Experiments in Fluids (2006) 40: 491–499 DOI 10.1007/s00348-005-0090-9
R ES E AR C H A RT I C L E
J. S. Park Æ K. D. Kihm
Three-dimensional micro-PTV using deconvolution microscopy
Received: 28 July 2005 / Revised: 18 November 2005 / Accepted: 21 November 2005 / Published online: 17 December 2005 Springer-Verlag 2005
Abstract A three-dimensional micro-particle tracking velocimetry (micro-PTV) scheme is presented using a single camera with deconvolution microscopy. This method devises tracking of the line-of-sight (z) flow vectors by correlating the diffraction pattern ring size variations with the defocusing distances of small particle locations. The working principle is based on optical serial sectioning microscopy, or equivalently deconvolution microscopy, that records images of an infinitesimally small particle, and generates a point-spread function of the three-dimensional diffraction patterns. A new imageprocessing algorithm has also been developed to digitally identify the center locations and measure the radii of the diffraction rings, which allows simultaneous tracking of all three-vector components. The developed PTV technique uses a 40·, 0.75 NA dry objective lens with 500-nm fluorescent seeding particles of SG=1.05, and successfully measures the fully three-dimensional fields flowing over a spherical obstacle snuggly fitted inside a 100 lm · 100 lm micro-channel. The volumetric measurement resolution of the present system is equivalent to a 5.16 lm · 5.16 lm · 5.16 lm cube, and the overall measurement uncertainty for single-point velocity vector detection is estimated to ±7.58%.
1 Introduction Since the innovation of micro-particle image velocimetry, so-called micro-PIV (Santiago et al. 1998), a volume of research has been done to apply the technique to map two-dimensional velocity vector fields for diverse microfluidic applications. Despite the intrinsic chalJ. S. Park Æ K. D. Kihm (&) Micro/Nano-Scale Fluidics and Energy Transport (MINSFET) Laboratory, Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee, Knoxville, TN 37996, USA E-mail:
[email protected] Tel.: +1-865-9745292
lenges occurring from the optical diffraction and background noise, the micro-PIV system associated with a high numerical aperture (NA) objective, in particular, has achieved substantial progress to enhance the image definition with a reduced depth-of-field thickness. A number of studies have been published to correlate the particle image intensity with the amount of defocusing or depth-of-field for the case of two-dimensional microPIV (Olsen and Adrian 2000; Meinhart et al. 2000). For a macroscale measurement, the use of multiple cameras arranged at different viewing angles, so-called stereoscopic or multiscopic PIV, has been proposed to simultaneously map three-dimensional velocity fields, but the success is largely limited for macro-scale applications to date (Prasad and Adrian 1993; Raffel et al. 1995; Gaydon et al. 1997; Ka¨hler and Kompenhans 2000). For the case of microscale stereoscopic PIV, limited success in implementing two cameras on a single microscope has been reported only for relatively lower magnifications (Bown et al. 2005; Lindken et al. 2005). Besides, there remain some open questions on its cumbersome calibration procedure, and the lowered imaging accuracy due to the significant optical aberration and astigmatism. The true ‘‘optical slicing’’ capability of confocal laser scanning microscopy (CLSM) shows noticeable enhancement for the micro-PIV techniques with superbly enhanced image definition (Park et al. 2004). The stack of confocally sliced images can be cross-correlated to complete mapping of three-dimensional velocity vector fields for limited flow conditions such as steady or periodic flow conditions. On the other hand, the CLSMPIV expresses planar velocity vector fields with just twovector components, so it is only useful for mapping three-dimensional vector fields of a horizontal flow. The major shortcoming of the CLSM-PIV comes from its relatively low frame rates that limit the dynamic range of velocity measurements. The current state-of-the-art is known to allow 1,000 fps at significantly reduced pixel resolution (Yokogawa 2005).
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Willert and Gharib (1992) used a triangular threehole aperture to resolve defocused particle images and determine the three-dimensional locations of particles. The level of distortion of the triangle formed by the three particle images increases with the defocusing distance and the corresponding line-of-sight location of the particle is detected. This technique is referred to as defocusing digital particle image velocimetry (DDPIV). Pereira et al. (2000) later applied this technique to a macroscale two-phase flow using the gas bubbles as tracers. More recently, Yoon and Kim (2005) miniaturized the aperture to fit to a microscope objective and successfully demonstrated the feasibility of DDPIV for microfluidic applications. Rohaly et al. (2001) used an off-axis single-hole rotating aperture to detect the line-of-sight particle locations based on the eventually identical principle to DDPIV. Since the finite time delay is essential to record multiple images with the aperture rotating, this technique is not suitable for dynamic flow measurements. In addition, this technique, likewise for DDPIV, requires small-holed apertures that substantially reduce the image brightness. Deconvolution microscopy was first developed to enhance the image contrast by eliminating the background noise by analyzing the point spread function (PSF) as comprehensively reviewed in McNally et al. (1999). One of the original purposes was to make the deconvolution microscopy comparable to or eliminating the need of confocal microscopy primarily for biological applications. The digital in-line holographic PIV (HPIV) stemmed from the same principle as deconvolution microscopy and HPIV can measure three-dimensional velocity components (Pan and Meng 2001). The line-of-sight particle locations are inversely reconstructed by analyzing the measured PSF patterns. HPIV is considered as a useful tool to map three-dimensional flow fields for macroscale applications as the detrimental effect of aberration on resulting images is usually negligible for larger scale imaging. For the case of microscale imaging, however, the aberration is inevitable and substantial due to the relatively short working distance in comparison with the lens diameter, the non-zero specimen thickness, and mismatch of refractive indices of different medium layers placed on the optical axis. Speidel et al. (2003) explored the idea of using the defocused diffraction patterns to identify the line-ofsight locations of stationary particles embedded in a highly viscous medium. They called the technique optical serial sectioning microscopy (OSSM) in lieu of more widely used deconvolution microscopy. Their correlation between the diffraction ring size and the defocusing distance relies entirely on their experimental measurements while the theory developed by Gibson and Lanni (1991) can make the correlation more comprehensive. Using the comprehensive PSF analysis of Gibson and Lanni (1991) as a correlation function, Park et al. (2005) measured the three-dimensional Brownian motions of
nanoparticles and determined the temperature for the surrounding fluid region based on the well-known Einstein’s Stokes diffusion theory (Einstein 1905). Wu et al. (2005) also demonstrated the feasibility of three-dimensional tracking of bacteria swimming in water using the Gibson and Lanni’s predictions. This paper introduces a formidable idea of threedimensional PTV based on the optical measurements of the diffraction ring patterns. Unlike the conventional deconvolution microscopy, the present technique measures the out-most diffraction ring diameters and determines the line-of-sight particle locations by comparing against the calculated PSF of Gibson and Lanni (1991). In order to digitally identify and analyze the ring diameters and the center locations, a novel computer algorithm has been constructed and used for an example application flow over an interrupting sphere in a square microchannel.
2 Diffraction and aberration in optical imaging of a small particle Optical diffraction occurs as a result of the wave-nature of light in microscopic imaging of a small particle when light waves emitted (scattered) from a point source-like particle and passing through a rear aperture of the objective lens, and then superposed constructively or destructively depending on the optical path difference (OPD) of waves (Born and Wolf 1999). The diffraction pattern is, in the absence of aberration, extended periodically above and below a focal plane along the optical axis (z) as well as radially around the axis (x and y). The resulting three-dimensional diffraction is expressed as a symmetric pattern in the absence of optical aberration or as an asymmetric pattern with a presence of optical aberration (Cagnet et al. 1962). PSF is an analytical expression of diffraction patterns and the square of PSF is equivalent to the corresponding intensity distribution of an image. Either symmetric or asymmetric patterns of PSF can be resulted from the structure of test section. Figure 1 schematically illustrates both a designed system and a non-designed system for the test section. The designed microscopy system has a negligibly thin specimen thickness and uses a cover glass and an immersion medium that have the thickness and refractive indices designated for a specific objective lens. This is also called as an aberration-free system. In the non-designed system, the specimen is thick (finite) and the use of substances and immersion medium having off-design refractive index values and thicknesses causes optical aberration in focusing an image plane and makes the depth of focus extended. Gibson and Lanni (1991) presents a useful prediction for PSF that was derived based on Kirchhoff’s scalar diffraction integral formula, and was expressed with convergent series of Bessel function and a complex exponential term that included an aberration function.
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Fig. 1 Two different structures of test sections having a a designed system, and b a non-designed system
observing the radial diffraction patterns. The authors have first employed the doconvolution concept to dynamically track three-dimensional locations of nanoparticles that are sufficiently small to accommodate the point source approximation for diffraction patters. Figure 2 illustrates the line-of-sight directional tracking of nanoparticles using the principle of deconvolution microscopy. When a focus of objective lens is fixed on the cover glass interface with test fluid, the first column (Fig. 2b) shows the intensity distributions of diffraction images of nanoparticles, calculated from (1), placed at different defocus locations from the focal plane. The second column (Fig. 2c) shows corresponding graphical diffraction images of concentric fringes for the identical optical and physical conditions as for the first column and also as for the main experiment to be discussed in the next section1. Figure 2d shows the calculated correlations of the out-most fringe’s radius (ROMF) with the defocus distance (Dz). The defocus distances of suspended particles can now be determined
The three-dimensional intensity distribution, equivalent to the square of PSF, is expressed in terms of the numerical aperture of objective lens (NA), the magnification of microscopy system (M), dimensionless lateral locations (xd, yd), and the defocusing distance (Dz) as: Z 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NA Iðxd ; yd ; DzÞ ¼ C J0 k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q x2d þ yd2 exp½jW ðDz; qÞq dq ; M 2 NA2 0 where k is the wave number (2p/k), q (=r/a) is the normalized radius in the exit pupil of objective, and W(Dz, q) is a phase aberration function, which is the product of wave number (k) and optical path difference (OPD). The OPD is a key factor accounting for optical aberration and is a function of microscopic parameters including the level of defocus, the cover glass thickness, the specimen thickness, and refractive indices of all dielectric mediums comprising the test section. More complete expression for (1) is available elsewhere (Gibson and Lanni 1991; Park 2005). The conventional in-focus two-dimensional microPIV suffers from the optical aberration occurring from practically inevitable non-designed experimental configurations. Indeed analytical treatment is popularly adopted to construct an intensified particle image only at the focal plane by filtering out defocused (off-focused) particle images. In contrast, the present micro-PTV technique analyzes the defocused diffracted images in terms of fringe size and defocus distance to locate the line-of-sight positions of particles relevant to the focal plane. This technique is commonly referred to as OSSM, or equivalently deconvolution microscopy. Deconvolution microscopy was originally developed to construct the three-dimensional image of a thick biological specimen by analyzing PSFs of in-focus and off-focus images (Agard 1984). In particular, the analysis method of off-focus images could be renovated to obtain three-dimensional locations of granules embedded in a thick fluidic specimen. The deconvolution principle is based on the determination of the radial intensity profile as a function of defocus. In other words, the relative line-of-sight (z) location of a particle measured from the focal plane can be determined by
ð1Þ
by optimally fitting the measured out-most fringes to the calculated PSF profiles and/or the regressed polynomial fitting2.
3 Experimental setup The experimental system consists of an upright epifluorescent microscope (Olympus, BX 61) with a mercury lamp illumination source, a digital CCD camera (UNIQ Vision Inc., UP-1830) equipped with a Sony 2/ 3¢¢ Exview HAD CCD chip (1,024·1,024 pixels, 6.45-lm pixel width), and a frame grabber (QED-Imaging Inc.), which records images as 8-bit signals with a frame rate of 30 fps. A 40· dry objective lens (0.75 NA and WD0.51 mm) is used for an image-capturing lens. The 100-lm2 square test channel is made of borosilicate glass (nc=1.475) with a nominal wall thickness of 50-lm (Fig. 3). The test channel is attached on the 1-mm thick slide glass and a 150-lm thick cover glass (ng=1.522) is glued on the top. In order to compensate the refractive index mismatching between the test channel wall, the cover glass, and particularly the thin 1
M (overall magnification of microscopy)=40, NA=0.75, k (emission light wavelength)=515 nm (Ar-ion laser source), tg (cover glass thickness)=0.21 mm, ti (immersion thickness)=0.483 mm, ng (refractive index of cover glass)=1.51, ns (refractive index of specimen medium)=1.33 (water), and ni (refractive index of immersion)=1.0 (air). 2 A regression of the calculated correlation provides a third-order polynomial approximation of Dz ¼ A r3 þ B r2 þ C r þD with fitting coefficients of A=0.0003, B = -0.028, C=1.0747, and D=0.4326. This functional form of the correlation will be useful in computational determination of the ring radii as will be discussed in the later section.
494 Fig. 2 The concept of deconvolution microscopy application for threedimensional micro-PTV technique; a Critical parameters of experimental setup such as specifications of objective lens, defocus distance between particles and lens’s focus, thicknesses, and refractive indices of cover glass and immersion medium, b PSF of diffracted particle image in each indicated defocus distance, c Graphical diffraction images, which are drown based on the corresponding PSF, and d A relation between a defocus distance (Dz) and a radius of the out-most fringe (ROMF)
air gap existing between the two glass contacting surfaces, a drop of immersion oil (no=1.516) is dropped to smear into the gap and surround the channel. The working distance between the objective lens and the cover glass is set to 483 lm because of the non-designed system. A 95-lm diameter polystyrene micro-sphere is snuggly fit inside the square test channel to generate a threedimensional flow surrounding the sphere that can simulate the liquid flow over a spherical bubble trapped in a microchannel3. Yellow-green (EX: 505 nm, EM: 515 nm) fluorescent polystyrene micro-spheres (500-nm diameter) with 1.05 SG (Molecular Probes Inc.) are used as tracers mixed at 0.001% volume concentration in deionized water. As previously discussed in Sect. 2, the objective lens is focused on the top inner surface of the test channel and the defocused images of particles in the flow are recorded. The interrogation volume is extended to 25 lm beyond the focal plane of the top inner surface as the illumination is quickly attenuated with increasing z-distance.
4 Results and discussion Figure 4a shows raw images of diffraction patterns for particles at different line-of-sight locations in the flow over the obstruction of a micro-sphere (shown by the dashed circle) inside the nominal 100-lm2 microchannel. 3
For instances, it can represent a practical situation such as a hydrogen bubble trapped in a PEM fuel cell operation, or an air bubble trapped in a lab-on-a-chip microfluidic device for various bio-processings.
When the number density of seeded particles is high and the inter-particular distance is shorter than the outermost fringe ring radius, the resulting overlap of diffraction patterns can make the fringe definition obscure. The present 0.001% volume concentration provides the number concentration of 1.46·105/lL, which is equivalent to the mean inter-particular distance of 19 lm. Since the maximum line-of sight detection range is limited to 25 lm due to the light attenuation, the corresponding out-most fringe radius (ROMF) is smaller than 10 lm (refer to Fig. 2), or approximately one-half of the mean inter-particular distance. Thus, the image blur occurring from the diffraction pattern overlapping is assumed negligible. 4.1 Computer algorithm to identify circular fringe rings New custom image analysis software is developed to identify the out-most fringe rings and to determine their center locations (Figs. 4b and 4c). The key innovation is the use of a ‘‘circle-fitting’’ concept in that a sufficiently large circle is started around a given pixel as centered, and then its radius is progressively shrunk to examine if the majority of the circularly distributed pixel points fall onto ‘‘bright’’ category ensuring that they are located on the fringe ring. The examination is conducted serially for all the pixels to identify all existing out-most rings on the same image and the examination is repeated for the successive images. Once a fringe ring is identified with its radius ROMF, the center of the ring is assigned as the x–y location of the particle and the corresponding z-location is determined by the Dz-ROMF correlation shown in Fig. 2d. Comparison of the x–y–z locations of pairing
495 Fig. 3 Geometry of test section that is constructed with a square-microchannel, a cover glass, a slide glass and a microsphere, which is trapped in the square channel
particles on two neighboring frames gives their threedimensional displacements and velocity vectors with a known lapse time. Two pre-processing schemes have been implemented prior to the circle-fitting algorithm: (1) the original image is divided into smaller interrogation sections such that each interrogation section can cover a portion of particle’s diffraction image and get information like an average, a maximum and a minimum of intensity levels, and (2) the original 8-bit raw image (0–255 pixel gray levels, Fig. 4a) is filtered to conform to a 1-bit image (0– 1 pixel gray levels, Fig. 4b) using appropriate normalization and filtering. In the first step, the original image is divided into 2,304 of 16·16 pixels (2.58 lm · 2.58 lm) interrogation sections. Given the averaged inter-particular distance of 19 lm, probability of the presence of multiple particle images in a single 2.58 lm-square interrogation section is estimated to be less than 2%, and the number of particle images to be present in a single interrogation section will be wither 1 or none with more than 98% confidence. In the second step, the original 256-gray levels are normalized to take fraction numbers ranging from 0 to 1, and then each normalized gray level is assigned by ‘‘0’’ or ‘‘1’’ based on threshold filtering. The threshold filtering dramatically enhances the signal-tonoise ratio based on the following protocol: a. For an interrogation section containing a maximum normalized gray level higher than 0.9, ‘‘1’’ is assigned to all pixels with gray levels higher than 0.75 to be identified as fringe ring image, and ‘‘0’’ for the remaining pixels with gray levels below 0.75 to be considered as the background or noise. b. If an interrogation section does not satisfy the protocol (a) AND the section shows the differential be-
tween its maximum and minimum gray levels being less than 0.15, i.e., for the case of a relatively blurry interrogation section, ‘‘0’’ is assigned to the entire pixels in the section. c. If an interrogation section does not satisfy both the protocol (a) and (b), ‘‘0’’ is assigned to all the pixels falling below the section-average gray level and ‘‘1’’ is assigned to those with gray levels above the average.Note that the threshold gray level values, such as 0.9, 0.75, and 0.15, are the most commonly used ones in our study and they occasionally had to be adjusted to optimize the filtering efficiency and accuracy depending on individual images mostly by ‘‘trial-anderror’’ basis. The idea of circle fitting is illustrated in Fig. 4c. For a given pixel point of a selected interrogation section, the circle-fitting algorithm starts with a sufficiently large circle with 45-pixel radius and 24 angular circumferential pixel points to examine the average of the 24-pixels4. If the average value is higher than a specified threshold, the circle is recognized as a fringe ring, otherwise, the circle is reduced to the next 44-pixel radius and thereafter until a ring is identified or the radius reduces to zero. The interrogation is repeated serially for the whole pixels of the interrogation section to eventually determine the existence of a fringe ring in the section. 4 The average gray level of ‘‘1’’ means that the entire 24 pixels fall on the fringe ring, and ‘‘0’’ means that the entire 24 pixels fall on the background. The criterion for identification of a ring is differently specified in-between ‘‘1’’ and ‘‘0’’ depending on the ring size and the image quality. For a larger ring, a relatively low value is used since large circular fringes often conform to incomplete circles. For a smaller ring, a relatively high value close to ‘‘1’’ since most of small circular fringes are well defined with completed circles.
496 Fig. 4 a A raw diffraction image captured with an intensity resolution of 8-bit and a shutter speed of 30-ms. The dashed line indicates an interrupting micro-sphere. b After an image processing the 8bit grayscale image is converted to a binary image that contains only 0’s and 1’s. c An identification of threedimensional particle position is conducted by using axsymmetric 24-point data. d A few identified three-dimensional particle positions: A(x-, y-, ROMF-pixel numbers: 158, 178, 16), B(140, 245, 22), C(240, 200, 40) and D(264, 261, 29)
Figure 4c shows an identification of a ring of 22-pixel radius as an example, and Fig. 4d shows a few identified rings of different shapes and (X, Y, ROMF) pixel-unit formation implying three-dimensional particle positions. X and Y positions are basically obtained from a conversion of unit from pixel to micrometer. Then similar steps are repeated for the whole interrogation areas composing the first image and repeated for the successive images to complete the x–y–z locations of all imaged particles. The pre-processing algorithm is summarized in the flow chart shown in Fig. 5. Once the ring sizes and their centers are identified for successive frames, the use of the nearest-neighborsearch method for particle tracking determines three-component velocity vector fields, and ensemble averaged field is determined from all recorded images.
As expected, virtually very few interrogation sections contain more than one vector. To eliminate erroneous vectors, using a three-dimensional median filtering with 3·3·3 surrounding vectors performs post vector processing5. Additionally any empty interrogation section identifying no vector is filled with the median vector of the surrounding vectors to smoothen the flow field without affecting the flow field magnitudes and directions. 5 Before a median filtering, the wrong directional vectors are eliminated. If the center vector deviates beyond a tolerable range from the median vector of the surrounding 26 vectors, the center vector is replaced by an average of both center vector and median vector. The tolerance is ranged from 0.1 to 0.5 depending on the image quality.
497 Fig. 5 A detail flowchart of a digital image processing
4.2 Microscale 3-D vector field measurements and discussion Figure 6 shows measured three-dimensional vector fields for the creeping flow over a snuggly fit 95-lm diameter sphere inside a nominal 100-lm2 square channel. 500nm fluorescent spheres seed the flow and the tested Reynolds number is approximately 0.003 assuming the effective viscosity6 of the particle-laid suspension to be equal to that of the base fluid (water). The presented vector field covers 25-lm deep from the top surface of the microchannel conforming the measurement volume of 165 lm · 93 lm · 25 lm where 2,304 vectors are shown with a spatial resolution of 5.16 lm · 5.16 lm · 5 lm in volume. The developed micro-PTV technique successfully maps the fully threedimensional flow vector fields. The flow acceleration is shown in the gap region between the sphere and the channel wall and the two different vertical motions of 6
The effective viscosity (leff) of a particle-laid suspension is given by (Deen 1998); leff=(1+2.5/)l0, where / is the volume fraction of spheres and l0 is the viscosity of the suspending fluid (959·10-6 Ns/ m2 for a water at 22C). The tested low volume fraction of 0.001% does not alter the effective viscosity from the viscosity of the suspending fluid. Thus, the applied flow rate of 1 lL/h at a mean velocity of 28 lm/s yields the Reynolds number of approximately 0.003.
ascending and descending are clearly identified at the front and the rear region of the sphere, respectively. The measurement accuracy of micro-PTV depends on several factors and one non-trivial factor associated with the small length scale comes from the rendering effect of Brownian diffusion of seeded particles. Santiago et al. (1998) proposed to estimate the error using one-dimensional Brownian motion and the present study extends the similar concept to account for the three-dimensional mean square displacement (MSD) of Brownian motion. Namely, rffiffiffiffiffiffi 2 1=2 r 1 6D eB ¼ ð2Þ ¼ u Dt Dx where Ær2æ denotes three-dimensional MSD, Dx is the average or representative displacement of seeded particles, u is the corresponding characteristic velocity, D indicates Brownian diffusion coefficient of the suspended particles in the fluid7, and Dt is a time interval between observations. For the characteristic velocity is estimated to 60 lm/s and the given time interval is 0.033 s, the 7
The diffusion coefficient is given by (Einstein 1905), D ¼ j T =6pl rp where j is the Boltzmann’s constant (1.3805·1023 J/K), T is the suspension temperature in absolute, l is the effective viscosity of the suspension, and rp is the particle radius.
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wDt ; is estimated to be 0.001 s. The uncertainty of the velocity vector is now given by: "
2
2 #12 @~ u @~ u wDs þ wDt ¼ 2:68 l m/s w~u ¼ @ Ds @ Dt ð4Þ The relative uncertainty of the velocity vector, for the case of a single point detection, is w~u 2:68 l m/s ¼ 7:6 % ð5Þ ¼ ~ 35:33 l m/s u For the present case of N=2, which is determined from the number of images (60) multiplied by the number of diffraction patterns (60) per image, and then divided by the number of effective interrogation volumes (@1,800), the uncertainty for an ensemble averaged data is estimated: w~u 2:68 l m/s pffiffiffi w~uN ¼ pffiffiffiffi ¼ ¼ 1:9 l m/s ð6aÞ N 2 w~uN= ¼ 1:9 l m/s ¼ 5:4 % ð6bÞ ~ u 35:33 l m/s
5 Concluding remarks
Fig. 6 Full-field mapping of vector profiles; a orthogonal projections on x–y plane and x–z plane and b a three-dimensional view
error due to the Brownian motion is estimated to be approximately 21% for the case of a single particle detection. For ensemble averaged pffiffiffiffi realization this error dramatically reduces by eB N ; where N is the number of independent samples. The present case of N=20, thus, yields the uncertainty of approximately ±4.8%. The measurement uncertainty is estimated by using the analysis by Kline and McClintock (1953). Since the velocity vector is calculated from the three-dimensional displacement of particle movement divided by the time interval, ~ u ¼
Ds Dt
ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ds ¼ Dx2 þ Dy 2 þ Dz2 : The average of the individual displacement measurements of Dx, Dy, and Dz are 1.023 lm, 0.393 lm and 0.399 lm, respectively, and thus, Ds=1.166 lm and ~ u ¼ 35:33 l m/s: The elementary uncertainties of wDx, wDy, and wDz are given by one half of the pixel resolution, i.e., ±0.081 lm. Thus, the measurement uncertainty for the threedimensional displacement is also estimated to be ±0.081 lm. The observation time interval is specified as Dt=0.033 s, and the uncertainty of the time interval,
A single-camera micro-PTV technique using deconvolution microscopy has been developed to simultaneously map the three-component velocity vectors for threedimensional micro-scale flow fields. From the comparison of measured diffraction images and calculated PSF functions, a calibration curve is obtained to correlate the out-most fringe’s radius and the defocus distance from a focal plane to find out the z-location of a seeded particle. An image-processing algorithm has been constructed to digitally identify the diffraction rings and their centers to simultaneously determine the x–y–z locations of the corresponding particles. The algorithm consists of four main steps: (1) binary image processing, (2) threedimensional particle positioning, (3) vector processing, and (4) post vector processing. The developed system uses an epi-fluorescent microscopy with a dry objective lens (40·, 0.75 NA) and images 500-nm fluorescence particles of SG=1.05. The tested field constitutes a three-dimensional flow field over a micro-sphere fitted in a 100 lm2 micro-channel. The single-camera micro-PTV system shows a successful reconstruction of the three-dimensional vector fields for the region of 165-lm (L) · 93-lm (W) · 25-lm (H) in the upper part of the sphere with a volumetric measurement resolution of 5.16 lm3. The maximum data biasing due to the Brownian diffusion of seeded particles is estimated to ±4.8%, conservative estimation for the overall measurement uncertainty for the velocity vector is ±7.6% for the case of for single-point detection, and ±5.4% for the case of ensemble average of 60 imaging frames.
499 Acknowledgements The authors wish to acknowledge financial support provided partly by the University of Tennessee Research Initiation Grant and partly by the Korea Institute of Science and Technology Evaluation Policy (KISTEP).
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