WO2004005941A1

WO2004005941A1 - Two-point ensemble correlation method for piv applications

Info

Publication number: WO2004005941A1
Application number: PCT/NL2003/000498
Authority: WO
Inventors: Jerry Westerweel
Original assignee: Stichting voor de Technische Wetenschappen STW
Current assignee: Stichting voor de Technische Wetenschappen STW
Priority date: 2002-07-05
Filing date: 2003-07-04
Publication date: 2004-01-15
Anticipated expiration: 2005-01-05
Also published as: AU2003253497A1

Abstract

The invention relates to a PIV method for measuring velocity of a plurality of particles in a streaming fluid. The method includes: registering at least one image pair successively in time; estimating a two-point correlation for the at least one image pair; deriving from the estimated two-point correlation the velocity of the plurality of particles in the plane of the at least one image pair. Also: a plurality of N image pairs is registered successively within a predetermined time period, in which predetermined time period the flow in the streaming fluid is substantially constant; and the two-point correlation is estimated for each image pixel based on all image pairs of the plurality of N image pairs. The basic idea is that the interrogation window for estimating the correlation can be reduced to 1 pixel if a large number, eg. 1000, of image pairs is recorded for a stationary flow.

Description

FLOW VELOCITY DETERMENATION USING PIV AND TWO-POINT ENSEMBLE CORRELATION

FIELD OF THE INVENTION

The invention relates to a method for measuring velocity of a plurality of particles in a streaming fluid, comprising: registering at least one image pair successively in time; estimating a two-point correlation for the at least one image pair; deriving from the estimated two-point correlation the velocity of the plurality of particles in the plane of the at least one image pair. The invention further relates to a particle image velocimetry apparatus and the use thereof.

PRIOR ART

The use of particle image velocimetry (PIV) to flows at microscopic scales is commonly referred to as μPIV. With PIV, the velocity of a streaming fluid is measured indirectly by imaging particles suspended in the fluid. This implementation of PIV has several characteristics that make it different from conventional PIV. For one, the measurement domain in the direction perpendicular the light-sheet plane is limited by the finite depth-of-focus of the imaging optics, as opposed to conventional PIV where it is limited by the finite thickness of the light sheet. Also, for very small tracer particles that are observed at high magnification, the motion of the tracer particles is influenced by the effect of Brownian motion, hi addition, the image density in μPIV is generally much lower than in conventional PIV. To cope with the effects of Brownian motion and of low image density, the interrogation of μPIV image frames is commonly done by means of so-called ensemble correlation. This implies that the spatial correlation is computed in identical positions for a set of 10-20 consecutive image pairs, and then ensemble averaged (DELNOΠ, E., WESTERWEEL, J., DEEN, N.G., KUIPERS, J.A.M. & VAN SWAAIJ, W.P.M. (1999) 'Ensemble correlation PIV applied to bubble plumes rising in a bubble column.' Chem. Eng. Set. 54, 5159-5171; MEINHART, CD., WERELY, S.T. & SANTIAGO, J.G. (2000) 'A PIV algorithm for estimating time-averaged velocity fields.' J. Fluids Eng. 122, 285-289; WERELEY, S.T., GUI, L. & MEINHART, CD. (2002) 'Advanced algorithms for microscale particle image velocimetry.' To appear in: AIAA Journal 40).

This procedure yields a substantial improvement of the signal-to-noise ratio in μPIV, as the ensemble averaging yields an increase of the effective image density proportional to the number of images included in the ensemble, and the averaging reduces the measurement errors associated with the Brownian motion of the tracer particles (Meinhart et al. 2000). For conventional PIV the use of ensemble averaging would not be of use as the time scales of the flow would be much shorter than the time between the recording of successive frame pairs, but in many microscopic applications the flow is mostly stationary.

SUMMARY OF THE INVENTION

The invention relates to a method as specified in the preamble characterised in that - a plurality of N image pairs is registered successively within a predetermined time period, in which predetermined time period the flow in the streaming fluid is substantially constant; and the two-point correlation is estimated for each image pixel based on all image pairs of the plurality of N image pairs. In this application the ensemble averaging is taken one step further by using an extended ensemble, and applying it to compute the two-point ensemble correlation. The two-point correlation is the theoretical backbone of the PIV method (WESTERWEEL, J. (1993) Particle Image Velocimetry - Theory and Application. Delft University Press (Delft)), but it can not be utilized as the flows under study in conventional PIV are generally not stationary. Instead, only a single frame pair is available, and thus the two- point correlation is estimated from a spatial correlation over a finite interrogation domain under the assumption that the flow within the interrogation domain is uniform. This also limits the maximal spatial velocity gradient that can be measured reliably. When the flow under study is (nearly) stationary, it is possible to record a very large ensemble of image pairs. In that case it would not be necessary to compute a spatial • correlation as an estimate of the two-point correlation, but directly compute the two- point ensemble correlation. For example, instead of computing the spatial correlation over a 32x32-pixel domain, the correlation between two pixels is computed over an ensemble of 1,024 image pairs. This would include the same information content (in terms of the number of processed pixels), and the results would have the same quality in terms of signal-to-noise ratio. One obvious advantage of the two-point ensemble correlation is that there is no restriction in regard to the maximum spatial variation of the velocity flow field. Hence, the spatial resolution of the results obtained by the two- point ensemble correlation is limited by the pixel size. So, a 32x32-pixel spatial correlation in a 256 <256-pixel image would yield only 8x8 = 64 independent velocity vectors, whereas the two-point correlation would yield a displacement field of up to 256x256 = 65,536 'independent' vectors (the number of truly independent displacement vectors would actually be determined by the number of pixels divided by the particle-image area in pixel units).

Of course, this would only work properly provided that the flow remains stationary over the full recording period. An implementation of this method would require a high- speed image acquisition system with a frame rate of at least 1 kHz. The reduced image format of such cameras is then compensated by the very high spatial resolution of the two-point ensemble correlation analysis. The use of a 1024 1024-pixel image sensor (with a typical frame rate of 15 Hz) would typically yield 64x64-vector displacement field. Maintaining the same signal bandwidth, a high-speed image sensor with 64x64 pixels would have a frame rate of 4 kHz. So, at comparable spatial resolution an image ensemble of 256 frame pairs can be collected. The method would be applicable to flows that can be considered stationary within the recording of the image ensemble. In view of this example the maximum allowable flow time scale would be about 60 ms. So, the condition of 'stationary' flow is quite acceptable for many practical application (for example, the typical heartbeat of a person at rest takes about 1 second).

In this application we explain the fundamental differences that exist between the conventional spatial correlation and the new two-point ensemble correlation. The theoretical aspects are explained. The generation of synthetic μPIV images for different displacement fields for a proof-of-principle of the two-point ensemble correlation method is described. Also the results are described, and the potential for this method in micro-fluidic applications is discussed. BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be discussed in more detail using a number of exemplary embodiments, with reference to the drawings, which are only intended to illustrate the present invention and not to limit its scope which is only limited by the appended claims:

figure la shows a spatial correlation in conventional PIV over an NxN-pixel interrogation domain that is computed for a single image pair (top); figure lb shows the ensemble correlation computed over 1 -pixel domains averaged over multiple image pairs (bottom); figure 2a and 2b show examples of a pair of 64x64-pixel synthetic PIV images for the flow near a stagnation point, where the image contains about 80 particle images;

figure 2c shows the vector plot of the standard PIV result using 32x32-pixel interrogation windows with 4-pixel spacing (viz., 87.5% overlap), where the dotted line represents the position of the wall, and the circles indicate vectors at 16-pixel spacing;

figure 3a, 3b, 3c, 3d, 3e and 3f show a result for the stagnation flow as a function of the number of frames used for the two-point ensemble correlation;

figure 4 shows the fraction of valid vectors as a function of the number of frames;

figure 5 a, 5b and 5 c show the measured horizontal displacement (in px) as a function horizontal position (hi px) for the two-point ensemble correlation with 128, 1024 and 8192 frames respectively, where the diagonal lines indicate the reliability level for the measured displacement; figure 6 shows the results for an infinitely thin shear layer: two-point ensemble correlation (fig. 6a), spatial correlation using 32x32-pixel interrogation windows with 4-pixel spacing (fig. 6b), and the profile of the displacement as a function (fig. 6c, O two-point ensemble correlation; • spatial correlation);

figure 7a, 7b and 7c show the results for a boundary laminar layer: two-point ensemble correlation (fig. 7a), spatial correlation using 32x32-pixel interrogation windows with 4-pixel spacing (fig. 7b), and the profile of the wall-parallel displacement component as a function of the distance from the wall (fig. 7c, O two-point ensemble correlation; • spatial correlation; — analytical profile);

figure 8a and 8b show the results for the flow around a circular cylinder: spatial correlation using 32x32-pixel interrogation windows with 4-pixel spacing (fig. 8a), and two-point ensemble correlation over 1024 frames (fig. 8b) and

figure 9 shows a preferred embodiment of a particle image velocimetry apparatus.

DETAILED DESCRIPTION The underlying theoretical description of velocity reconstruction in standard PIV was first described for continuous image fields by Adrian (ADRIAN, R.J. (1988) 'Statistical properties of particle image velocimetry measurements in turbulent flow.' In: Laser Anemometry in Fluid Mechanics - III (ed. R.J. Adrian et al.) LADOAN Instituto Superior Tecnico (Lisbon), pp. 115-129) and later extended to digital images by Westerweel (1993). Adrian (1988) considers the ensemble of all possible realizations of the tracer pattern for a given (fixed) flow field u(X,t). For this ensemble, the two-point conditional correlation of the image fields ^(X) and/_(X) yields:

i-(s) _≡ <7₁(X)/₂(X + s)|u) <χ _JF_o(Δz) -(CΔz₀/ ₀ ²)- (_S-s_D) , a)

where C is the number density of the tracer particles, Δz₀ the light-sheet thickness, ₀ the image magnification, Eo(Δz) the loss-of-correlation due to an out-of-plane displacement z (i.e., the fraction of particles that enter of leave the light sheet during the exposure time delay), F^s) the particle-image self-correlation, and S_D the in-plane particle-image displacement. The expression implies that the two-point ensemble correlation yields a single correlation peak for which the amplitude is proportional to the number density of the particle-image pairs, and for which the location corresponds to the in-plane displacement S_JJ of the particle images.

When PIV is applied to an actual experiment, the flow field u(X,t) is not fixed, so that only one realization of I\(X) and 7₂(X) is available. In order to evaluate the images the ensemble averaging is replaced by a spatial averaging over an interrogation domain W, i.e.:

-$(s) = |/,(X)/-(X + s)dX . b) w

The expectation of this estimator yields a biased estimate of the two-point ensemble average:

(R(_S)

∞ F₁(_S) - F₀(AZ) - N_! - F (S -_SD) , with: N, = (CAZ₀/M₀ ²) - A , C)

where N₇ is the number of particle-image pairs in the interrogation domain, or image density, A is the characteristic area of the interrogation domain W, and Fj(s) is the in- plane loss-of-correlation. The bias due to Fj(s) is a known function for a given interrogation domain W. The spatial correlation is a reliable estimate for the ensemble correlation provided that the image field is a homogeneous random field (ROSEΝFELD, A. & KAK, A.C. (1982) Digital Picture Processing (2nd Ed.) Academic, Orlando; KEAΝE, R.D. & ADRIAN, RJ. (1992) 'Theory of cross-correlation analysis of PIV images.' Appl. Sci. Res. A9, 191-215). The principle of interrogation of a single image pair using a spatial-average estimate for the correlation is illustrated in Figure 1.

A contemporary PIV system consists of a digital camera with a typical resolution of 1024x1024 pixels, which can record image pairs at a rate of 15 Hz. The typical size of the interrogation domain is 32x32 pixels, so that the typical number of vectors per image is (63x63=) 3,969 (for interrogation domains with 50% overlap). When the PIV measurement is designed with respect to the 'rules' prescribed by Keane & Adrian (1992), the success rate for measuring the correct displacement is at least 95%. For a typical domain size for W of 32x32 pixels the measurement precision for the displacement is better than 0.1 px (WiLLERT, C.E. (1996) 'The fully digital evaluation of photographic PIV recordings.' Appl. Sci. Res. 56, 79; WESTERWEEL, J. (2000) 'Theoretical analysis of the measurement precision in particle image velocimetry.' Exp. Fluids 29 (Suppl.), 3-12).

So, the consequence of using a spatial-average estimator for the two-point ensemble correlation is that a camera image that contains more than a million pixels yields (typically) on the order of 10⁴ measurements. In μPIV the image density Nj is generally very low (Ni ~ 1), so that it is not possible to obtain reliable estimates of the two-point correlation by means of computing the spatial correlation between two consecutive image pairs. The common procedure to enhance the quality of the spatial-average estimator is to average the spatial correlation of corresponding interrogation domains in successive image pairs. This procedure is generally referred to as ensemble correlation (Dehioij et al. 1999, Meinhart et al. 2000), although in view of the present paper we would prefer to call it: ensemble averaging of the spatial correlation estimate. Meinhart et al. (2000) used this approach to enhance the results for flow in a micro-channel, and Dehioij et al. (1999) used a very similar approach to detect simultaneously correlation peaks due to tracer motion and bubble motion in a bubble plume. In both cases about 10-20 image pairs were used for the averaging procedure.

The result is that the effective image density is improved in proportion to the number of image pairs used for the averaging of the spatial-correlation estimate. The amplitude of the displacement-correlation peak is directly proportional to the image density, so that averaging over Np image pairs would yield a displacement-correlation peak with an amplitude Np-Ni. For example, when the image density for a single image pair is Nj = 1, then ensemble averaging over 20 image pairs would yield a spatial correlation with a quality that would correspond to N_/ = 20. The correlation noise i?'(s), i.e. random correlation peaks defined by: R'(s) s R(s) - (R(s) | u) , is effectively suppressed relative to the amplitude of the displacement-correlation peak.

The disadvantage of the standard method discussed above is that we use spatial interrogation windows to reconstruct the velocity. The consequence is that we can not use the full pixel resolution of the camera for the measurement of the velocity field. For applications in μPIV this is serious limitation. Therefore, we present here a new approach that has the potential of yielding a much higher spatial resolution, but that can only be applied to a limited class of (quasi-) stationary flows.

Let us return to the original two-point ensemble correlation. Assume that we have a sample of N image pairs of a stationary flow. In that case it would be possible to compute an estimate of a two-point ensemble correlation, using: R^*(s) = -∑/₁ ⁽'^,(X)/ (X + _S) . d)

The expectation of this estimate yields the exact two-point ensemble correlation. This type of approach is referred to as two-point ensemble correlation, and its principle is illustrated in Figure 1. Provided that the number of images is extended substantially, it would be in theory possible to completely avoid the spatial averaging. Suppose it would be possible to capture a very large number of images, say 10 (each containing a different set of tracer particles), then it would be possible by using R (s) to estimate for each individual image pixel the two-point correlation. In other words, each pixel would yield a displacement measurement. When the number of image pairs is 1,024, then the amount of information that is available for estimating the two-point correlation is in theory equal to that for a 32x32 pixel interrogation domain using a spatial-average estimator. Hence, we can expect the same success rate for valid measurements and precision for the displacement; evidently this conjecture needs to be validated by means of simulations and in measurements.

For the method to work sufficiently accurate, the flow velocity of the streaming fluid must be substantially constant during the time period in which the N image pairs are captured. No restriction is necessary with regard to the spatial variation of the flow velocity.

In order to validate the two-point ensemble correlation method described above, we generated a series of synthetic PIV images for different flow types. The objective is to validate the reliability and precision of the new method, and also to demonstrate its capabilities with respect to standard PIV and ensemble averaging of the spatial correlation estimate. We consider four flow types that represent different aspects of the method: (1) the flow near a stagnation point; (2) shear-layer flow; (3) boundary-layer flow, and (4) the flow around a circular cylinder.

In most μPIV applications the tracer particles are fluorescent, and a filter is used to record only the fluorescent light emitted by the tracer particles. For an exposure that fully saturates the fluorescent dye, the amount of light emitted by the tracer particles would be proportional to the particle surface area or particle volume. We assume monodisperse tracer particles, and thus it would be a valid assumption that all particle images have the same (integrated) intensity. Also, the tracer particles are very small (typically 200-300 nm), so that all particle images are diffraction limited, and therefore all particle images have the same diameter. We also assume that all particle images are in focus (although our program for generating the synthetic PIV images is capable of generating also images with out-of-focus particle images), and that the tracer particles are ideal (i.e., we ignore Brownian motion, as this would be effectively cancelled in the ensemble averaging; see: Meinhart et al. 1999). All particles that move out of the simulation domain are re-inserted at a random position on the in-flow boundary (depending on the flow type; see below). For flow types that include a plane boundary, we place the boundary at a distance of 8 pixels from the lower edge of the image. All images are generated with a format of 64x64 pixels with 256 gray levels (viz., 8-bit quantization). The mean number of particle images in each frame is 80. This would imply image densities of 5 and 20 for a 16x16-pixel and a 32x32-pixel interrogation region respectively. The particle-image diameter in all synthetic images is 4 pixels, which would correspond to a relatively large diffraction-limited spot for high magnification imaging. A typical image pair (for the flow near a stagnation point) is shown in Figure 2.

The two-point ensemble correlation was implemented in a Matlab program (version 5.3), running on a PC with a AMD Athlon 900 MHz CPU under Linux (RedHat version 7.2). The time for processing a 1,024-image ensemble is 186 s for computing the two- point ensemble correlation for all 64x64 pixels. The computation of two 32x32-pixel FFTs replicated 61x61 times under Matlab takes 4.1 seconds. (For comparison, our own standard PIV code uses 3.8 s for the processing of a single lkxlk image pair in 61x61 positions using 32x32-pixel interrogation regions.)

We compare the results of the two-point ensemble correlation against the spatial correlation result using 16x16-pixel and 32x32-pixel interrogation regions.

We generated an ensemble of 8,192 consecutive images of the flow near a stagnation point. The displacement field is given by:

Ax = a ■ x and: Ay - -a-y with: a = 1/32 px. e)

This represents a situation with a strong gradient in the displacement field of (1/32 =) 0.03 pixels displacement per pixel distance (px/px). We evaluated the result as a function of the number of images in the ensemble. The results are shown in Figure 3, which clearly demonstrates the effect of the ensemble size. For a small number of frames it is possible to recognize the individual trajectories of the particle images, but as the number of frames increases this effect rapidly vanishes. The effective image density for the two-point ensemble correlation is given by the mean number of particle images per pixel multiplied by the number of frames that was used for the computation of R (s). We used an average of 80 particle images for a 64x64-pixel image, i.e. a density of 0.020 particle images per pixel. In order to reach an image density that is comparable to a 32x32-pixel interrogation region, we need at least 1,024 frames.

In Figure 3 we only plotted vectors for which R (s) contained a well-defined displacement-correlation peak. In Figure 4 is shown the number of valid displacement vectors as a function of the number of frames. When the number of frames is more than about 200 (i.e., an effective image density of about 4) the number of spurious vectors (in our simulations) is effectively zero. However, the random error in the data is clearly a function of the number of frames, as illustrated in Figure 5. This figure shows the displacement data parallel to the wall. The 95-98% reliability interval is indicated by the red dashed lines, and the rms random error could be estimated as 1/4 of this interval which is about 0.1 px for a 1024-image ensemble. This is comparable to the expected random error for 32x32-pixel interrogation regions with an image density of 20 (Willlert 1996; Westerweel 2000).

We generated an ensemble of 1,024 consecutive images of a two-layer flow, i.e. a shear layer with zero width. A flow with a discontinuity in the flow field is not a physical reality, but is an excellent test for the spatial sensitivity of the method. The displacements in the upper and lower layers are 2 px and 4 px respectively. Results for the displacement fields obtained with two-point ensemble correlation and spatial correlation are shown in Figure 6. In this figure we also compare the velocity profiles of the two-point ensemble correlation result against the results for standard PIV with 32x32-pixel interrogation regions. The results clearly demonstrate that the two-point ensemble correlation is capable of resolving the discontinuity, whereas for the standard-PIV results the width of the velocity change is clearly determined by the size of the interrogation domain. Also note that the result for the spatial correlation is biased towards the low-velocity region, i.e. the spatial correlation results at x = 32 px is 2.6 px in stead of 3.0 px (viz., the mean of displacements in the two layers). This velocity biasing is a direct consequence of using the spatial correlation estimator (Adrian 1988; Westerweel 1993), and is absent in the two-point correlation result. Also not that the gradient of the result for the two-point ensemble correlation is only about 2 px, whereas the particle-image diameter in the synthetic μPIV images is 4 px.

A practical flow encountered in flow through micro-channels is a boundary layer. For this type of investigation it is very important to have a high spatial resolution near the wall, in order to validate the whether or not the no-slip boundary condition applies (STONE, S., MEINHART, CD. & WERELEY, S.T. (2001) 'Using μ-PIV to probe wall shapes with nanoscape resolution.' In: Proc. 4th Int. Symp. on Particle Image Velocimetry (Gδttingen, Germany) Sept. 17-19, paper 1143). To demonstrate how two- point ensemble correlation can resolve near-wall motion, a set of 1024 synthetic micro-PIV images were generated with a displacement field given by:

Ax =

0 with: & = 5 px, f)

where y is the distance from the wall. The results for the two-point ensemble correlation and spatial correlation are given in Figure 7. Note that the spatial correlation results is now biased towards a higher displacement. This is caused by the fact that part of the interrogation domain overlaps with the boundary, so that the true velocity at the interrogation position is always smaller less than the displacement measured with spatial correlation PIV. The error is quite substantial: the measured displacement at the wall is about 25% of the free-stream displacement. This demonstrates that the use of spatial correlation can lead to incorrect conclusions in regard to the existence of a non-zero wall slip velocity .

Our final flow type is the flow around a circular cylinder. This flow type combines aspects of the three earlier flow types, and would be representative of the flow around an object, e.g. a bubble or a blood cell. Again, a 1,024-image ensemble was generated. The results for the two-point ensemble correlation and the standard PIV are shown in Figure 8. The 1 -pixel resolution of the two-point ensemble correlation makes it possible to resolve the detailed flow pattern at any point around the cylinder, whereas the standard-PIV result only represent a rough picture of the flow and has difficulties in resolving the flow near the cylinder wall. Especially the near-wall flow and the front and back stagnation points are not resolved be the spatial correlation.

The evaluation of the two-point ensemble correlation requires a new approach. For the conventional spatial-average estimate we can make use of the efficient fast Fourier transform, or FFT, algorithm to compute the correlation. In the case of the ensemble correlation we can not rely on such highly efficient algorithms. Therefore, new analysis algorithms need to be developed. Fortunately, this type of interrogation is ideally suited for implementation on a computer with parallel processor architecture, so that we can make use of recent advancements in affordable high-performance computing.

Of course, there are a couple of drawbacks associated with two-point ensemble correlation. First, this would only work when the flow is stationary during the full period when the images are recorded. This is unappealing for a conventional PIV system, which can only record at a rate of 15 Hz. However, the development of new, high-speed cameras make it possible to record images at a rate of more than 10 kHz. For a slowly evolving flow it would then be possible to record many images before the flow itself has changed. For example, the typical human hart rate is 60 beats per minute, i.e. 1 Hz, so the flow through an artery during 0.1 seconds may be considered as almost stationary. When recording at 10 kHz, it is possible to capture a thousand image pairs during that 0.1 second interval.

The typical resolution for these high speed cameras is 256x256 pixels; a reduction of the spatial resolution is necessary in order to reach such high framing rates. However, the ensemble correlation proposed here would enable us to extract a displacement vector for each pixel, so that the number of velocity measurements would also be 256x256, which is still higher than the data resolution for conventional PIV and that would enable us to capture the very steep velocity gradients in near-wall regions (e.g., for the measurement of wall shear stress).

In order to get an appreciation for the length scales and velocity scales that can be measured we consider the following configuration: Assume that the image field-of- view of a 256x256-pixel CCD is lxl mm², so that a displacement of 5 px corresponds to 20 micron. At a framing rate of 10 kHz, the corresponding velocity would be 0.2 m/s. Under the assumption that the displacement can be determined with a precision better than 0.1 px, the anticipated measurement error would be less than 0.004 m/s. Given that the typical size of a pixel on a CCD sensor array is lOμm, the two-point ensemble correlation would in principle make it possible to reach a lμ resolution at a lOx magnification. This new approach would also create new options for the future. For example, a further reduction of the CCD format, e.g. from 256x256 to 64x64 pixels would make it possible to capture images at a framing rate of 160 kHz (while maintaining the same signal bandwidth). Hence, a set of 1,000 images can be captured in less than 10 milliseconds, and flows with a typical time scale of around 0.01 s could be captured.

A preferred embodiment of the invention is shown in figure 9. The particle image velocimetry apparatus 1 comprises processing means 11 and imaging means 12, where the imaging means 12 are arranged to communicate with the processing means 11. The imaging means 12 are arranged to register a plurality of images and to send information about the registered images to the processing means 11. The images can be images of particles in a streaming fluid, such as a channel 1.

The processing means 11 are arranged for executing the method as described above.

The imaging means 12 can for instance be a sensor array, such as a CCD imaging device or any other known camera known to a person skilled in the art. Such imaging means 12 can for instance be arranged to produce a 64x64 pixel image.

The particle image velocimetry apparatus 10 can also comprise integrated optics for maximizing optical performance of the imaging means 12. The imaging means can be also be arranged to register images with a predetermined focus range.

The invention makes it possible to produce relatively small PIV- and μPIV-systems. It will be understood by a person skilled in the art that such small PIV- and μPIV-systems can, for instance, be used for biological and/or medical applications. Small probes, using e.g. a 0,5x0,5mm CCD array with integrated optics, can be created that can easily, for instance, be placed on the body of an animal, such as a human animal or be inserted in a blood vessel or artery of such an animal, to measure a blood flow. The PIV method has been applied with much success to various fluid mechanical experiments. In a examplary embodiment of the present invention this method is adapted to the measurement of flow phenomena in a compliant biological system that take place on the micrometer scale. The adaptation to microscopic scales involves changes in the optical set-up of the μPIV-system. The principal limitations are related to fundamental aspects of diffraction limited optics, such as the illumination with a very thin light sheet and the image formation of a very small tracer particle.

Let us first consider what the physical limitations are to the application of this measurement technique. The observation of a small particle limited by diffraction is given under some assumptions by the equation

where d_f is the diameter of the diffraction spot, M the magnification, f ^# the numerical aperture and λ the wave length of the light.

The situation that we are faced with in our biological experiment is to measure the blood flow in a vessel of a chicken embryo with a diameter d_vessei = 0.43 mm. This diameter determines our field of view. With a typical size of a CCD-chip given by dcc_D = 8.6 mm the magnification should be in the range of M = 20. The size of a pixel on the CCD-chip is 6.7 μm. Under this condition, but without taking into account the diffraction limit, the spatial resolution of the measurement is better than 1 μm.

The laser wavelength is λ = 0.532 μm. For the microscope we estimate f* =1-2. With these values for M, λ and f* the diffraction limited spot becomes 28-55 μm in the image plane and this corresponds to a spatial resolution of 1.4 to 2.3 μm in the measurement plane and this is sufficient for our purpose. It should be stressed that this is a theoretical upper limit to the spatial resolution and that the real resolution depends on the method of velocity reconstruction as discussed above.

The second limitation of known PIV systems in microscopic flows is the thickness of the laser light sheet. In standard PIV all particles in the light sheet are illuminated and recorded because all of the illuminated particles are within the focal depth of the imaging optics. In our case we will use microscopes and these have a typical optical depth of a few μm. The minimum thickness of the waist of the laser sheet δ can only be maintained over a finite distance (Rayleigh length) Λ, where the ratio δ ²/Λ is a constant depending on the wavelength of the light. For example, a light sheet with a thickness of 10 μm can only be maintained over a distance of 0.3 mm for a wavelength of light of λ = 0.532 μm. Therefore, the use of a laser light sheet for μPIV seems not a good option, even if we would neglect the problems of constructing such light sheet in our biological experiment. Therefore, we resort to the use of diffuse lighting. For our case we propose to use Nd:YAG laser in low power configuration for a uniform spatial illumination in backward scattering mode. The use of the Nd:YAG laser is of advantage since the PlV-cameras have to be illuminated with pulsed light. Because the object domain is defined by the focal depth of the optics one will also record out-of- focus particle images. With the proper image processing one can eliminate the influence of these smeared out-of-focus particle images on the measurement.

The next problem to be considered is whether particles need to be added to the flow in order for the PIV technique to work. In a biological experiment with a living organism we should refrain from adding particles to the flow because that would possibly interfere with normal development. In the first stage of embryonic development to which we restrict our study, the blood vessels are transparent for visible light. Therefore we will start with the particles that are already present in the bio fluid. These are for instance the red blood cells, which have a size of 7.5+0.3 μm.¹

Another problem to be expected is that the lighting set-up with the Nd:YAG laser may cause too many reflections. In that case we can not work with light scattered from native blood cells. The alternative is to use small fluorescent particles, with a size of about 0.2 μm, in combination with optical filters. The Nd:YAG laser in combination with a sensitive camera should be sufficient to observe these particles despite their small scattering cross section. Experience with dyes, carbon particles and 20-30 nm colloidal gold particles in biological experiments shows that such small particles do not interfere with normal development or function of the organism for at least 20 hours. The invention can also be used to create relatively light PIV- and μPTV-systems that can cost-effectively be used in space applications, such as scientific experiments in a space station under microgravity.

For the purpose of teaching the invention, preferred embodiments of the method and devices of the invention were described. It will be apparent for the person skilled in the art that other alternative and equivalent embodiments of the invention can be conceived and reduced to practice without departing from the true spirit of the invention, the scope of the invention being only limited by the annexed claims.

Claims

1. Method for measuring velocity of a plurality of particles in a streaming fluid, comprising - registering at least one image pair successively in time; estimating a two-point correlation for the at least one image pair; deriving from the estimated two-point correlation the velocity of the plurality of particles in the plane of the at least one image pair; characterised in that - a plurality of N image pairs is registered successively within a predetermined time period, in which predetermined time period the flow in the streaming fluid is substantially constant; and the two-point correlation is estimated for each image pixel based on all image pairs of the plurality of N image pairs.

2. Method according to claim 1, in which the two-point correlation is calculated using the formula

in which R (s) is the estimate of the two-point correlation, s is the displacement vector, Ii^W and I₂ ^(l) are the plurality of N image pairs.

3. Method according to claim 1 or 2, in which each image is a 64x64 pixel image.

4. Method according to claim 1, 2 or 3, in which N is at least 1000.

5. Method according to one of the claims 1 to 4, in which the plurality of image pairs are registered with a rate of at least 1 kHz, preferably at least 10 kHz.

6. Method according to one of the claims 1 to 5, in which the streaming fluid is a microscopic scale flow.

7. Particle image velocimetry apparatus (10) comprising processing means (11) and imaging means (12) connected to the processing means (11) for registering a plurality of image pairs, the processing means (11) being arranged for executing the method according to one of the claims 1-6.

8. Particle image velocimetry apparatus (10) according to claim 1, in which the imaging means (12) comprise a sensor array, e.g. a CCD imaging device.

9. Particle image velocimetry apparatus (10) according to claim 7 or 8, in which the imaging means (12) are arranged to provide a 64x64 pixel image.

10. Particle image velocimetry apparatus (10) according to claim 7, 8 or 9, in which the imaging means (12) comprise integrated optics for maximizing optical performance of the imaging means (12).

11. Particle image velocimetry apparatus (10) according to one of the claims 7-10, in which the imaging means (12) are arranged to register images with a predetermined focus range.

12. Use of the particle image velocimetry apparatus (10) according to one of the claims 7-11 in biomedical diagnostics, e.g. for measuring blood flow in a vessel or artery.