8

Automatic spatial characterization of low-speed streaks

from thermal images

M. Zacksenhouse, G. Abramovich, G. Hetsroni

Abstract Flow visualization is an important tool for in-vestigating turbulent ¯ow, and, speci®cally, for character-izing low-speed streaks in the boundary layer. The span-wise spatial characteristics of these streaks are commonly extracted by human visual inspection, which is time con-suming and subject to human errors and biases. Attempts to develop automatic methods have relied exclusively on spectral techniques, using mostly the autocorrelation or its Fourier transform, the spatial spectrum. However, the autocorrelation tends to get ¯attened with the amount of data analyzed and has been reported to provide biased estimates. Furthermore, it estimates only the mean spacing and does not provide a direct measure of its distribution. In this paper, an alternative automatic method is devel-oped based on edge detection, and is applied to thermal images obtained by infrared thermography of a heated wall exposed to a turbulent ¯ow. The method presented yields not only the spacing between the low-speed streaks but also their width and separation. The analysis indicates that the spacing (120 ‹ 52 wall units) is divided almost evenly between the width (65 ‹ 33 wall units) and the separation (55 ‹ 40 wall units) between the streaks, and that the width and separation are statistically independent. We also present a statistical model for the data, and demonstrate that when the spatial parameters of the streaks are so widely distributed, the spectral methods are not reliable. List of symbols

a,q parameters of the gamma density function k span-wise streak spacingkmean streak spacing k‡ _{mean streak spacing expressed in wall units} j average density of a point process

l mean distance m kinematic viscosity w coef®cient of variation r standard deviation

s lag of autocorrelation function

h ¯uid depth

h(t) impulse response of a shot noise process

H(0) integral of impulse response

p distance in pixels

R(s) autocorrelation function

Reh Reynolds number

u mean ¯uid velocity

u _{wall shear velocity}

1

Introduction

Low-speed streaks are regarded as a universal characteristic of turbulent boundary-layer ¯ow. The low-speed streaks appear very close to the wall, and are considered to be the trails of stream-wise vortices that pass close to the wall (see Smith 1996). The turbulent nature of the ¯ow is related to the transport of ¯uid from the low-speed streaks away from the boundary in a process termed ``bursting'' by Kim et al. (1971). During this process, some of the ¯uid from the low-speed regions is ejected into the main stream, and some ¯uid from the outer region is swept toward the wall. Thus, the characteristics of the low-speed streaks are important in investigating turbulence and in characterizing surface-transport processes such as heat transfer.

A variety of ¯ow visualization techniques have been developed for imaging the low-speed streaks as recently reviewed by Johnston and Flack (1996). Key techniques include laser-Doppler anemometry (LDA), visualization techniques based on bubbles or passive particle tracers, particle-image velocimetry (PIV), and infrared (IR) ther-mography. Most studies focused on extracting the span-wise spacingkbetween the low-speed streaks. A summary of such studies is presented in Table 1, with emphasis on recent works. The estimated mean span-wise spacingk is normalized with respect to the viscous length and given in wall unitsk‡ˆku_=m.

Streak spacing provides a new tool for investigating turbulence and answering a variety of questions. Kline et al. (1967), Achia and Thompson (1977), Oldaker and Tiederman (1977), Nakagawa and Nezu (1981), Smith and Metzler (1983), and Hetsroni and Rozenblit (1994) inves-tigated the effect of Reynolds number on the mean spac-ing. These studies support the conclusion that at least in low Reynolds numbers the non-dimensional mean streak spacingk‡ is invariant. In particular, Jimenez and Moin (1991) and Hamilton et al. (1995) suggested that a char-acteristic spacing of 100 ‹ 20 wall units is a critical value for sustained turbulence. Oldaker and Tiedreman (1977) and Hetsroni et al. (1997) used the mean spacing as a tool

Experiments in Fluids 31 (2001) 229±239ÓSpringer-Verlag 2001

Received: 19 December 2000/Accepted: 26 January 2001

M. Zacksenhouse (&), G. Abramovich, G. Hetsroni Faculty of Mechanical Engineering

Technion ± Israel Institute of Technology, Haifa, Israel e-mail: [email protected]

This research was supported by the Basic Research foundation administered by the Israeli Academy of Science and Humanities, and by the Fund for the Promotion of Research at the Technion. This research was also supported by the Israeli Ministry of Science.

to investigate the effect of drag-reducing solutions on the streaky structure, while Grass and Mansour-Tehrani (1996) used it to investigate the effect of surface roughness.

The answers to the above questions rely not only on the visualization method but also on the method for extracting the streak spacing from the measured images. There are two prevailing methods: (1) human visual inspection and (2) spectral techniques based on the autocorrelation or its Fourier transform, the spatial spectrum. Visual inspection may be performed directly on the video screen, as in Kaftori et al. (1994), or on a sequence of still images, which has been extracted from a high-speed video, as in Smith and Metzler (1983). Smith and Metzler (1983) established a systematic visual inspection method based on a number of examiners (three in their case) that sep-arately mark the location of the streaks and later resolve their discrepancies. A set of rules has been developed to guide the examiners in localizing the streaks in pictures taken by hydrogen bubble-wire ¯ow visualization. Gener-ally, a well-de®ned stream-wise concentration of bubbles that extend far enough had to be present, and the location of the low-speed streak was marked at its center some distance from the bubble wire. An additional criterion was developed to determine the eligibility of weak or poorly developed streaks. The distances between the marked streak centers were measured and used to generate the corresponding histogram. Table 1 indicates that even re-cent investigations rely heavily on human visual inspec-tion for estimating streak spacing.

Efforts to automatically estimate the streak spacing have relied exclusively on spectral methods. The auto-correlation method estimates the characteristic streak spacing by the lag of the ®rst peak in the averaged auto-correlation. Alternatively, the characteristic spatial fre-quency can be determined by the peak frefre-quency of the spatial spectrum, which is the Fourier transform of the autocorrelation. However, Fortuna and Hanratty (1972) and Gupta et al. (1971) reported that as the amount of data (i.e., the number of data frames) increases, the averaged autocorrelation tends to get ¯attened, making the detec-tion of the characteristic spacing, or periodicity, unreli-able. To overcome this problem, Achia and Thompson (1977) developed an ad-hoc method by which the lag of the ®rst peak in each individual autocorrelation function is extracted and averaged. In the present paper, we suggest that the observed smearing be due to the width of the

underlying spatial distributions and demonstrate that in this case the autocorrelation-based methods are poor estimators.

Another important drawback of the autocorrelation-based method is that it does not provide an estimate of the distribution of the streak spacing, only of the mean. While the shape of the ®rst peak of the autocorrelation is related to the probability density function of the spacing, other aspects also affect it, as described in Sect. 5. Thus, the autocorrelation method does not provide a full character-ization of the streak spacing. Visual inspection methods, in contrast, can yield the histogram of the spacing, which, when normalized to a unit area, provides an estimate of its probability density function. In particular, Oldaker and Tiederman (1977) and Smith and Metzler (1983) reported that the streak spacing is distributed with a coef®cient of variation (the ratio of the standard deviation to the mean, see Appendix) in the range of 0.3±0.4. Indeed, it will be argued in this paper that the relatively wide distribution of the spacing impedes the autocorrelation-based methods. We present an alternative automatic method to estimate the distribution of the spacing in such a way that the other span-wise spatial parameters (i.e., the width and separa-tion) are also extracted and characterized.

Our work is based on IR thermometry of a heated foil inserted in the wall of the ¯ow, a method that has been developed and investigated extensively by Hetsroni and Rozenblit (1994) and Hetsroni et al. (1997). The local ve-locity of the ¯ow affects the heat convection and thus the temperature of the thin foil, which is imaged on the dry side. The resulting two-dimensional image of the tem-perature ®eld provides indirect visualization of the veloc-ity ®eld at the wall. The thermal images (see Fig. 1 for an example) are characterized by bright streaks correspond-ing to the high-temperature streaks separated by dark regions corresponding to the low-temperature streaks. Iritani et al. (1983) con®rmed that the high-temperature streaks are mostly associated with the low-speed streaks and low-temperature streaks with high-speed streaks.

We focus on the span-wise spatial characteristics of the bright streaks, including their width, separation, and spacing. The streaks are localized by detecting their stream-wise edges, and their width, separation and spacing are measured as the span-wise distances between the corresponding edges. The resulting histograms provide an estimate of the probability density functions of the cor-Table 1. Visualization and feature extraction methods and the resultingk‡ _estimates

Reference Year Visualization Extraction k‡ _value

Kline et al. 1967 Bubbles Inspection 100

Achia and Thompson 1977 Hologram-interferometry Spectral 79±93

Nakagawa and Nezu 1981 Anemometry Conditional averaging 100±103

Smith and Metzler 1983 H2bubbles Inspection 87±104

Lian 1990 H2bubbles Inspection 104±107; 68±69

Rashidi and Banerjee 1990 O2bubbles Inspection 100 ‹ 7

Hetsroni and Rozenblit 1994 Infrared thermography Inspection 79±98

Kaftori et al. 1994 LDA+O2bubbles Inspection 100

Sabatino 1997 PIV and TLC Spectral 68±120

Itoh et al. 1997 H2bubbles ± incl. surfactants Inspection 114±194

responding spatial characteristics and thereby a full characterization of the span-wise spatial distribution of the streaks. A texture generation model is developed for the thermal images and used to evaluate the autocorrelation-based methods. It is shown that when the underlying probability densities are wide, as in the thermal images studied here, the spectral methods are not reliable. 2

Methods and experiments 2.1

Experimental facility

The experiments were performed in a ¯ume similar to the one described by Hetsroni and Rozenblit (1994). The re-circulating test rig also includes a tank and a centrifugal pump. The ¯ume is 4.5 m long, 45 cm wide and the water is 3.5 cm deep (accuracy of 1 mm). The experimental re-gion is located 2.5 m downstream from the inlet to the ¯ume where a fully developed ¯ow is established. The experimental region consists of a 50-lm-thick constantan ¯oor heated by a direct current source and is imaged on the dry side by an IR camera.

2.2

Imaging technique

Hetsroni and Rozenblit (1994) demonstrated that the foil heater is thin enough so that the temperature measured on the outside re¯ects well the temperature of the heater ex-posed to the ¯ow, as long as the frequency of the tem-perature ¯uctuations is below 15 Hz. Thus, the time constant of the measurements is adequate for the present study, where the mean time interval between bursts is around 0.5 s. Heating is suf®cient to create the necessary thermal distribution for imaging, with wall-temperature variations of up to 2°C, but has been shown to have a negligible effect on the ¯ow. A sample thermal image is shown in Fig. 1. The low temperature (dark) regions are traces of the high velocity regions, where the forced heat convection is high and the temperature is low. The bright streaks are footprints of the low-speed streaks.

The detector of the IR camera is cooled in a closed circuit to 77 K to maximize the temperature sensitivity

and spatial resolution. In gray level mode the resolution is 256 gray levels and the accuracy is 0.1°C. Image scanning is performed at 4 kHz horizontally and at 50 kHz verti-cally. Image processing is carried on a region of interest of 185´ 250 pixel covering an area of 10.4´ 13.3 cm in the ¯ume.

2.3

Flow conditions

A constant ¯ow rate was maintained by the pump and measured by ori®ces connected to manometers. The mean velocity wasuˆ0:25 with an accuracy of 2%. The Rey-nolds number in the ¯ume is based on the depth hof the water and is given byRehˆuh=m(m, the kinematic vis-cosity, is 10)6_m2_{/s at the mean water temperature of} 20°C). Under the above conditions, the Reynolds number is 8,750, and the wall shear velocity, given by

u_ˆ0:167uRe 0:125

h (Hetsroni and Rozenblit 1994), evalu-ates tou_{ˆ0.0134 m/s. The spatial dimensionkmeasured} in meters can be converted to wall units according to: k‡ˆku_{=mˆ13;425k. In terms of image pixels, the} con-version to wall units isk‡ˆ7:55pwherepis the span-wise (vertical) dimension in image pixels. Given the accuracy of the ¯ow rate (2%), water depth (0.3%) and viscosity (1%), the accuracy of the conversion constant is 2.25%.

3

Spatial characterization methods

The thermal images have a characteristic texture of elon-gated stream-wise streaks (the streaks are aligned parallel to the direction of the ¯ow, i.e., horizontal in the present images), as evident in the image shown in Fig. 1. The span-wise spatial characteristics of the streaks are measured perpendicular to the main axis of the ¯ow (vertically in the present images). Our approach is to detect the edges of the bright streaks and measure the pixel-wise distance between corresponding edges along selected columns. The edges are marked as positive or negative edges according to the direction of the local gradient, and thus can be paired to determine the width, separation or spacing of the streaks. The histograms of the measured spatial characteristics from selected columns of different images provide esti-mates of the corresponding probability density functions. 3.1

Image processing and edge detection

Edges are localized on the basis of gray level discontinu-ities, which are detected using either the ®rst or second derivative operator. The local ®rst derivative operator should reach a local maximum at the edge and thus the second derivative operator should change sign and cross the zero. Different edge-detection methods have been de-veloped based on either the magnitude of the ®rst deriv-ative operator, or the zero crossing of the second derivative operator (see for example, Jain 1989).

Gradient-based methods apply a threshold to the magnitude of the local ®rst derivative to distinguish be-tween sharp, and presumably meaningful, transitions in the gray level from weak, and presumably meaningless, transitions. The level of the threshold should satisfy two contradicting requirements: it should facilitate detection of Fig. 1. Thermal image (185´250 pixels, 10.4´13.3 cm)

show-ing high-velocity temperature dark regions separated by low-velocity high-temperature bright streaks. The ¯ow direction is horizontal, from left to right

existing edges while avoiding detection of noisy edges. Zero-crossing methods by-pass the need to apply a threshold but introduce other problems associated with the high sensitivity of the second derivative operator. Canny (1986) developed a gradient-based method that improves the trade-off between detection and noise-re-jection and includes hysteresis thresholding. We adopt this method and tailor it to the speci®c texture of the thermal images, as described below.

Hysteresis thresholding involves two threshold levels, a high threshold level and a low threshold level, whose se-lection is discussed below. As in other gradient-based methods, pixels at which the magnitude of the ®rst de-rivative is higher than the high threshold level are marked as strong edge-pixels and assigned to the edge-image (as shown in Fig. 2, top panel, for the image of Fig. 1). Pixels at which the magnitude of the ®rst derivative is higher than the low threshold level are marked as weak edge-pixels. Hysteresis thresholding refers to the method of assigning weak edge-pixels that are connected to strong edge-pixels, either directly or via other weak edges, to the ®nal edge-image (as shown in Fig. 2, bottom panel, for the image of Fig. 1).

Alternatively, edges may be determined by the zero-crossing method as the pixels at which the second gradient crosses zero. The resulting edge-image for the image of

Fig. 1 is shown in Fig. 3, top panel, where the sensitivity to noise is clearly evident. We adopt a hybrid method by which edges should pass both the hysteresis thresholding of the ®rst gradient and the zero crossing of the second derivative. The resulting edge image is shown in Fig. 3, bottom panel.

The basic algorithm presented above is augmented by pre-processing and post-processing stages. The pre-pro-cessing stage involves standard contrast enhancement using histogram equalization, described for example in Gonzales and Wintz (1987). The post-processing stage involves standard and special morphological operators to enhance the image of the edges. The standard morpho-logical operators include cleaning of short edges and thinning of double edges. The specially designed mor-phological operator is tailored at linking horizontal edges to bridge over short sections connecting two well-aligned edge-segments. The effect of the post-processing stage is demonstrated in Fig. 4 showing the ®nal edges for the image of Fig. 1.

The high threshold is selected to guarantee a desired percentage of edge-pixels at the initial edge image (those passing the high threshold). The low threshold is set to a third of the value of the high threshold. It is therefore necessary to determine a single parameter, the desired percentage of edge-pixels at the initial edge image. This parameter is application speci®c and was determined

Fig. 2. Hysteresis thresholding. Initial edge-image with strong edges (top) covering 15% of the image. Edge-image passing hysteresis thresholding (bottom) include also weak edges that are connected to strong edges (either directly or via other weak edges)

Fig. 3. Zero crossing. Edges based on zero crossing of the second derivative (top) and edges based on both zero crossing and hysteresis thresholding (bottom)

according to two rules: (1) the percentage of initial edges should be set low to ensure good noise-rejection, and (2) the ®nal results should be insensitive to the desired percentage of initial edges. Sensitivity analysis was performed by applying the complete procedure and re-cording the resulting mean spacing at different percentage levels. The results, shown in Sect. 4, motivate the selection of the high threshold level so that the initial strong edges cover 15% of the image.

3.2

Spatial parameters estimation

Given the elongated nature of the streaks, there is a high level of correlation between the span-wise temperature signals along adjacent columns. To ensure that different sample measurements are independent, a single column is selected from each image and the width and spacing of the bright streaks along that column are determined as de-tailed below. Furthermore, the different images considered in the present study were extracted every 0.5 s to avoid temporal correlation given the estimated 0.5 s lifetime of these structures, while providing a large number of sample measurements.

The detected edges are marked as either positive or negative edges, depending on the direction of the local gradient. Given the alternating nature of the bright streaks and dark regions, it is expected that positive and negative-marked edges would appear alternately along columns of the edge-image. However, given the rich texture of the thermal images, and acknowledging the imperfections of the edge detection algorithm, this order may not be maintained along each column. Two approaches may be considered: (1) selecting only valid columns, or (2) se-lecting a pre-determined column and ignoring interme-diate edges. Abramovich (1998) applied the ®rst approach by scanning the columns of each image to identify valid columns with alternating positive and negative-marked edges, and selecting a single valid column for further analysis. Here we apply the second approach by selecting a pre-determined column and ignoring any edge that cannot be paired properly. Focusing on bright streaks, positive edges that are immediately followed by negative edges are paired together as the upper and lower borders of a bright streak. Other edges (i.e., positive edges followed by posi-tive edges or negaposi-tive edges following a negaposi-tive edge) are

ignored. Most of the results presented in this paper are based on the second approach, while results based on the ®rst approach are stated for comparison.

The distances between the alternating positive (nega-tive) and negative (posi(nega-tive) edge-pairs are recorded and stored as samples of the width (separation) of the bright streaks. Similarly, the distances between adjacent positive edges, which are separated by a single negative edge, are recorded and stored as samples of the spacing between bright streaks. Finally, all the samples taken from the 78 images are considered in constructing separate histograms for the width, separation and spacing and in computing the corresponding statistics. Each histogram is normalized to a unit area to provide an estimate of the corresponding probability density function. The resulting distributions are characterized by computing (see Appendix) the mean l, standard deviationr, coef®cient of variation wˆr=l, skewness and ¯atness.

Two standard families of probability density functions are considered for ®tting the measured histograms: the lognormal (where the logarithm is normally distributed) and the gamma distributions. Nakagawa and Nezu (1981) and Smith and Metzler (1983) suggested that the streak spacing be distributed according to a lognormal proba-bility density function. We also consider the gamma probability density function since it facilitates analytical investigation in the context of the texture generation model developed in Sect. 5. In each case, the parameters of the analytic distribution (see Appendix) are computed to yield the same mean and standard deviation as the esti-mated probability density function.

3.3

Conditional statistics

Given the edge images, it is possible to explore different relations between the spatial and thermal features of the streaks. In the present paper, we investigate the relation between the width of the bright streaks and the separation between them. The mean separation on either side of streaks of a given width is computed as a function of the width of the center streak. The slope of the resulting conditional mean function provides a measure of the span-wise serial dependence between the separation and the width of the streaks. In particular, a ¯at conditional mean function would indicate that there is no serial correlation between the width and separation of the streaks.

3.4

Spectral methods

Spectral methods may be used to estimate the periodicity of a signal. The characteristic period may be estimated as the lag of the ®rst peak in the average autocorrelation. Alternatively, the peak frequency in the power spectrum density (PSD), provided by the Fourier transform of the averaged autocorrelation, estimates the characteristic spatial frequency.

However, the above features of the autocorrelation and PSD are related to the periodicity only when the under-lying process is highly periodic. In contrast, when the underlying events (centers or edges of streaks, in our case) are irregularly spaced, these features are not related to the Fig. 4. Detected edges overlaid on the original thermal image

characteristic spacing. Consider, for example, a shot noise process (see for example, Papoulis 1984), which is the response of a linear system [with an impulse responseh(t)] to a train of Poisson impulses. In this case, the autocor-relationR(s) is given by:R…s† ˆj2_H2_{0† ‡jh…s† h… s†,} wherej is the average density (the reciprocal of the av-erage spacing), and H(0) is the integral of the impulse response (its average component). The above expression for the autocorrelation indicates that the location of its ®rst peak is related to the impulse response and not to the average spacing between the impulses.

A similar argument was made by Nakagawa and Nezu (1981) who showed that with a coef®cient of variation of wˆ0.5, the curve of the conditional mean velocity does not show any periodicity. However, there they assumed that the instantaneous pattern of velocity is harmonic. Here, we develop a variant of the shot noise model, which is better tailored to model the span-wise temperature pro®les of the thermal images. The model is developed in Sect. 5 and used to further evaluate the spectral estimators and the proposed image processing method.

4

Spatial characterization results 4.1

Edge detection

The application of the proposed algorithm to the thermal image shown in Fig. 1 is demonstrated in Fig. 4. The temperature pro®le and detected edges along a speci®c column (column 125) are shown in Fig. 5. Most edges can be paired as boundaries of streaks and are marked by solid lines. Only a single edge, marked by a dotted line, cannot be paired appropriately and is ignored for further statis-tical analysis. The width of the former streak is measured up to the preceding negatively sloped edge.

4.2

Spatial distributions

The histograms of the span-wise spatial parameters, shown in Fig. 6, are very well ®tted by lognormal density func-tions, and, albeit to a lesser degree, by gamma density functions (matched to have the same mean and standard deviation). The corresponding statistical parameters are speci®ed in Table 2. The mean spacing is 119.8 wall units and is divided almost evenly between the width of the streaks (65.1 wall units) and the separation between the streaks (54.8 wall units).

All the spatial parameters are widely distributed with coef®cients of variation of 0.51, 0.73 and 0.44 for the width, separation and spacing, respectively (Table 2). For the streak spacing, the coef®cient of variation is at the upper range of 0.3±0.4 reported in the literature by Oldaker and Tiederman (1977) and Smith and Metzler (1983) However, the streak width and streak separation are distributed with

Fig. 5. Edges detected along a cross-section (column 125) of the thermal image of Fig. 1. Most edges (marked assolid lines) are paired as the boundaries of streaks. The unpaired edge marked by adotted lineis ignored

Fig. 6. Scaled histograms of the span-wise spatial parameters of the streaks: width (top), separation (middle) and spacing (bottom). Overlaid are matching lognormal (solid) and gamma (dashed) density functions with the same mean and standard deviation

Table 2. Parameters describing the histograms of the span-wise spatial measures of the thermal streaks

Case Meanl r CVw Skewness Flatness

Estimated probability density functions ± thermal images

Width 65.1 33.4 0.51 1.6 6.4

Separation 54.8 39.6 0.73 1.5 6.1

Spacing 119.8 52.5 0.44 1.1 4.5

Simulation (with lognormal distributions)

Narrow 60 7 0.11 0.3 3.3

Wide 60 36 0.60 2.0 11.0

Estimated probability density functions ± simulated images (simulation with the wide lognormal distribution)

Width 61.8 35.1 0.57 1.99 9.57

Separation 60.6 36.6 0.61 1.81 7.71

Spacing 122.8 49.8 0.41 1.26 5.14

coef®cients of variations above 0.5. Since streak spacing is the sum of the streak width and streak separation, its distribution is expected to be characterized by a smaller coef®cient of variation, assuming that the width and sep-aration are independent variables.

Indeed, the standard deviation of the sum of two dependent identically distributed random variables in-creases by the square root of two, and its coef®cient of variation should decrease by the same (compared with the corresponding parameters of the individual variables). When the two independent variables are distributed dif-ferently, with standard deviation ofr1andr2, the standard deviation of their sum is given byr2

sumˆr21‡r22. The fact that this relationship holds very closely for the spatial parameters of the low-speed streaks (given in Table 2), supports the assumption that the width and separation are statistically independent.

4.3

Sensitivityanalysis

The results described above are based on using a high threshold level set so that the strong edges cover 15% of the image. The sensitivity of the span-wise spatial pa-rameters of the streaks to the high threshold level, which is determined by the desired percentage of strong edge-pix-els, is shown in Fig. 7. The mean width, separation and spacing decrease sharply at low percentage levels and mildly at intermediate levels. The desired percentage coverage (marked by the vertical line at 15% coverage) is selected close to the minimum percentage coverage at which the sensitivity of the results (i.e., the slope of the curve) is still low. At this percentage coverage, the edge detection algorithm provides good noise rejection but may miss some edges. Using a threshold level set to 25% cov-erage facilitates the detection of weaker edges and result in a mean spacing of 105.7 wall units, a decrease of 11%.

Regardless of the high threshold level, the bottom panel of Fig. 7 indicates that the coef®cients of variation of all the span-wise spatial characteristics are high, with a mean of 0.45, 0.57 and 0.71 for the spacing, width and separa-tion, respectively. Thus, the conclusion that the spatial parameters are widely distributed is independent of the speci®c high threshold level used in the edge detection algorithm.

Similar observations are made when the statistical analysis is based on a single valid column (with alternating positively and negatively sloped edges) from each image. Using the same threshold level, the statistical analysis based on valid columns results in a mean spacing of 104.9 wall units, a decrease of 12%.

4.4

Span-wise correlation

On the average, the streak width is mostly independent of the separation from the adjacent streaks, as demonstrated by the conditional mean function at the top panel of Fig. 8. Similarly, the separation between the streaks is mostly independent of the width of the adjacent streaks (bottom panel, Fig. 8). These results further con®rm our previous assumption (Sect. 4.2) that the width and separation are independent variables. Thus, it seems that the effect of the streaks, or the underlying vortical structures, is local and does not extend to affect the location of adjacent streaks. 4.5

Autocorrelation

The average autocorrelation of the span-wise cross-sec-tions (columns) of the thermal images, shown in Fig. 9, does not exhibit any signi®cant peak. The averaging is over 78 columns, one from each of the 78 images. Each autocorrelation was computed after extracting the mean with unbiased normalization and further scaling to achieve a unit autocorrelation at the origin (zero lag). The average autocorrelation decreases rapidly at small lags and

Fig. 7. Sensitivity analysis. The mean (top) and coef®cient of variation (CV,bottom) of the spacing (asterisks) width (plus signs) and separation (open circles) are derived by applying the statistical estimation procedure on edge images obtained using different high threshold levels, according to the desired per-centage of strong edge-pixels

Fig. 8. Conditional statistics demonstrating a lack of serial de-pendence between the span-wise width and separation of the thermal streaks. Thedotted linesare the average width (top) and average separation (bottom) across all samples

remains low with no signi®cant peaks thereafter. Thus, the ®rst peak cannot be detected reliably to estimate the characteristic period. Furthermore, as mentioned above, the location of the ®rst peak does not provide a good estimate of the characteristic period when the period varies over a wide range of values. This issue is further demonstrated in the next section.

5

Texture modeling and simulations 5.1

Texture generation model

The texture of the thermal images consists of elongated streaks aligned horizontally. Their span-wise spatial characteristics are re¯ected along the column-wise gray level pro®les. Zacksenhouse et al. (1998) and Abramovich (1998) have developed a texture generation model to de-scribe these cross-section pro®les and relate them to the statistics of the streak spacing, as shown in Fig. 10. Brie¯y, a stochastic point process (see, for example, Cox 1962) is used to generate a sequence of points that alternately mark the location of the positive and negative edges. The points are spaced according to an appropriate probability density function, as described in the next paragraph. Alternating positive and negative unit pulses are placed at the marked locations, and the resulting pulse train is passed through an integrator to generate a box train. Finally, the box train is ®ltered to simulate the gray level pro®le along an image column (up to a scaling and shifting factor).

Based on the spatial characteristics of the thermal images, lognormal or gamma probability density func-tions, of appropriate parameters, may be selected to sim-ulate the texture of the thermal images. As is evident from Fig. 6, lognormal probability density functions ®t the measured histograms better, so they are used for subse-quent simulations. However, the gamma probability den-sity functions provide reasonable ®ts too, and facilitate

analysis, as detailed in the next section. For simplicity, a single probability density function is used to determine the intervals following either a positive or a negative pulse, independent of the previous interval. This is justi®ed by the observations that (1) the probability density functions describing the width and separation between the streaks are similar, and (2) there is no signi®cant span-wise correlation. Using a single density function simpli®es an-alytical and numerical evaluation of the model but dif-ferent density functions with any dependence structure may be used in general.

5.2

Spectral methods

We apply the texture generation model to evaluate the performance of spectral methods in extracting the un-derlying probability density function from the gray-level pro®les. Analytically, the autocorrelation of the output gray-level pro®le is a ®ltered version of the autocorrelation of the box train signal, and for that Abramovich (1998) derived the following conclusions:

1. For gamma density functions withaˆ1 (Poisson process), the autocorrelation function is decaying and there is no peak (other than the one in the origin). 2. For gamma density functions withaˆ2, the ®rst peak

of the autocorrelation function (away from the origin) is atsˆ1.75p/q, while the mean spacing is 2lˆ

2a/qˆ4/q. Hence,soverestimates the mean spacing by 37%.

3. For gamma density functions withaandqapproaching in®nity (so that their ratio, the mean period l, is constant), the density function approaches an impulse at the mean spacing 2l. In the limit, the simulated signals are purely periodic and their average autocor-Fig. 10. Steps in simulating the texture across columns of the thermal images. The bi-polar pulse train (top) is integrated into a box train (middle) and ®ltered into a one-dimensional texture signal that simulates, up to a scaling and shifting factor, the span-wise gray-level signal (bottom). The intervals are distributed according to a lognormal probability density function with the parameters speci®ed in Table 2 (wide distribution)

Fig. 9. Average autocorrelation of span-wise cross-sections (col-umns) of the thermal images. The average is over the autocor-relation of 78 columns from 78 different images taken 0.5 s apart

relation function peaks at the corresponding constant spacing.

The parameteracontrols the coef®cient of variation (wˆr=lˆ1=pa), of the probability density function. Hence, aswincreases (adecreases) the location of the ®rst peak of the autocorrelation increasingly overestimates the characteristic spacing. Furthermore, asw increases, the autocorrelation tends to smear out and so the detection of the ®rst peak becomes less reliable.

These effects are demonstrated by analyzing the sim-ulated signals at the different stages of the texture gener-ation model. The average autocorrelgener-ation function of the simulated pulse-train (top), box-train (middle), and gray-level (bottom) signals are shown in Fig. 11, based on a narrow (right) and a wide (left) lognormal probability density functions (with parameters speci®ed in Table 2). When the underlying probability density function is nar-row (wˆ0.11), the ®rst peak in the autocorrelation is well de®ned, and its location agrees well with the characteristic spacing (2k‡_{ˆ120 wall units). However, when the} un-derlying probability function is wide (wˆ0.6), the ®rst peak (away from the origin) becomes shallow and its location poorly de®ned.

5.3

Proposed edge detection-based method

We have also evaluated the performance of the proposed method in extracting the underlying probability density function from the simulated images. To this end, we have converted each simulated gray-level pro®le into an image with the same pro®le along all its columns. The gray-level pro®les have been generated with a point process whose intervals are distributed according to the wide lognormal probability density function speci®ed in Table 2. The edge

detection-based method has been applied as described in Sect. 3. The estimated probability density functions, shown as scaled histograms in Fig. 12, and their matching lognormal distributions, closely match the underlying probability density functions used in simulating the tex-tured images. The estimated mean of 62, 61 and 123 and the estimated coef®cient of variation of 0.57, 0.61 and 0.41 (Table 2, last three rows) for the width, separation and spacing, respectively, closely match the actual character-istics of the underlying distributions.

6

Discussion

An automatic method for reliably extracting the span-wise spatial characteristics of the streaks from thermal images is presented and demonstrated. The method is powerful in extracting and fully characterizing all the relevant features, including the width, separation and spacing, and their dependence structure. The key aspect of the method is the localization of the edges of the low-speed streaks based on the detection of abrupt changes in the measurements. The proposed method is most suitable for extracting the spatial parameters of the streaks from the thermal images, as edge detection is a well-developed image processing technique. Using the texture generation model developed in the second part of the paper, we have generated synthetic images and vali-dated the accuracy of the measurement technique. However, the method could also be tailored to other ¯ow visualization techniques.

Applying the proposed method to the measured thermal images, a mean span-wise spacing ofk‡ ˆ120 wall units was obtained, in good agreement with previous studies. A novel result of this study is that the streak spacing is almost evenly divided between the width and separation of the streaks, and that the later parameters are widely distributed with coef®cients of variation above 0.5. This is to our

Fig. 11. Average autocorrelation of simulated pulse-train (top), box train (middle), and output signal (bottom) at the different stages of the texture generation model based on a narrow (right column) and a wide (left column) lognormal probability density function, respectively. Parameters speci®ed in Table 2. Average based on 78 samples of 185 long simulated vectors. Thedotted vertical linesmark the actual mean spacing

Fig. 12. Scaled histograms of the span-wise spatial parameters of simulated texture: width (top), separation (middle) and spacing (bottom). Overlaid are the actual (dashed) and matching (solid) lognormal density functions

knowledge the ®rst work to directly measure the distribu-tions of the streak width and separation. The streak spacing, which is given by the sum of the streak width and streak separation, is distributed less widely, as should be expected for the sum of two independent variables.

Since the span-wise width and separation of the streaks are widely distributed, spectral methods are not adequate for estimating the characteristic spacing. This claim is supported theoretically for shot noise processes, and both analytically and numerically for the texture generation model developed herein. The main problem stems from the fact that the lag of the ®rst peak in the autocorrelation deviates from the mean underlying spacing. In addition, its detection becomes unreliable, as the autocorrelation becomes ¯at. Thus, ad-hoc methods to localize the ®rst peak do not provide meaningful es-timators.

The edge detection method facilitates the investigation of span-wise relations between the spatial and thermal (velocity) features of the streaks. In the present work, we demonstrate that there is no evidence for span-wise spa-tial dependence between the width and spacing of the streaks. This is further supported by the observation that the standard deviation of the streak spacing relates to those of the streak width and streak separation as if the streak spacing is the sum of independent width and sep-aration. The lack of serial dependence suggests that the vortical structures giving rise to the low-speed streaks are local and do not affect adjacent structures. Future work may also investigate how the mean temperature of the streak, which measures its mean velocity, relates to its width. By automatically tracking the location of the edges across time, it would also be possible to investigate the span-wise ¯uctuations of the streaks and their develop-ment across time.

Appendix ± Statistical analysis

LetXbe a random variable with meanland the standard deviationr. The coef®cient of variation is the ratio wˆr/l. The random variableXis distributed according to a lognormal distribution when its probability density function is given by (see Papoulis 1984):

f…X† ˆexp 1 2 w10ln

X X0

2

Xw₀…2p†1=2

where the parameters lnX0 andw0are the mean and co-ef®cient of variation of ln Xand are related to the meanl and the coef®cient of variationwof Xaccording to:

X0 ˆl…1‡w2† 1=2 andw0ˆln…1‡w2†1=2

The skewness,Sˆ Pn

iˆ1

…Xi l†3

…n 1†r3 , of the lognormal density is

Sˆe3r2 3er2_‡2

…er2 _1†3=2 ; while the ¯atness,Fˆ

Pn iˆ1…Xi l†

4

…n 1†r4 , is given by

Fˆe6r2 4e3r2_‡6er2 ₃ …er2 _1†2 .

The random variable Xis distributed according to a gamma distribution when its probability density function is given by:

f…X† ˆq…qX_C†…a_a1†_†e qX

whereGand e are the gamma function and the base of the natural logarithm, respectively. The parametersaandqof the gamma distribution function are related to the meanl and the coef®cient of variationwaccording to:

aˆw 2 _andqˆw 2_l 1

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