco and Krothapalli (1995). The primary difference between the two studies is that the present study uses correlations calculated from the digital images of the particles, while Lourenco and Krothapalli (1995) generated the correlation function from the digitized Young's fringe pattern (using an FFT). The second method is a higher resolution technique which would directly lead to an improved resolution of the displacement peak (relative to the correlation of the digital images). Thus the accuracy of the

Whittaker peak ®nding algorithm degrades as the width of the displacement peak decreases towards the size of 1 pixel. Another way of viewing the differences between the two studies is that the present study has a smaller particle image size to pixel size ratio (de =dr ˆ 1:4 as stated in Sect. 3). The curve-®t of the Whittaker bias error was used in an attempt to remove the bias error from the instantaneous data. For an instantaneous measurement, the bias error can be calculated using the curve shown in Fig. 5a, using the instantaneous sub-pixel displacement for evaluation of the bias error. The measured displacement sample is then adjusted by subtracting the bias error. The bias error for the corrected Whittaker method is shown in Fig. 5b. The trend of the displacements being biased towards the nearest integer pixel value has been removed. Obviously, since the Gaussian curve-®t algorithm has low bias error, correcting the Whittaker data does not appear to be advantageous. There may be experimental conditions where the Gaussian curve-®t has poor uncertainty characteristics (e.g. large and irregular particle images), where the correction curve generated as described above will improve the accuracy of the measured displacements using any of the sub-pixel algorithms. The use of the Whittaker algorithm with bias correction in a real experiment can be compared to the Gaussian algorithm results to further the understanding of how the uncertainty of the sub-pixel algorithms may depend on ¯ow conditions; this will be addressed in Sect. 5. The uniform particle displacement data can also be used to quantify the precision error of the various peak ®nding algorithms. For the purposes of this study, the precision error will be de®ned as the standard deviation of the displacement samples over the ®eld of view. The precision error for the Parabolic and Gaussian curve-®t algorithms for 128 ´ 128 pixel interrogation regions are shown in

Fig. 5. Bias error for the Whittaker's reconstruction sub-pixel algorithm using 128 ´ 128 pixel interrogation region: a original Whittaker, b corrected Whittaker

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Fig. 6 (each point represents the standard deviation of the 300 displacement samples for one image). For the Parabolic algorithm, the precision is maximum at the half pixel location, and decreases as the sub-pixel component moves away from the mid-pixel location towards 0 or 1.0 subpixel displacement. This trend was also observed by Huang et al. (1997). This fact along with the observations made about the bias error of the Parabolic algorithm suggests there to be a connection between the bias and the precision errors. For example, if a set of displacement samples had a mean near 0.5 pixels and a standard deviation of 0.05, the displacements below 0.5 pixels would be biased towards smaller sub-pixel displacements and displacements larger than 0.5 pixels would be biased towards larger sub-pixel displacements. This results in the data being spread out to a larger range, increasing the standard deviation. If the mean value is near an integer number of pixels (near 0 or 1 for sub-pixel displacement), then the bias error will tend to reduce the precision error because the data range is compressed by the bias error. The precision error for the Gaussian curve-®t algorithm is also shown in Fig. 6. The variation in precision error with subpixel displacement is much smaller, which is expected if the connection between bias and precision errors is valid, as the Gaussian algorithm has low bias error. The precision error for the Centroid peak ®nding algorithm is shown in Fig. 7. The Centroid algorithm has slightly higher precision error than the Gaussian algorithm. This suggests that the Centroid algorithm is more sensitive to the noise in the correlation function. The increased scatter in the precision error of the Centroid algorithm also suggests higher sensitivity to particle concentration variations. Figure 8 shows the precision error of the Whittaker algorithm, with and without the bias error correction. The bias error correction for the Whittaker data tends to remove the precision error dependence on the sub-pixel displacement. The bias error correction generated from the mean displacements must be applied to the instantaneous displacements to remove the precision error dependence on sub-pixel displacement. This fact along with the bias and precision error trends of the other peak

Fig. 6. Precision error for the Parabolic and Gaussian curve-®t sub-pixel algorithms using 128 ´ 128 pixel interrogation region

Fig. 7. Precision error for the Centroid sub-pixel algorithm using 128 ´ 128 pixel interrogation region

®nding algorithms support the proposed connection between bias and precision errors. The validity of correcting instantaneous data with a calibration generated from mean data is substantiated by the results presented in Figs. 5 and 8. The uncertainty of a single displacement measurement in the x direction is (for 95% con®dence interval):

q dDxi ˆ b2 ‡ …2r†2

…2†

where b is the residual bias error (represented by the error bars on Fig. 5b), which for this calibration is 0.015 pixel, and r is the standard deviation (i.e. precision error) of the uniform shift displacement data. The precision error of the corrected Whittaker algorithm is similar to that of the Gaussian algorithm. dDxi for the corrected Whittaker is approximately 0.06 pixels, while the uncorrected Whittaker is as high as 0.14 pixels.

4.3 Interrogation region size As shown in the previous section, the Gaussian curve-®t algorithm has low (yet systematic) bias error for an in-

Fig. 8. Precision error for the Whittaker's reconstruction subpixel algorithm with and without bias correction using 128 ´ 128 pixel interrogation region

curve-®t algorithm. The overall magnitude of the bias error (for all interrogation region sizes) for the Gaussian algorithm is still relatively small. Since we attempt to correct the bias error through direct calibration, the bias error of the Whittaker algorithm as a function of interrogation region size is also of interest. Figure 10 shows the bias error for the Whittaker method for square interrogation regions of sizes 64 ´ 64 and 128 ´ 128 pixels. The window bias is seen to be present (the curve is displaced downward) and have a similar magnitude as for the Gaussian algorithm. The amplitude of the bias error is larger for the smaller interrogation region, which is the opposite trend observed for the Gaussian algorithm results. Again, the results appear to be connected to the changing resolution in the Young's fringe plane. Since it has been established that the Gaussian curve-®t algorithm with a 64 ´ 64 pixel interrogation region has the least variation in the bias error with sub-pixel displacement (Fig. 9), this processing con®guration can be used to study the spatial variation in shift induced by the spinning mirror. Figure 11 shows the shift induced by the spinning mirror per 1 lsec of pulse separation as a function of the x location in the image plane (for a mirror rotational fre-

terrogation region of 128 ´ 128 pixels. The behavior of the bias error may change depending on the interrogation region size. Repeating the work described in Sect. 4, the bias can be quanti®ed for different interrogation region sizes. The actual displacements calculated from the 128 ´ 128 pixel interrogation region data were used as the baseline comparison for the various interrogation region sizes. This will give insight into the window bias for different interrogation region sizes. Figure 9 shows the bias error for the Gaussian curve-®t algorithm (only the polynomial curve-®ts are presented, the data which had scatter similar to that shown on Fig. 3 are excluded for clarity purposes). As previously described, the window bias tends to push the displacement peak towards the origin, thus the bias is expected to be negative for displacements in the positive direction. The ®gure shows that for square interrogation regions ranging from 128 ´ 128 to 65 ´ 65 pixels, the bias error has a very similar shape, but this shape is displaced downwards on the ®gure as the interrogation region is reduced which is the effect of the window bias. The shape of the bias error for the 64 ´ 64 pixel interrogation region is quite different from that of the others, but has similar window bias as the 65 ´ 65 interrogation region (i.e. resides at the same average bias level). The observation that the window bias magnitude increases with decreasing interrogation region size agrees with Keane and Adrian (1990). The window bias is no more than )0.01 pixels for interrogation regions down to 64 ´ 64 pixels (for this particle image size). The bias error for the 64 ´ 64 pixel interrogation region has very little variation with sub-pixel displacement as compared to the other interrogation region sizes. The Insight software package calculates the correlation functions using the FFT approach. If the interrogation region has more than 64 pixels in either the horizontal or vertical direction, the interrogation region is padded with zeros to ®t a 128 ´ 128 region. This directly results in a doubling of the resolution in the Young's fringe plane. Thus it appears that the lower resolution in the Young's fringe plane caused by the 64 ´ 64 pixel interrogation region suppresses the bias error variation with sub-pixel displacement for the Gaussian

Fig. 10. Bias error for the Whittaker's reconstruction algorithmeffect of interrogation region size

Fig. 9. Bias error for the Gaussian curve-®t algorithm-effect of interrogation region size

Fig. 11. Mirror shift vs. the x-coordinate in the image plane

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quency of 30 Hz). No trend is observed in the data, thus mean spatial gradients in the displacement ®eld due to mirror rotation have not contaminated the study of the bias and precision errors. The data shown in Figure 11 show that the displacements generated with the spinning mirror are accurate to ‹0.005 pixels, which is signi®cantly lower than the bias and precision errors being measured. The effect of room drafts must also be considered with regards to the bias and precision errors presented in Figs. 3±10. Both the bias and precision errors may be in¯uenced by room drafts. The precision error would be affected by room drafts by the presence of velocity gradients in the captured region of the light sheet, which would cause larger standard deviations within the displacement sample set for the image. If this were the case, one would expect to see the precision error increase with pulse separation, since the displacement gradients are equal to the velocity gradients times the pulse separation. It was found that the precision error is independent of the pulse separation, which suggests that the displacement gradients due to room drafts are insigni®cant. This is expected, since the image captures a region of the size approximately 2 cm ´ 2 cm, and room drafts would be expected to have gradients on a much larger scale than this. Large scale gradients could bias the mean displacement calculation for an image. If the average draft velocity varies from image to image, that will lead to increased uncertainty of the mean displacements. Two images were captured at each mirror speed/pulse separation combination. The mean displacements for the two images can be compared to give insight into the accuracy (temporal variation) of the mean values due to room drafts. Using the differences between the two images, the mean displacements have an uncertainty of ‹0.015 pixels, which is represented by the error bars on Figs. 3±5.

the interrogation region, it would be expected that the bias of the peak ®nding algorithms would also be dependent on displacement ®eld gradients. For the present study, a jet ¯ow was used to study the effect of displacement ®eld gradients on the bias error. This is an attractive ¯ow for this purpose because it contains regions of very low spatial velocity gradients, the potential core, and regions of high spatial velocity gradients, the shear layer. As shown in Fig. 1, the experimental con®guration was setup to measure the velocity ®eld of a rectangular jet which was operated at an average exit velocity of 65 m/sec. Images of the jet were taken with a pulse separation of 0.5 lsec and a mirror rotation frequency of 67.5 Hz. Images without jet ¯ow were also taken to evaluate the mirror shift displacement, which was found to be 5.12 pixels using the Gaussian curve-®t algorithm with a 64 ´ 64 pixel interrogation region. Over 400 images were taken of the jet ¯ow at a constant rate of approximately 7 pictures/min. over the streamwise domain of 3 < x/H < 5 and crossstream domain of )1 < y/H < 1. A sample digital image and displacement vector ®eld are shown in Fig. 12; only the upper half of the jet is shown. The measured displacements are the sum of displacements caused by the spinning mirror and the ¯uid ¯ow. The displacement caused by the ¯uid motion is calculated by subtracting the known mirror displacement from the measured total displacement. The vectors in Fig. 12 are the ¯uid motion displacements, that is they are proportional to the velocity ®eld at this particular instant. The measured displacements ranged from approximately 5 to 6.8 pixels providing a relatively low dynamic range of »1.8 pixels. However, the bene®t of using such a low dynamic range is the reduction in spatial displacement gradients and out-of-plane displacements, which increase with pulse separation. Hence the short pulse separation leads to improved signal-to-

5 Displacement gradient effects on bias error The effect of displacement gradients within the interrogation region must be considered in the evaluation of the uncertainty of peak ®nding algorithms. Keane and Adrian (1990) discussed a velocity gradient error which biases the displacement towards lower magnitudes because the larger displacements in the interrogation region tend to have higher probability of particle pair loss due to the ®nite size of the interrogation region. It was proposed that velocity gradient bias is negligible for MDtDu p < 1 de2 ‡ dr2

…3†

where dr is the pixel size, Dt is the pulse separation, and Du is the velocity variation from the mean velocity contained in the interrogation region (i.e. the interrogation region contains velocities in the range u  Du). The quantity MDtDu is the displacement ®eld variation present in the interrogation region, which is simply the velocity variation multiplied by the pulse separation and magni®cation. Thus the simplest way of satisfying Eq. (3) is to have suf®ciently short pulse separation. Since the source of the velocity gradient bias is a distortion of the correlation Fig. 12. Free jet DPIV image and displacement vector ®eld 3 < x/ peak caused by gradients in the displacement ®eld within H < 5, 0 < y/H < 1

noise ratios, i.e. stronger correlation, and more complete instantaneous displacement vector ®elds (fewer erroneous vectors). An additional motivation for reducing the pulse separation is that it simpli®es the application of PIV to highly three-dimensional ¯ows which have been traditionally dif®cult to study using planar particle tracking methods. The calibration procedure presented in Sect. 4 would allow for the use of very small pulse separations while maintaining high relative accuracy. Since the dynamic range is 1.8 pixels and the particle image size is 1.4 pixels, the maximum MDtDu=…de2 ‡ dr2 †1=2 will be 0.52, which satis®es the criterion stated in Eq. (3). Thus the velocity gradient bias as described by Keane and Adrian (1990) is expected to be negligible. Because of the high signal to noise ratios caused by the small dynamic range, the free jet experiments yielded approximately 99% valid displacement vectors for mean velocities ranging from 65 to 0 m/ sec. As can be seen from the displacement ®eld in Fig. 12, a signi®cant portion of the ¯ow ®eld contains the uniform potential core, while the upper half is in the turbulent shear layer. For the uniform displacement ®elds, the Gaussian curve-®t algorithm was seen to have much lower bias error than the Whittaker algorithm. For the jet ¯ow experiment, the actual velocity ®eld is not known and cannot be readily extracted from the PIV measurements without assuming one of the algorithms to be an accurate ``baseline''. It is expected that the Gaussian algorithm will be the most accurate, thus it will be used as the baseline algorithm for the moment. The free jet digital images were processed to yield a 41 ´ 32 grid of displacements with 50% overlapping using 64 ´ 64 pixel interrogation regions. Histograms of the x direction displacement for the free jet are shown in Fig. 13. The histograms have bin sizes of 0.01 pixel, and were generated from more than half a million samples (430 images). The histogram for the Gaussian curve-®t

algorithm tends to show the expected distribution of data samples for an algorithm having low bias error. There is a peak at approximately 6.6 pixels, which is the data contained in the potential core. The pixel displacement equivalent for the jet exit velocity of 65 m/s as measured with a pitot probe is also shown on Fig. 13, which is aligned with the peak at 6.6 pixels. The histogram data below 6.4 pixels is fairly ¯at, which is the data in the shear layer. The data probability does not seem to be higher at the integer pixel locations, which would be expected for high bias errors (Westerweel, 1993). On the other hand, the Whittaker algorithm does have the expected trends for large bias errors. There is a peak in the probability for displacements near 5.0 and 6.0 pixels, with probability de®ciencies near 5.5 and 6.5 pixel displacements. Intuitively, there is no reason to expect this from the given ¯ow ®eld. From the earlier observations made with the Gaussian algorithm for uniform displacement ®elds, it is expected that the Gaussian data in the potential core has low bias error because displacement gradients are not present in this particular region of the ¯ow. Thus the peak at 6.6 pixels in the histogram for the Gaussian algorithm is expected to be representative of the true data probability, which is supported by the agreement with the pitot measurement. For the Whittaker algorithm results, this peak is still present, but is seen to have been shifted closer to the 7 pixel location. The peak is also seen to be lower in amplitude and broadened, which is a trend expected when bias error is present. The fact that the peak is broader agrees with the proposed connections between bias and precision error that were made in Sect. 4.2. Overall, there is seen to be large discrepancies between the Gaussian and Whittaker results, with the Whittaker having expected characteristics of an algorithm which has large bias errors. The correction generated from the uniform displacement ®eld images for the Whittaker algorithm with a 64 ´ 64 pixel interrogation region (polynomial curve-®t

Fig. 13. Histogram of the x-component for the free jet experiment

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advantageous because it incorporates the actual particle image intensity pro®les which will be used during the experiment, thus any uncertainty dependencies on the particle image intensity pro®le will be accounted for in the calibration. The calibration displacement images are generated using a spinning mirror; the mirror itself was shown to not introduce measurable spatial displacement variations. The mirror speed and pulse separation were varied to obtain a wide range of total displacements and a complete sampling of the sub-pixel displacement domain. Four sub-pixel peak ®nding algorithms were studied to examine their in¯uence on measurement uncertainty including Parabolic and Gaussian curve-®ts of 3 ´ 3 pixel regions centered on the displacement maximum, a Centroid algorithm, and a Whittaker's reconstruction algorithm. The bias error present for all of the algorithms tended to bias the displacements towards the nearest integer pixel value. In general, the Gaussian curve-®t algorithm was found to have the lowest bias and precision errors. The Whittaker method was found to have large bias errors which was not the case for an analog/digital technique employing the digitization and analysis of the optically generated Young's fringe pattern (Lourenco and Krothapalli, 1995). For the present study, the bias error was found to be slightly dependent on the interrogation region size. The cause of this dependence is hypothesized to be the changing resolution in the Young's fringe plane (since the correlation is calculated with FFT's of the digital images of the particles). Large bias errors were seen to cause the precision error to be a strong function of the sub-pixel displacement, thus the two errors are not independent. The bias error for the Whittaker method evaluated from the calibration procedure was used to attempt a correction of the Whittaker results. The bias error calibration was used to calculate the bias error for each instantaneous displacement measurement sample. The bias was then subtracted from the instantaneous displacement sample. The entire set of the Whittaker data was corrected so that the uncertainty of the corrected displacements could be determined. It was found that the bias error of the Whittaker algorithm results could be effectively removed using this correction for the uniform displacement calibration images. The uncorrected Whittaker results suffered from large bias errors, which produced a precision error which was dependent on the sub-pixel displacement. The correction of the Whittaker results removed this sub-pixel displacement dependence of the precision error. The results of the calibration procedure for the four subpixel algorithms is valid only for a speci®c experimental setup, but does not account for displacement gradients within the interrogation region. A rectangular free jet was used to test the correction on a typical experiment which contains displacement gradients which was within the 6 current criterion for accurate PIV measurements (Keane Conclusion A calibration procedure has been described which quan- and Adrian, 1990). Histograms of the x direction displacement were used to explore the relative bias error beti®es the uncertainty of DPIV. For a particular experimental setup, a sequence of digital images are taken with tween the Gaussian and Whittaker algorithms. The the ¯ow turned off and the quiescent test section seeded uncorrected Whittaker histogram had characteristics that with approximately the same particle concentration as will suggested the presence of bias error which were not present in the Gaussian histogram. The Gaussian and the corrected be used for the experiments with ¯ow. This method is shown in Fig. 10) was used to correct the jet ¯ow measurements made with the Whittaker algorithm. The Whittaker algorithm with bias correction is included in Fig. 13. Overall, the corrected Whittaker compares well with the Gaussian ®t histogram distribution. The correction was expected to be most accurate in the potential core of the jet, because like the calibration images, this portion of the image does not contain displacement gradients. This is the case, as can be seen from Fig. 13, the peak in the Whittaker data at 6.8 pixels has been shifted to the 6.6 pixel location with both the amplitude and width of the peak in good agreement with the Gaussian algorithm histogram. There begins to be a noticeable deviation between the Gaussian and corrected Whittaker for the displacement below approximately 6.5 pixels, these are the data that are in the shear layers. If we assume for the moment that the uncertainty characteristics of the Gaussian algorithm is unaffected by the presence of displacement gradients, it appears as if the application of the correction has actually overcorrected the original Whittaker data. That is, there appears to be a de®ciency in the probability of the displacements near 6.0 pixels. In order to check this point, the original Whittaker data was corrected with a new correction curve, which was simply the curve shown in Fig. 10 with a multiplying factor. It was found by trial and error that the curve generated for the Whittaker algorithm with a 64 ´ 64 pixel interrogation region had to be multiplied by 0.6 in order to get a good match between the Gaussian and corrected Whittaker in the histogram near displacements of 6.0 pixels. This 60% corrected Whittaker result is also shown in Fig. 13. Interestingly, the comparison in the potential core was signi®cantly degraded, showing signs of undercorrection. The bias error for the Whittaker algorithm does appear to depend on the presence of the displacement gradients. Furthermore, it appears that the bias error for the Whittaker algorithm has been attenuated by the presence of displacement gradients. This of course assumes that the bias error of the Gaussian algorithm is negligible for displacement ®elds containing gradients, which may not be the case. It is possible that the Gaussian algorithm begins to have increased bias error in the presence of displacement gradients. The important conclusion is that the bias error for one or both of the algorithms have shown a dependence on displacement gradients. It is interesting to note that even though the gradients are small (displacement variations » 2% of the interrogation size), they create signi®cant changes in the bias error. This is in contrast to the results of Fincham and Spedding (1997), which showed weak effects of displacement gradients on the bias error when a smoothed spline ®t is used for sub-pixel resolution.

Whittaker results had good agreement for the potential core portion of the jet, which was an expected result since this portion of the ¯ow is similar to the calibration images used in generating the bias correction. The shear layer measurements suggested that the application of the correction to the Whittaker data tended to overcompensate for the bias error. It was shown that if the Whittaker bias calibration curve was attenuated by 40%, the agreement between the Gaussian and corrected Whittaker was greatly improved in the shear regions of the ¯ow®eld. It must be stressed that this does not imply that the Whittaker algorithm bias error is attenuated by the presence of displacement gradients because the same observation would be made if the Whittaker bias error remained ®xed while the Gaussian bias error was ampli®ed by the presence of displacement gradients. The conclusion is that the differences between the Gaussian and Whittaker results depend on the displacement gradients in the interrogation region, which is caused by a dependence of one or both of the algorithms' bias errors on the displacement gradients. This dependence is likely to be reduced if the particle image size is increased, since the effect of the gradients on the correlation peak would be less signi®cant. The commonly used criterion for avoiding velocity gradient biases was found to be insuf®cient for avoiding a dependence of the bias error on displacement gradients. Although the trends of the uncertainties may be similar for a broad range of experimental setups, the uncertainty of DPIV should be handled in situ. The calibration procedures must be repeated to determine which peak ®nding method is best suited for a particular experiment. Additionally, one must be cautious when using peak ®nding algorithms which have low bias errors as measured in uniform displacement ®elds since they may have bias errors in the gradient ®elds present in actual experiments.

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