On the effect of particle image intensity and image preprocessing on the depth of correlation in micro-PIV

The analytical model used to derive the DOC formula in (2) assumes that the intensity distribution \(\hat{J}_{0}\) of a particle image can be modeled as a Gaussian function (Olsen and Adrian 2000). This can be generally written as:where r is the distance from the particle center, d _e the particle image diameter and A and β two constants. The particle image diameter d _e corresponds to the point where the intensity distribution falls below a certain threshold, given by \(\exp(-\beta^2)\) times the maximum intensity at the center. The value of β² is typically equal to 3.67 (Adrian and Yao 1985).

The particle image diameter d _e is the result of three contributions: one due to the geometrical dimensions of the particle, one due to diffraction and one due to the distance z of the particle from the focal plane (defocusing). Under the assumption that these three contributions can be modeled as Gaussian functions, the resulting particle image diameter is given by (Olsen and Adrian 2000):

The single thin-lens optical geometry depicted in Fig. 1 was used by Olsen and Adrian (2000) to derive the particle image diameter. The lens corresponds to the objective lens of the microscope, D _a is the objective lens’ diameter and s _o is the working distance of the lens.

The particle geometrical dimension term in Eq. 4 is simply obtained by the particle diameter d _p times the magnification of the system. The diffraction term is given by the diameter of the point response system of a diffraction-limited lens (Adrian and Yao 1985), given by:

Considering the definition of f ^# = f/D _a , the expression of M = s _i /s _o and the Gaussian lens formula:Eq. 5 can be rewritten as:From trigonometry, we have that:so substituting Eq. 8 in 7 and using the definition of \({\text{NA}}=n_0\sin\theta,\) results in the diffraction term as a function of NA:

This expression, derived for a general case, coincides with the one derived by Meinhart and Wereley (2003) for infinity corrected systems.

The defocusing term was derived using geometrical optics for the single thin-lens configuration and is given by:Assuming s _o ≫ z and substituting Eqs. 8, 9 and 10 in 4, the particle image diameter as a function of z and NA is given by the following expression:

When the refraction index of the immersion medium of the lens n ₀ is different from the one of the working fluid n _w , there will be a deflection of the rays at the interface due to refraction. A ray is deflected depending on its angle of incidence according to Snell’s law: \(n_0\sin\theta_0=n_w\sin\theta_w. \) This means that a ray that would reach its focal plane at a distance l from the interface when n ₀ = n _w , will reach it at a distance of \(k\cdot l\) when n ₀ ≠ n _w . The factor k depends on the incidence angle θ as shown in Fig. 2.a and can be determined from Snell’s law and trigonometry:

The shift of the focal plane has to be taken into account when one determines the defocusing of a particle image. Without refraction, the defocusing term for a particle at a distance z ₀ from the focal plane is given by Eq. 10. When refraction is present, the same defocusing is obtained at the distance z _w from the focal plane, which can be determined from geometry (see Fig. 2b):

For z ₀≪ s _o , we can assume θ≈θ′ and \(z_w=k(\theta)\cdot z_0.\) Now, it remains to be determined which θ must be used to calculate the shift of the focal plane. In this case, an experimental approach was used to determine the k of lenses with different NA. The microscope was initially focused on particles that were stuck to the bottom wall of the microchannel using the selected lens. The microscope stage was then moved until the focal plane of the microscope was displaced to the particles stuck on the top wall. The focal position was determined by maximizing a focusing sharpness function. The variance of the pixel gray level of the image was used as the focusing sharpness function (Yeo et al. 1993).

The graph in Fig. 3 shows the theoretical values of k for the axial rays \((\sin\theta=0)\) and the marginal rays \((\sin\theta=\hbox{NA}/n_0)\) as a function of the NA of the lens, and the experimental value of k, determined for 4 types of lenses with NAs, respectively, equal to 0.3, 0.4, 0.6, 0.75. As can be observed in the graph in Fig. 3, the experimental k is in good agreement with the theoretical value for the axial rays, corresponding to the expression k = n _w /n ₀. This result might be different when different types of lenses are used; therefore, it is advisable to experimentally characterize this parameter for each experimental setup.

Bourdon et al. (2004a, b) showed that the weighting function in (1) for a certain depth coordinate z _k is equal to the curvature of the local correlation function evaluated at the local maximum:Under ideal conditions, this reduces to (Olsen and Adrian 2000):

The DOC formula is obtained solving for z _corr the following equality: \(W(z_{\rm corr})=\varepsilon. \) Using Eqs. 11 and 15 yields:

The multiplicative factor n _w /n ₀ is introduced to account for refraction. This formulation will be adopted in the following sections of the paper to obtain the theoretical values of the DOC.

In the most practical applications, the μPIV measurements are performed in commercial inverted or upright microscopes. A microscope, in its basic configuration, is composed of a 2-lens optical system: an objective that focuses an image in an intermediate image plane and an eyepiece that looks at that image. When human eyes are used to observe a specimen through the microscope, the eyepiece is a negative lens that forms a virtual image in front of the lens. When a CCD camera is used, the eyepiece is a positive lens that projects the intermediate image in the CCD. Quite often, infinity corrected lenses are also adopted to allow for the introduction of auxiliary components such as filters or polarizer (Fig. 4). Additional optical elements can be present as well to account for aberrations, to provide optical zoom and so on.

As a consequence of the increased complexity of a real microscope in comparison with the single thin-lens model adopted to derive the DOC formula, the following observations can be made:

To properly address these issues, it would be necessary to know exactly the configuration of the microscopic system in use, which is often very difficult or not possible at all in case of proprietary commercial microscopes. In this paper, we suggest to account for these effects using an effective NA defined as:where C _d is a constant to be determined experimentally. This approach has the advantage to be extremely simple and does not require any change in the DOC formula. On the other hand, it does not account for the effect of asymmetries or more complex defocusing or distortion patterns. The accuracy of this approach will be tested and discussed in the following sections.

The images of defocused particles can give a first qualitative impression of the behavior of a real microscopic system. Particle images obtained from observing a 2-μm particle with M = 20× and M = 63× lenses, at different distances from the focal plane, are shown in Fig. 5. The corresponding theoretical diameters calculated with Eq. 11, represented by a white circle, are superimposed on each image. From a qualitative observation of the images, the theoretical diameters match the border of the particle images at M = 20× but fail for images at M = 63×.

To quantitatively address the problem, the particle image diameter has to be determined experimentally. Unfortunately, this approach cannot be applied to real images in a straightforward manner for several reasons. A real defocused particle image, specially at large magnifications and NAs, starts to stray from the Gaussian behavior, showing, for instance, the Airy patterns. Moreover, background noise is present in every image, which makes the unambiguous determination of the threshold problematic. Additionally, for small particles at small magnification, the particle image size is reduced to a few pixels, and the reconstruction of the intensity distribution function from a single image is problematic. A more robust diameter determination can be obtained from the image autocorrelation.

The measured diameters for the two cases in Fig. 5 are reported in Fig. 6 for different depth positions, together with the maximum image intensity and the corresponding d _e (z) calculated using (11). The diameters were measured by setting a threshold equal to 0.15 between the background noise level and the peak intensity of the image autocorrelation. The diameter values were divided by a factor \(\sqrt{2}\) to account for stretching in the autocorrelation space (this is exact strictly for Gaussian-like distributions of particle image intensities). It can be observed that for M = 20× the measured diameters follow the theoretical prediction, while in the case of M = 63× the diameters grow much more slowly than expected. Moreover, for M = 63×, a clear asymmetric behavior can be observed, in particular looking at the drop of maximum intensity that is much sharper moving to positive z.

The results in Figs. 5 and 6 show that the analytical model in (11) is inadequate to predicts the particle image diameter for the M = 63× objective and suggest that the DOC predicted by (16) will be biased as well. However, to properly evaluate the contribution of real defocused particle images to the DOC prediction, one needs to measure the curvature of the image autocorrelation (i.e., the weighting function W), as shown by Bourdon et al. (2004a, b). The measured W at different z positions for the two cases in Fig. 5 is plotted in Fig. 7. In the same graphs are also plotted the W obtained from synthetic images, the W obtained using the theoretical model (W _th, dotted line), and the W obtained using the theoretical model in which the effective numerical aperture NA_eff was used (W _eff, continuous line). For the case of M = 20×, the theoretical model slightly underestimates the width of the actual W, and a practically perfect match can be obtained using a NA_eff with C _d  = 0.75. For the case of M = 63×, the theoretical model strongly underestimate the width of the actual W, and an asymmetric behavior can be observed moving away to the focal plane. However, the asymmetric part is restricted to regions in which W < 0.4, and a good fit with the theoretical model can be achieved in the central part using a NA_eff with C _d  = 0.37.

Following this approach, the curvature of the image autocorrelation at the different distances from the object plane was determined for each combination of lenses and particle sizes reported in Sect. 3, apart from the case of d _p  = 0.5 μm observed with M = 10×, for which the particle images were too small to be properly resolved. The data are presented in the graphs in Fig. 8. Each graph shows the measured W for a given configuration together with the W _eff calculated using the corresponding NA_eff. The x-axis, showing the distance z from the focal plane, was normalized over the corresponding DOC calculated using NA_eff (DOC_eff), so that the W _eff was always represented by the same function. The parameter C _d was derived experimentally for each lens searching the value that provided the best fit with the experimental data. The C _d values for the different configurations together with the DOC values obtained from the nominal NA (denoted as DOC_th) and the effective NA (denoted as DOC_eff), are reported in Table 1.

For magnifications of up to 20× (and NA up to 0.4), the model using NA_eff is able to precisely predict the particle image diameter. For the 40× lens, the experimental curves start to deviate from the theory for negative z; however, a good match between experiments and theory is observed in the central part. The same behavior can be also observed for the 63× lens except for the largest particles with d _p  = 5μm. In this case, the width of W is significantly larger also in the central part, and the only use of an effective NA in the model is not sufficient to correctly predict the trend of W also at short distances from the focal plane. Besides this last configuration (that is however never used in practical applications), the theoretical model combined with an effective NA, experimentally determined, provides a good estimate of the weighting function in proximity of the focal planes for all the different tested configurations and can be used to determine the actual size of the DOC in the experiment.

The results show that the actual DOC is indeed larger than the one predicted using the nominal NA of the objective lenses, especially when high NA lenses are used. This is evident in the graph in Fig. 9, where the ratio between DOC_eff and DOC_th as a function of the NA and the d _p is presented. For NA up to 0.4, the DOC_eff is always approximately between 1.3 and 2 times the DOC_th. For NA = 0.6, the DOC_eff is approximately 3–5 times larger than the DOC_th. For NA = 0.6, this ratio ranges between 4 and 12. The largest errors are observed for large NAs and small d _p s.

To conclude this subsection, it must be added that all the results were obtained taking particle images in the central region of the CCD sensor. Particle images taken at the outer regions of the sensor, especially at higher NA, might suffer for nonlinear distortions that were not taken into account in this analysis. Since this paper intends to provide a general single value of DOC in real experiments, it was chosen to limit the investigation to the central region of the CCD sensor. Moreover, preliminary experiments suggested that this aspect is negligible for the lenses used in this investigation. However, it is worth to keep in mind this aspect when measurements with high NA objectives are planned on the entire CCD area.

A direct consequence of the DOC is a bias error in velocity measurements taken in a flow where velocity gradients in the z-direction are present (Olsen 2009). For instance, if velocity measurements are taken scanning through a microchannel in which a Poiseuille flow is established, the reconstructed velocity profile will present a systematic error due to the DOC: in the region close to the center of the channel, the velocity is underestimated due to contribution of particles with lower velocities and close to the wall the measured velocity is overestimated due to contribution of particles with larger velocities. The velocity measured at the wall will be in fact different from 0 due to contribution of defocused particles, and a non-zero velocity will still be measured when the focal plane is moved outside of the channel. Actually, as a consequence of the flattening of the weighting function, the measured velocity is expected to increase again as the focal plane moves away from the channel wall, so that a local minimum of velocity will be present at the z location corresponding to the wall surface. A typical DOC-biased velocity profile is shown in the left graph of Fig. 10.

The velocity V(0) measured at the wall directly relates to the size of the DOC, as shown in the right graph of Fig. 10. This approach can be in principle used as an indirect quantification of the DOC in operative conditions. This is also useful to test the effectiveness of image preprocessing in reducing the bias error in conventional μPIV measurements. However, it has to be noted that the bias error cannot be uniquely assigned to the DOC, since it also depends, to a minor extent, on other contributions such as the finite size of the interrogation windows (Keane and Adrian 1990; Westerweel 2000), or the tracer particle concentration, which is lower in the region very close to the wall.

The velocity profiles in a rectangular microchannel with Poiseuille flow were measured using different combinations of lenses and particle sizes, in particular with 20×/0.4, 40×/0.6, 63×/0.75 and d _p  = 0.5, 1, 2, 5 μm. The velocity profiles were taken in the center of the channel (0.5 times the width), scanning in the z-direction. Results are presented in Fig. 11. For each set of data, the measured velocity profiles were obtained from raw and preprocessed images, and the results were compared with the theoretical profile obtained using nominal (V _th, dotted line) and effective DOC (V _eff, continuous line).

In the graph of Fig. 12, the measured velocity at the wall, presented as a function of the NA and the d _p , is reported. The velocity values are reported normalized over the theoretical values obtained with nominal (gray markers) and effective DOC (black markers). It is apparent from the graphs that the bias error predicted using the nominal DOC strongly underestimates the actual error, whereas using the effective DOC the prediction shows an excellent match with the experimental results. In particular, for small particles (d _p  = 0.5 and 1), the ratio between measured and predicted velocity at the wall is always below 1.3, whereas for larger particles, it varies between 1.5 and 2. This difference can be also ascribed to the additional bias error introduced by the lower particle concentration close to the wall, which is also proportional to the particle size.

The velocity measurements confirm that the actual DOC in a practical μPIV application can be significantly larger than the one predicted using only the nominal NA provided by the manufacturer of the lens. The measurements confirm also that using an effective DOC, calculated using an effective NA experimentally determined from the curvature of the particle image autocorrelation, provides a good estimate of the actual size of DOC.

Finally, the effect of image preprocessing was investigated. Image preprocessing is widely used in practical application of μPIV to remove background noise and could in principle eliminate the bias error due to DOC when all out-of-focus particles were removed. To quantify this last effect, we introduce the parameter C _p defined as the ratio between the measured velocity at the wall with (V _prep (0)) and without (V(0)) image preprocessing. C _p  = 0 means that the image preprocessing was able to completely remove the bias error, whereas C _p  = 1 means that the bias error was not diminished at all. Velocity profiles measured using preprocessed images are reported in Fig. 11. In Fig. 13, the C _p as a function of the DOC_eff is reported. The results show that the contribution of image preprocessing is quite limited. In general, it helps to reduce the error by a factor between 0.6 and 1, showing slightly better performance for small DOCs, but it is never able to produce data in which the effect of DOC could be completely neglected. This can be explained by the fact that particles remain in-focus, with a nearly constant diameter and intensity, in a relatively large region, corresponding approximately to W > 0.8. This prevents the preprocessing algorithm to discriminate and reject particles in that region, so that a bias error due to DOC will be always present.

The C _p can be used to estimate the effective DOC achieved after image preprocessing is applied. In fact, as a first approximation, the velocity at the wall can be considered linearly proportional to the DOC (see right graph on Fig. 10). Under this approximation, the C _p represents also the ratio between the DOC with and without the image preprocessing and can be directly multiplied in front of Eq. 16.

In conclusion, this experimental analysis shows that is very difficult to get rid of the bias error due to DOC in practical μPIV applications. Increasing the NA of the objective lens does not necessarily correspond to a significantly smaller DOC, since the effective NA resulting at the CCD sensor can be significantly smaller than expected. The image preprocessing scheme used in this work could only marginally reduce the DOC. Other solutions could be trying different image preprocessing strategies, such the power filter suggested by Bourdon et al. (2004a, b). However, on the opinion of the authors to completely avoid the bias error due to the DOC, other measurement techniques, such as 3D particle tracking methods, should be applied. For instance, Cierpka et al. (2010, 2011a, b) recently showed a method based on astigmatic aberrations that is able to measure 3D velocity fields in microflows without bias error due to the DOC. The effectiveness of this technique has been already proven in comparison with conventional and stereoscopic μPIV on a backward-facing step (Cierpka et al. 2011a, b).