Fast by a

quantitative processing of particle whole-field filtering technique V. Palero, N. Andres, M. P. Arroyo, M. Quintanilla

Experiments in Fluids 19 (1995) 417 425 ~ Springer-Verlag 1995

image velocimetry photographs

Abstract A fast quantitative processing of particle image velocimetry photographs by a whole-field spatial filtering technique is described. Photographs are observed through a conventional filtering setup. This produces fringe patterned images with each fringe corresponding to a fixed value of one velocity component. These images are acquired with a CCD camera and digitally processed to retrieve the fringe centerline positions. The interpolation of these data provides the velocity value on a grid of regularly spaced points.

Photographs taken from a Rayleigh-B~nard convective flow have been processed with this technique and with a previously reported point-by-point technique. Results from both tech- niques compare well.

1 Introduction Processing of particle image velocimetry (PIV) photographs originated with point techniques based on the analysis of Young's fringes (Burch and Tokarski 1968) and a whole-field approach based on spatial filtering (Celaya et al. 1976). Sub- sequent research has concentrated more on point analysis techniques, producing a whole range of techniques (Meynart 1980a; Robinson 1983; Arroyo et al. 1988a, b; Kirita et al. 1988; Arroyo and Greated 1991; Prasad and Adrian 1993) that automatically measure the two and even the three components of the velocity at selected points of the flow, usually spaced along a regular grid.

In the point techniques, a small portion of the photograph is illuminated and analysed to get information of the local velocity at that point. The local velocity is always automatically obtained through a processing that can be fully digital (Westerweel et al. 1991; Willert and Gharib 1991), fully optical (Coupland and Halliwell 1988; Kompenhans et al. 1989) or a mixture of optical and digital (Adrian 1986a; Huntley 1986; Arroyo et al. 1988a, b). In a fully digital processing the illuminated portion of the photograph is projected onto a CCD for digitization, and the correlation function is calculated

Received: 2 February 1995/Accepted: 23 June 1995

V. Palero, N. AndrOs, M. P. Arroyo, M. Quintanilla Departamento de Fisica Aplicada Facultad de Ciencias Universidad de Zaragoza Ciudad Universitaria E-50009 - Zaragoza, Spain

Correspondence to: M. P. Arroyo

This work was supported by Diputaci6n General de Arag6n under Grant No. PCB6-90

digitally. This correlation function has some peaks at the most frequent distance between the image particles on the photograph. The search for the position of these peaks is also done digitally. In the other two processings, a first Fourier transform of the illuminated region is obtained optically by placing a converging lens after the photograph. A Young's fringe pattern is thus obtained on the focal plane of the lens (Burch and Tokarski 1968), with the spacing and orientation of the fringes being the parameters to be measured in order to calculate the local velocity. This requires a second Fourier transformation that is usually done digitally, after the fringe pattern has been detected and digitised with a CCD camera (Adrian 1986a; Huntley 1986; Arroyo et al. 1988a, b). However, it also can be done optically (Coupland and Halliwell 1988; Kompenhans et al. 1989), leaving the correlation function ready to be digitised in order to perform the peak finding.

In the whole-field approach (Celaya et al. 1976; Meynart 1980b), the whole photograph is illuminated and observed through a conventional spatial filtering setup. This produces fringe patterned images, with each fringe corresponding to a fixed value of a certain component of the velocity, controlled by the position of a filtering hole, placed in the Fourier plane of the setup. Usually, these filtered images were only used as a fast means of obtaining qualitative information about the spatial structure of the velocity field (Bernabeu et al. 1982; Suzuki et al. 1983; Giirtner et al. 1986; Arroyo et al. 1988a, b; Merzkirch and Wintrich 1990).

The purpose of this work is to extend this whole-field filtering technique to obtain quantitative information about the velocity field in a fast and automatic way. Fringe centerlines have been determined from each filtered image using digital image processing techniques adapted from the techniques developed for interferometric patterns (Yu 1988; Yatagai et al. 1982). Thus, the two components of the velocity are known at randomly spaced points in the flow. These data are then interpolated to have values of the velocity on a regular grid of points. It is shown that the accuracy of the data obtained with this whole-field technique (3% of the full-scale velocity) is not very different from the 1% accuracy obtained with the point-by-point techniques (Adrian 1991).

2 Whole-field filtering technique The technique is based in the processing of the PIV photographs with a conventional spatial filter (Fig. 1). A uniform parallel beam from a He-Ne laser illuminates the photograph. A filtering hole is placed in the focal plane (Fourier plane FP) of a converging lens L3, situated at focal

417

He-Ne laser } ~ ~

L 1

~, (x~176 i (xf ,yf) (Xi ,Yi)

L 2 P L 3 FP L 4

Fig. 1. Spatial filtering setup (L~, L2: beam expander; P: particle image velocimetry photograph; L3: Fourier transform lens; FP: Fourier plane with the filter; L4: CCD camera lens)

418

distance f~ from the photograph. The image of the photograph is observed with a CCD camera, which has its lens L 4 placed at focal distance f2 from the Fourier plane.

It can be demonstrated (Cathey 1974; Pedretti and Chiang 1978) that if t(ro) is the transmittance of the photograph, where ro is the position vector in the (xo, Yo) object plane, the amplitude on the Fourier plane, after the filter, may be expressed as

exp [i f~ 4n/2] U ( r f ) - i2f~ FT{t(ro)}P(rf) (1)

where rf is the position vector on FP, f~ is the focal length of L3, P(rf) is the transmittance of the filter and FT{t(ro)} is the Fourier transform of t(ro), defined as

1 , FT{t(ro)}=S t(ro) exp - - i s T r o ' r f dxodyo (2) ~(,

In the same way, the amplitude on the (x~, yi) image plane, where the CCD sensor is placed, may be expressed as

U(ri) exp [i~4~/2] - i2fi FT{U(rf)}

= exp [i 4~ fl ~(fl+f2)]ft(--~ri).FT{P(rf)} (3)

where ri is the position vector on the image plane, f2 is the focal length of the CCD camera lens, �9 means a 2-D convolution and FT{P(rf)} is defined as

FT{P( r f )}=~ i'~ P(rs) exp - l ~ r f ' r i dxfdyf (4)

If the filter is a circular hole (with radius R), whose center is located at p, the intensity in the image plane may be expressed by

I(ri)=C ~ ~ t ~ (r-ri) ]exp [ - i 2)~f2P'r ]

211 (2~Rr/2f2) dxdy 2 x 2~Rr/2f2 (5)

where C= 0zR2fjy~) 2 is a constant. The transmittance of the photograph may be expressed as

N

t ( ro )= ~ g[ro-(n-1)d] (6) t / - - 1

where g(ro) is the transmittance of the photograph when only one exposure is taken, d=d( ro ) is the displacement of the particle images between two successive exposures and N is the total number of exposures taken on the photograph.

Assuming that the particle displacements d (ro) and its spatial gradients are small, Eq. (5) can be rewritten as

I(ri) = F(ri) I' (ri) (7)

where

F (ri) = sin2 (N~p" d/2f l ) sin 2 (~p" d/2f~) (8)

and

I ' ( r i ) = C S i g ~ 2 ( r - r i ) l e x p [ - i ~ 2 P ' r 1

211 (2z~Rr/2f2) dy 2 x 2~Rr/2f2 dx (9)

The term I' (r~) represents the image of the photograph (with a single exposure) through the optical system affected by a speckle due to the interference between independent particle images, which is related to the factor 2J~ (27rRr/2f2)[ 2z~Rr/2 fd. This speckle is always very noticeable on the filtered images since its size (1.22 2fd2R) is usually bigger than the size of the particle images. The term F(ri) represents a fringe pattern that takes its maximum intensity, N 2, on the points where

p ' d ( r i ) _pdp m; m = 0 , • • • . . . . (10) ;#,

where dp represents the projection of the displacement on the direction defined by the filtering hole. Thus isovelocity contours are drawn by the spatial filter, with the value for each fringe being

vp = m 2fl/tMp (11)

where m is the order of the fringe, t is the time interval between two successive exposures and M is the magnification of the PIV photograph.

Finally, let's note that the two assumptions used to obtain Eqs. (7) (9) are equivalent to the following conditions: a) R must be smaller than the least spacing of the Young's

fringes in the Fourier plane (R < 2f~ld), and b) the fringe spacing, AI, in the filtered images must be larger

than the speckle size (A l> 2fdR) If these two conditions are poorly met, the fringe contrast will be low and the fringes may not be well resolved. Thus, the fringe spacing in the filtered image should be such that

AI >~ d (12)

Figure 2 shows an example of the filtered images obtained from a multiple exposure PIV photograph taken from a Rayleigh B6nard convective flow (Arroyo and Savir6n 1992).

419

Fig. 2a. Multiple exposure photograph of a Rayleigh-Bbnard convective flow in a small rectangular cell; b, c filtered images. The increment between consecutive fringes is d Vx = 25.0 ~m/s for b and A Vz = 34.0 ~tm/s for c

Figure 2a shows the PIV photograph, where 7 exposures of 140 ms at intervals of 1.4 s were taken from the cell midplane (dimensions of the plane: 25 x 12.3 mm) with M = 1. The fluid is going up by the center of the plane and going down near the cell walls. Filtered images have been obtained with the filtering setup shown in Fig. 1, where fl =300 ram, J~=65 mm and R ~ 0.5 mm. In Fig. 2b the filtering hole was placed at 4.9 mm from the origin along the horizontal axis, corresponding to Vx= m 25.0 ~m/s. In Fig. 2c the filtering hole was placed at 3.6 mm from the origin along the vertical axis, corresponding to Vz = m 34.0 ~tm/s. These filtered images allow a rapid whole-field examination of the features of the flow pattern. For instance, one observes that both the horizontal and the vertical velocity fields are pretty symmetric with respect to the axis of symmetry of the cell. The zeroth order fringes, corresponding to Vx = 0 in Fig. 2b or G = 0 in Fig. 2c, are easily identified by comparing with the photograph (Fig. 2a). It is also easy to identify which fringes correspond to negative velocity values

and which correspond to positive velocity values since we have already said that the fluid is going up by the center of the plane. By counting the number of fringes, it can be seen that the maximum value is similar for the positive and the negative horizontal velocities, while the value of the positive vertical velocity is about twice the value of the negative vertical velocity.

This technique is extremely simple and provides directly a velocity map of the 2-D flow pattern. The main advantages are: a) vp can be easily changed by changing the distance of the

filtering hole to the origin; b) the velocity component to be drawn can be easily selected

by placing the filter in the appropriate direction; c) the zeroth-order fringes are easily identified as they do not

change when changing the position of the filtering hole. This allow an easy assignment of other fringes' order, apart from the sign that can only be deduced from flow visu- alization.

420

The main drawbacks that until now have prevented this technique for being used to get accurate quantitative results are: lO

a) the intrinsic width of the fringes: since the intensity on the ~- image plane is a sinusoidal function, fringes will appear bright in a certain region around the maximum as far as the N intensity is above a certain value. Thus, fringes will be 5 broader for photographs with smaller number of exposures. Thus, the quality of the isovelocity fringes is very improved with a multiple exposure technique since multiple beam

a 0 interferences cause a noticeable sharpening and increase in brightness of the isovelocity fringes,

b) the size of the speckle affecting the fringes that produces 10 discontinuities, making more difficult the location of the fringe maxima. E

g When these filtered images are compared with inter- N

ferograms obtained from solid mechanic problems (Yu 1988; 5 Yatagai et al. 1982), it can be seen that they have similar features, the size of the speckle being the only significant difference. Quantitative information, namely fringe center- lines, is systematically retrieved in a fast and automatic way 0 from the interferograms. As we will show in the next section, b the same type of information can be retrieved from the filtered images using similar algorithms.

3 Quantitative processing of the PIV photographs The quantitative processing of the PIV photographs is based on the analysis of the fringe patterns generated by the whole-field filtering technique. The fringe analyzing procedure includes fringe peak detection and fringe order determination. This procedure is automatically done with a standard image processing system based on a PC486 computer using the appropriate software. The filtered images are read with a CCD camera. The video signal from the CCD camera is digitized with a frame grabber from Data Translation (DT-2583) giving digital picture of 512 x 512 pixels with 256 gray levels, which is stored on the PC to be analyzed.

The procedure to detect fringe peaks involves a first step where noise in the fringe pattern is reduced. This is done by using a digital low pass filter that reduces the high-frequency speckle noise. This filter consists simply on replacing the gray level at each point (x, z) by the average of the gray levels on a neighborhood of (x, z). The optimum size of this neighborhood depends on the fringe spacing and noise characteristics. It has to be bigger than the speckle size for the filtering to be effective but much smaller than the fringe spacing for the filtering to produce an almost negligible fringe broadening. Practically, 7 x 7 or 9 x 9 are convenient sizes of neighborhood. Further reduction of the noise is achieved with a spin filter (Yu 1988). The spin filter is a directional filter that consists on replacing the gray level at a certain point (x, z) by the average of the gray levels of n points along a line locally tangent to the fringe. The spin filter has the advantage of filtering out the noise without blurring the fringe pattern which allows to use bigger sizes of neighborhood. In practice an average of 19 points is used. However, due to the high frequency of the speckle noise, the spin filter does not work properly if the average filter is not used previously.

' ' ' ' I ' ' ' ' I ' J

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' ' ' ' I " ' I ' ' ' ' I ' ' I ' ' ' '

0 5 10 x (mm) 15 20 25

Fig. 3a, b. Comparison between fringe maxima positions (dots) inferred from the filtered images shown in Fig. 2 and isovelocity lines (solid lines) obtained from a point-by-point analysis. Vx - m 25.0 pm/s for each line in a and V z = m 34.0 pm/s for each line in b, being m the number placed near each line

The next step is to extract fringe centerline positions, i.e., fringe skeletons. This is done by comparing the gray level at each point (x, z) with the gray levels at the nearest points (Yatagai et al. 1982). This comparison is done along two directions: horizontal and vertical. The point (x, z) is con- sidered to be a maximum (minimum) if its gray level is higher (lower) than the gray level of the three or five neighbors in any of the directions. To reduce false fringe detection, due to the residual noise on the fringe pattern, the following criteria have been used. a) the maxima and minima should be alternating in space; b) the maxima (minima) should have an intensityvalue higher

(lower) than a prefixed value; c) the distance between successive maxima or minima should

be larger than a prefixed value, related to the fringe period. The fringe skeletons detected in the former step are very

often too wide to extract fringe centerline positions. Thus, the next step is the thinning of the skeleton. This is done by selecting, from all the maxima obtained in a direction locally perpendicular to the fringes, the point with the highest gray level.

The detection of maxima and minima has been done by rows or columns, assigning an order related to its position on the row or the column. The next step is the correct renumbering of the detected fringes. This is done automatically by comparing the position of the maxima from different rows or columns. Continuity and closeness criteria are used to determine all the maxima points corresponding to the same order fringe. A final check of the correctness of the renumbering of the fringes is done interactively looking at the display on the computer

screen of the fringe maxima for each order. In this step, the actual value of the order is set for each fringe, too.

This is the end of the fringe analyzing procedure. At this stage there are in the computer memory a set of data triplets, each giving the x, z coordinates of a fringe maximum and its corresponding velocity component value (inferred from the order of the fringe). The task remaining is to interpolate that data to obtain the velocity value on a grid of regularly spaced points.

An example of the output of the fringe analyzing procedure is shown in Fig. 3, corresponding to the filtered images shown in Fig. 2. The dots represent the points obtained as fringe maxima (fringe skeletons). The solid lines are the isolines obtained from pointwise measurements of the photograph. Fringe maxima agree well with all the isolines, except that corresponding to Vz = 102 pm/s. It can be seen that this fringe is poorly resolved in the filtered image (Fig. 2c). Since this fringe corresponds to a Vz value quite close to the absolute maximum velocity, Vz changes slowly in that region of the flow, not reaching the velocity that will produce the next minimum in intensity. This yields a broad and poorly resolved fringe. Thus, data obtained from fringes like this particular one i.e., from poorly resolved fringes corresponding to values near the maximum velocity, should be discarded.

4 Error analysis There are several sources of error in the measurements, which are related either to the technique itself or to the fringe analyzing procedure. The main source of error related to the technique is due to the uncertainty in the determination of the hole distance, 6p. This is mainly related to the accuracy in determining the origin of the filtering plane but also to the accuracy of the system used to displace the filtering hole. In our system, 6p~0.1 ram. From Eq. (11) and considering that the errors in the other parameters are negligible, we obtain that

6 r e _ 6 p

ve P (13)

The main source of error in the fringe analyzing procedure is due to the uncertainty in the determination of the fringe maxima. As seen in Fig. 2, the fringes are not continuous due to a noncontinuous distribution of particles in the fluid. Besides, not all the particles have the same brightness in the photo- graph. This means that the fringe centerlines in the filtered image may be biased towards brighter points of the fluid. This effect is directly proportional to the width of the fringes, meaning by width the distance between two points with the same intensity. To quantify this effect, lets assume that there could be a 20% variation in brightness on the photograph. This implies that we can detect as fringe maxima points where F(ri) is within 20% of its maximum. As we have already seen (Eqs. (8)-(11)) F(ri) takes its maximum value, N 2, on the points where

)Lf~ (14) vp = m "cMp

0.15

~- 0.10- 0

t~ 0.05-

0 ' ' ' ' [ ' ' ' ' I ' ' ' ' I ' ' ' ' 0 5 10 15 20

Number of exposures, N

F i g . 4 . D e p e n d e n c e o f t h e e r r o r f a c t o r , k , o n t h e n u m b e r o f e x p o s u r e s

o n t h e p h o t o g r a p h , N

In the same way, it can be deduced from Eq. (8) that F(ri) =0.8 N 2 o c c u r s in the points where

v'p= [m + k (N) ] --2 f' (15) tMp

where k(N) is a factor that depends only on the number of exposures taken in the photograph. It has been calculated for each N from N = 2 to N = 2 0 (Fig. 4). Thus, due to a 20% variation in the brightness of the particle images, the fringe centerlines may correspond to any velocity value between vp and v~. This implies an uncertainty, 6vp, that can be expressed as

&~=v'e- v~=k(N) - - (16) tMp

which leads to

~v; k(N) - - - - - - ( 1 7 ) ( V p ) m a x m m a x

where mmax is the order of the fringe corresponding to the maximum velocity. It can be seen that the uncertainty is directly proportional to the factor k(N), which decreases with N. Thus, a multiple exposure technique not only improves the quality of the filtered images but also considerably reduces the error in their analysis.

Typical errors for our data are 6vv/vp=O.020 and 0.028 for Figs. 2b and c respectively, according to Eq. (13), and r according to Eq. (17). This gives a total error of about 3 % of the maximum velocity.

5

V e l o c i t y f i e l d p r o c e s s i n g

When measuring the velocity field, it is convenient to have the velocity data regularly distributed over the flow field. Thus, an interpolation of the data over a regular grid is needed. This interpolation is done in the same way as for the data measured with a point-by point analysis (Arroyo et al. 1988a, b). In summary, the data are interpolated over a grid with 26 by 14 points (including the cell walls, where the velocity is assumed to be zero). The interpolation routine fits, by triangulation, a smooth surface to the velocity mesurements. The fit is applied, independently, to Vx and Vz. However, the data to be fit are more irregularly spaced than in the pointwise measure- ments. The fringe analysis procedure produces many data for

4 2 1

422

the same velocity value with little spacing between them (Fig. 3). Thus a sampling of the data before the interpolation is needed. In practice, about 300 points per fringe have been t0 obtained. Only one every 0.75 mm has been retained for interpolation, which means about 30 points per fringe. On the other hand, the spacing between fringes is quite large. Thus, we "~ have found it necessary to take data from several filtered 5 images corresponding to different vp. In practice, two or three images are used although not all the fringes are taken from each image. The zero-velocity line data are taken from the filtered image with narrower fringes (corresponding to larger a 0 p).

Finally, other information such as flow rate, vorticity, and to streamlines can be inferred from the interpolated velocity data to provide a better knowledge of the flow characteristics. This is done in exactly the same way as with the data obtained with ~ the point-by-point analysis (Arroyo et al. 1988a, b). Thus, after ~ 5 a bicubic spline is fitted to G and V~ independently, the 2-D velocity field is processed to obtain the velocity vector map, the iso-G and iso-V~ maps, the isovorticity (y-component) map and the 2-D streamlines map.

6 Results and discussion The whole-field filtering technique has been applied to some photographs taken from a Rayleigh-B~nard convective flow (Arroyo and Savirdn 1992). For comparison, the same photographs have also been analyzed with a point-by-point technique based on a 1-D averaging Young's fringe method (Arroyo et al. 1988a, b). Both analyses have been done using a compatible PC486 computer. The photograph shown in Fig. 2a will be used to illustrate the full process.

Five filtered images have been analyzed. Three of them correspond to iso-V~ contours, where the increments between consecutive lines are AVe=25.0 gm/s for image 1, 33.0 gm/s for image 2 and 42.2 pm/s for image 3. The other two images correspond to iso-V~ contours with A V~ = 34.0 pm/s for image 1 and 47.0 gm/s for image 2. Figure 5a shows the 253 G-data used for the interpolation. The Vx = 0 lam/s data have been taken from image 1 and only the highest order fringes (m = + 2) have been taken from image 3. Figure 5b shows the

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x �9 �9 �9 �9 0 A 0 [ ]

Vx(~tmls ) 0.0 25.0 42.2 50.0 66.0 --25.0 --42.2 --50.0 -66.0 Vz(pm/s) 0.0 34.0 47.0 68.0 94.0 --34.0 --47.0

187 Vz-data used for the interpolation. The V~=0 gm/s data have been taken from image 1, and the highest order fringe (m = 3) has not been used. The interrogation of the photograph on a 1 x 1 mm grid (24 • 12 points) with the point-by-point technique has given velocity measurements in 214 points. The analysis time is similar with both techniques.

A comparison between the interpolated velocity and vorticity fields obtained from the whole-field filtering techique and the corresponding fields obtained from the point-by-point technique is presented in Figs. 6 and 7.

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Fig. 7a-l . Systematic processing of the photograph shown in Fig. 2a using the whole-field filtering technique (images on the left) or the point -by-point technique (image on the right), a Velocity vector maps; b iso- Vx maps. V x = rn 12.5 jam/s for each line, rn being the n u m b e r placed near each line; c iso-V, maps. Vz= m 12.5 i~m/s for each line, m being the n u m b e r placed near each line; d isovorticity (y-component) maps. COy=m 12.5 x 10-3s -1 for each line, being m the n u m b e r placed near each line; e 2-D streamline maps

423

424

Figure 6 shows the data on the interpolation grid points obtained from one technique plotted vs the corresponding data obtained from the other technique. The r.m.s, deviations of the data points from the corresponding solid line are 2.9 ~tm/s for Vx (Fig. 6a), 3.0 ~am/s for V z (Fig. 6b) and 3.5 x 10 -3 s -~ for coy (Fig. 6c). This correspond to average differences of 2.7% and 5.3% full scale for the velocity and vorticity data respectively.

Figure 7 presents a 2-D comparison of the velocity and vorticity fields obtained with both techniques. From this figure we observe an overall agreement between the two sets of measurements. The velocity vector maps (Fig. 7a) show very clearly the spatial features of the flow. However, they are not very sensitive and will only show big differences in the velocity fields. The iso-Vx maps (Fig. 7b) and the iso-Vz maps (Fig. 7c) are a better way of comparing the velocity fields. The high similarity between the maps on the left and the maps on the right is a consequence of the good agreement between the two sets of data. The isovorticity maps (Fig. 7d) should be more sensitive to differences in the velocity fields due to its derivative characteristics. A very good agreement is found again between the map on the left and the map on the right. However, the isolines obtained from the point-by-point measurements look smoother than the isolines obtained with the whole-field filtering technique. This reflects the different accuracy of the two techniques. Finally, the flow features presenting the highest sensitivity to velocity changes are the 2-D streamlines (Fig. 7e). They are even more sensitive to systematic errors which will accumulate on the integration procedure necessary to calculate the 2-D streamlines. For example, we have analyzed the effect of a ~ 2 misalignment on the horizontal or vertical velocity component selected in the whole field filtering technique. This causes the streamline to drastically change its spiraling pitch (it increases in one of the streamlines and decreases in the other) while leaving almost unchanged the other 2-D maps.

Finally, we note that the whole-field filtering processing is more suitable for photographs with a high number of exposures and a high image particle concentration, which will produce narrow and continuous fringes. The image particle concentration increases with the fluid seeding concentration but also with the number of exposures. It is also convenient to have zeroth order fringes on the filtered images to be able of determining the order of the fringes. Thus, the recorded image particle displacement field should go from negative to positive values.

The spatial resolution of the technique is related to the minimum fringe spacing that can be resolved. As deduced from Eq. (12), this spacing should be somewhat bigger than the maximum displacement on the photograph, when measured in the same coordinates as on the photograph. In practice, we have observed a lower limit of about 1 mm for the spacing of well resolved fringes. This would be the spatial resolution of the technique if only one filtered image is used, However, when three images per component are used, the spatial resolution would be roughly three times better, meaning a resolution of 300 ~m. This means that the data obtained in the reduction of a photograph will be that close in the regions of fast changes in the velocity but will be more spaced in areas of slow velocity

changes (see Fig. 5). Since the spatial resolution of the whole-field technique is similar to that of the pointwise techniques, the authors think that both techniques would perform similarly in flows with large velocity gradients. In flows with a non-zero mean velocity, the whole-field would perform better if an image shifting technique (Adrian 1986b) is used in the recording of the photograph to subtract such a mean velocity.

One of the advantages of the whole-field filtering technique is that the filtering hole, instead of the photograph, is being displaced. Since only a very few displacements, which are usually small (less than about 10 mm), are needed to analyze the photograph, they can be done manually. This makes unnecessary the use of expensive computer controlled displacement systems, accurate over long distances as used in the point-by-point techniques. Another advantage is that the processing of each photograph requires the analysis of only a few filtered images, instead of hundreds of Young's fringe images. If human intervention is required for positioning the photograph to be analyzed, the processing of several photographs cannot be fully automatic with a pointwise method. However, the filtered images corresponding to many photographs can be stored in the computer and analyzed afterwards without any human intervention in the middle of the process. Both methods also require human intervention at the end of the process either for the assignment of fringe orders (in the whole field technique) or for the assignment of the velocity sense (in the point-by-point technique).

7 Conclusions A fast quantitative processing of particle image velocimetry photographs by a whole-field filtering technique has been described. Fringe patterned images have been obtained from the observation of the photographs through a conventional filtering setup and acquired with a CCD camera. Some digital image processing techniques previously used for inter- ferometric patterns have been adapted to automatically determine the fringe centerlines. Fringe centerlines correspond to a value of one velocity component fixed for each fringe. By keeping only one every 0.75 mm of the fringe centerline points determined from several filtered images, two velocity com- ponents are determined at randomly but conveniently spaced points. An interpolation of these data gives the velocity field over a regular grid of points.

In performance the full scale accuracy of the technique is estimated to be 3%. In the processing of photographs taken from a Rayleigh B6nard convective flow, the technique provides quantitative measurements in very good agreement with results obtained from a digital/optical point-by-point analysis technique and with a similar processing time. The whole-field filtering technique is a fast and simple alternative to the point-by-point techniques for the processing of photographs with a high number of exposures, a high image particle concentration and image particle displacements going from negative to positive values. In these conditions, the fringe patterned images should have the good quality necessary for obtaining measurements with similar spatial resolution as with the pointwise techniques.

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