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Abstract
Interferometric particle imaging (IPI) is a robust and popular technique for measuring particle size and velocity. A method based on a template matching algorithm and an auto-correlation method is proposed to simultaneously extract the location and the separation of doublet images of a particle from an IPI focused image. The position coordinate (x,y) of the particle can be determined with high accuracy, as evaluated by using a serial particle mask. Furthermore, the method can be employed to achieve sub-pixel spacing extraction when combined with Gaussian interpolation. The algorithm is tested using synthetic and experimental data. The results suggest that the method presented here is promising for its application to a high-density particle field, in accurately measuring both the particle size and its location.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Interferometric particle imaging (IPI), based on Mie scattering theory, is a popular method for instantaneously measuring the size and spatial distribution of the spherical particle, utilizing the interference fringes or glare-point pairs generated by the imaging of reflection and first-order refraction rays scattered from a particle illuminated by the laser light on a defocused or focused image plane. It has been widely applied to sprays [1–10], flow fields [11,12], and other situations [13–17]. In this technique, the size of a particle is evaluated by the glare-point separation or the fringe number of the interferogram, the measurement accuracy mainly depending on the image processing technique. Generally speaking, the image processing of IPI consists of two steps: locating each particle image and extracting the fringe frequency/the glare-point separation. It is important to accurately locate each glare point pair or the fringe image in subsequent processing, especially for the subsequent determination of the movement and velocity using particle tracking velocimetry technique.
Compared with the defocused fringe, the focused image of IPI is more promising to be applied to a high-density spray field due to the fact that the glare-point image decreases overlapping. However, it is much harder to determine the position of glare point image than that of an interferometric image. At present, a few processing algorithms have been suggested for focused image processing of IPI. Zama et al [8,9] proposed the dynamic threshold and the three-point Gaussian peak fitting technique to locate the image of particles, the auto-correlation and a Gaussian interpolation method to extract the glare point spacing, which was applied to size and 3D velocity of droplet in a spray field. Hardalupas et al [10] employed continuous wavelet transform with a small scale along each horizontal line of the image using Mexican Hat wavelet, and each maximum obtained from the wavelet transform spectrum corresponds to the location of a glare point. The droplet size, velocity and gas phase velocity in a spray are obtained by combining the out-of-focus imaging with the in-focus imaging of IPI technique. In this paper, we present a method of extracting location and separation of glare-point image based on template matching and auto-correlation algorithm, the sub-pixel accuracy of spacing can be acquired with the Gaussian interpolation. The performance of the method is demonstrated through simulation and experiment, the location and size of particle are estimated simultaneously, the experimental results match well with those from the simulation.
2. Extended glare point image technique
The scattering characteristics of the particle are utilized in the technique. When a transparent spherical particle is illuminated with a laser sheet, the reflected and first-order refracted rays scattered from the particle are dominant, compared with higher order refraction for Mie scatting, so those two scattered light rays provide two bright spots on the particle, which is called glare points [18,19]. The pair of glare points is imaged on a focal plane, as shown in Fig. 1(a). Using geometrical optics, particle diameter d can be expressed as the following (according to the size parameter x = πd /λ >10~20 [19], then particle diameter d > 1.69 µm for wavelength λ = 532 nm) [8]:
Using trigonometric relations, Eq. (1) can be written to bewhere d, ∆l, M, θ and m are the diameter of a particle, the glare-point separation on the image plane, magnification of the optical system, scattering angle and the relative refractive index of the particle in medium, respectively.From Eq. (2), the particle diameter d is a function of M, θ, m and ∆l. For the scattering angle θ≠90°, the magnification is different, which affects the accuracy particle sizing. When θ = 90°, the intensities of zero-order and first-order scattering lights are different (the relative intensity of zero-order is almost 103 times that of first-order for perpendicular polarized light), as a result, the brightness of two glare points are different, which makes it difficult to identify and extract the glare point separation with high accuracy in the image processing of IPI. Therefore, the extended IPI system with opposite two-sheet illumination is used, as shown in Fig. 1(b). The two glare points with the same brightness are formed, which correspond to two reflection lights. The distance between the two externally reflected glare points in the image plane is given by
where θ = 90°, then the particle diameter d can be expressed as the followingFrom Eq. (1)/(2) and Eq. (4), for a given experimental system, that is, for θ, m, and M given, the particle diameter d is proportional to the glare point separation ∆l, and can be calculated by measuring ∆l. Consequently, the glare point image processing method determines the accuracy of particle sizing using IPI, in which the automatic extraction of glare-point separation is among the crucial aspect of this technique.3. Algorithm
Figure 2 shows the algorithm flowchart of the automatic IPI focused image processing we present, which consists of two steps: (1) locate each particle, which is the first and most important step in the glare point image analysis of IPI to accurately locate the particle center position. The glare point image of a particle I(x,y) and the mask image P(x,y) are operated by cross-correlation function:
where ⊗ denotes correlation, the peak value coordinate (x,y) of which indicates the geometric center of the particle focused image. (2) Extract the glare-point separation of each particle. The focused images of each particle is extracted from raw image according to the center coordinate (x,y) and the shape and size of the mask image, subsequently auto-correlation operation and Gaussian interpolation are performed for each extracted glare point image to evaluate the separation of the doublet images, namely the spacing between the zero-order peak and first-order peak. The sub-pixel accuracy of particle diameter can be calculated by Eq. (1)/(2) and Eq. (4). The particle center coordinate is given through the geometric center coordinate (x,y) of the focused image using geometric relationship.4. Simulation results
4.1 The mask image
In IPI, the mask image of a focused image is completely different from that of a defocused one [20]. Because the glare-point separation ∆l of a particle is a function of the diameter d for a given experimental system (M, θ and m), it means that the particle mask image for the focused image may not be the only one. For example, the spray field is composed of droplets with various diameters; thus, it is not a matter of simply applying the particle mask correlation method, but a matter of selecting the reasonable template. Figure 3 shows an example of numerical simulation. Figure 3(a) is the simulated two point images with ∆l = 8 pixels. Figure 3(b) is a series of mask images with the separation of 4, 5, 6, 8, 10, 11 and 12 pixels, respectively. The cross-correlation function between Figs. 3(a) and 3(b) is calculated and the result is shown in Fig. 3(c), each pair of glare point results in a different number of peaks. For the separation ∆l of 4, 5, 11 and 12 pixels, four cross-correlation peaks are observed, and the peaks are almost the same for ∆l = 4 and 12 pixels; the middle peaks are slightly lower than the side peaks for ∆l = 5 and 11 pixels. For the separation of 6, 8 and 10 pixels, there are three cross-correlation peaks, and the peaks are different, the middle peak being highest for the separation of 8 pixels. By selecting an appropriate threshold, it is possible to obtain the particle location. Figure 3(d) shows the results of detected particle location denoted with the red spot. As shown in Fig. 3(d), the particle center location can be accurately obtained by the peak value by using the mask image of the separation of 6, 8 and 10 pixels. For the mask ∆l = 5 and 11 pixels, two peaks are extracted, and the two position coordinates (x,y) are obtained, but not the particle image location. For the mask of ∆l = 4 and 12 pixels, Either four peaks or zero are extracted, and the four position coordinates (x,y) are obtained, but similarly, not the particle image location. Therefore, for the glare point image with ∆l = 8 pixels, the cross-correlation is operated with the mask images of the separation varying from 6 to 10 pixels, then the geometric center coordinate (x,y) can be accurately extracted. That is, for the glare point image with ∆l varies from 6 to 10 pixels, the mask image for the separation of 8 pixels can be used to achieve the particle location. From the above analysis, we can draw a conclusion that for the glare point image with the separation of ∆l, the algorithm is effective for the mask image with the separation varying from (∆l-2) to (∆l + 2) pixels.
Fig. 3 Simulation results of a glare-point image locating: (a) a glare-point image with the separation of 8 pixels; (b) a series of mask template with the separation of 4, 5, 6, 8, 10, 11 and 12 pixels from top to bottom, respectively; (c)1D distribution of correlation value; and (d) the results of the center detection.
4.2 Locating the center of glare point image
We here use simulation data to assess the performance of the algorithm we present. The IPI numerical simulator was developed by Shen [21], in which the scattering light by the particles is calculated using a Debye series expansion of the Mie theory, glare points can be Dirac functions, assimilated as a punctual light source. Considering that we study detection position algorithm and time consuming, we directly yield two point pairs as the particle image. Figure 4 shows the results of simulation. Figure 4(a) is the simulated glare point images with 50 spherical particles randomly distributed in a field of view of 512 pixels × 512 pixels, the glare point separation varies from 4 to 20 pixels. According to the analysis of Section 4.1, four mask images with the separation of 6, 10, 14 and 18 pixels are selected. The cross-correlation function between the glare-point images and the four mask images are calculated and the result of detected particle center is displayed in Fig. 4(b), denoted with the red spot. Comparing Fig. 4(a) with 4(b), it can be seen that the center positions of the glare points images in Fig. 4(a) are completely and accurately identical in Fig. 4(b).
Fig. 4 Simulation results of glare-point images center detecting: (a) glare-point images; (b) the results of the center detection.
4.3 Estimation of glare point separation
In this section, we test the performance of the auto-correlation and Gaussian interpolation algorithm. From Eq. (1)/(4), the particle diameter d is proportional to the glare-point spacing ∆l. The glare point image presented here is the simulated using the numerical simulator of IPI developed by Shen [21], the simulation of which is completely different from Section 4.2. Figure 5(a) shows the simulated glare point image of a particle 10 μm in diameter according to the setup in Fig. 1(b), where Mie theory is used to calculate the scattered intensity of the reflected rays scattered by the particle. The imaging system consists of an aperture and an equivalent thin lens and is modeled by using Fourier optics. The parameters used in the numerical simulation are: λ = 532 nm, the imaging lens is a circular aperture with a focal length f = 180 mm and = 3.5, the magnification M = 1.0, the field area is 101 pixels × 101 pixels with a pixel size of 0.2 μm × 0.2 μm, and the refraction index m = 1.2. Figure 5(b) shows the auto-correlation value. Figure 5(c) shows the results of first-order peak and Gaussian interpolation. The spacing between the zero-order peak and first-order peak of the auto-correlation value is the glare point separation, and before the Gaussian interpolation, estimated separation ∆l = 36 pixels, and then the particle diameter is calculated using Eq. (4), and d = 10.18 μm; after the Gaussian interpolation, estimated separation ∆l = 35.51 pixels, and d = 10.04 μm, the absolute error and relative error being 0.04 μm and 0.4%, respectively. The result proves that a higher precision of glare point separation is obtained using the autocorrelation combined with Gaussian interpolation.
Fig. 5 Simulation results of the glare point separation estimation: (a) glare point image; (b) auto-correlation value of (a); (c)first-order auto-correlation value and Gaussian interpolation.
Furthermore, we also numerically explore the performances of estimated spacing by means of Monte Carlo simulation for different particle diameter at Gaussian noise of 5 dB. Figure 6 shows the error curve of the estimated spacing obtained by 1000’s of Monte Carlo simulations for Gaussian interpolation, m-Gaussian interpolation, cubic B-spline interpolation, and Gaussian fitting. It can be seen from Fig. 6 that these interpolation algorithms hold superiority in the RMSE (root mean square error) and MAE (mean absolute error) except Gaussian fitting. Gaussian interpolation is chosen in this paper.
5. Experimental results
We have performed a simulated result to verify the accuracy of glare point image positioning and extraction separation. Now we test the method experimentally, for both particle location detection and particle size measurement accuracy. The particle positioning experiment includes simulation and standard particle experiments.
5.1 Detection of glare-point image center
5.1.1 Simulation experiment
It is known that the exact particle positions are unknown for a real particle field. We first need to utilize simulation experiment to test the algorithm of the particle center detection, the experimental method is exactly the same as the previously proposed one in Ref [20]. which is used to test the algorithm of the particle center detection for fringe pattern. Figure 7 shows the experimental set-up designed, the detailed experimental setup and method can be described in Ref [20], here the attenuator is used instead of a filter with a high transmission wavelength of 632.8 nm, and the white light is used in our presented experiment. Note that the glare-point images presented here are made with the same method of section 4.2, and each glare point pair as a particle image in the in-focus image plane.
Figure 8 shows the results produced by the processing algorithm. Figure 8(a) is an example of a simulated 30 glare-point pairs randomly distributed in a field of view of 1024 pixels × 768 pixels, in which all the particle locations are known a priori, and the glare-point separation varies from 6 to 16 pixels. Figure 8(b) is the image of Fig. 8(a) in the CCD, which is the amplification of the inverted real image. The center of each glare point pair in Fig. 8(b) is extracted by mask matching algorithm, denoted with red dots, also shown in Fig. 8(b), three mask images with the separation of 8, 12, and 16 pixels are used in locating. Let the preset glare point pair center coordinate in Fig. 8(a) be, where n is the particle serial number, namely, the first disk coordinate is, and the second of and so on. Take any pair position in Fig. 8(a) as starting position, and the distance between the first pair and any other is given by. Similarly, the corresponding glare point pair center coordinate extracted in Fig. 8(b) is, and the center coordinate of the corresponding starting position is, and thus we can obtain . For every particle pair, the ratio of the distance, , is calculated, and theoretically should be equal. The value of = 19.41 is calculated for the experimental system after calibration. Figure 8(c) showsfor particle counts of 15, 30, and 50. The valuesare 19.38 ± 0.009, 19.36 ± 0.012, and 19.37 ± 0.011 for particle counts of 15, 30, and 50, respectively. The maximum absolute error and relative error are 0.052 and 0.27%, respectively. The mask matching algorithm clearly has quite high center-detection accuracy in the experiment.
Fig. 8 (a) A simulated glare-point images; (b) the image of (a); (c) the result of the center location for different particle counts.
Although the experiment here is not identical to the real particle field experiment and particle is not a true particle, we believe that the result may have proved the validity of the location of extraction algorithm.
5.1.2 Standard particle measurement
To further evaluate the method presented experimentally, we complete the standard particle measurement, the experimental setup shown as in Fig. 1(b) is assembled in Ref [22]. A 532 nm CW semiconductor laser with the maximum power of 1.5 W is used as a light source, and a laser beam, expanded, spatially filtered, collimated, compressed by a pair of cylindrical lenses, becomes a vertical sheet with a thickness of 0.9 mm, and is divided into two light beams of the same intensity by a splitter. The CCD is a 10-bit digital CCD sensor (GRAS-14S5M/C) with 2448 pixels × 2048 pixels and pixel size of 3.45 μm × 3.45 μm. The imaging lens is a EF 180 mm f/3.5L Canon lens. In experiment, the particles are immersed in deionized water, the scattering angle is set at θ = 90°, the measuring magnification M = 3.746. The particles used in the experiment are of the nominal diameter 14.9 μm (NT29N11699), 25.0 μm (GBW(E)120027) and 51.0 μm (GBW(E)120046), and the refraction index of 1.589.
Figures. 9(a)–9(c) are respectively a captured glare point images with the diameters of 14.9, 25.0 and 51.0 μm. As shown in Figs. 9(a)–9(c), the glare point separation ∆l is the same for each figure, due to the fact that θ = 90°, M is the same, so for Figs. 9(a)–9(c), using Eq. (4), the mask images with the separation of 11, 19 and 39 pixels are respectively selected for particle locating, and the locating results are correspondingly shown in Figs. 9(d)–9(f), respectively. The particle images which are seen with naked eyes are almost identified. As the exact particle positions are unknown, we now cannot say that the particles are precisely positioned. Figure 10(a) shows a synthesized focused image which is produced by adding Figs. 9(a), 9(b) and 9(c), pixel by pixel. Three mask images with the separation of 11, 19 and 39 pixels are simultaneously used, the result of particle locating is shown in Fig. 10(b). Figure 10(c) displays the spatial distribution of Fig. 10(b) extracted, and the spatial distribution of Figs. 9(d)–9(f) extracted is also shown in Fig. 10(c). It can be seen that the particles in Figs. 9(d)–9(f) detected are completely extracted in Fig. 10(b), and the location coordinates are identical completely. This result indirectly indicates the performance of our locating algorithm, which not only can accurately locate the single particle size but also can accurately locate the wide distribution of particle size, such as spray field.
Fig. 9 Results of center location of standard particles: the images record (top row) and the corresponding positioning results (bottom row), the diameter of (a), (d) 14.9 μm; (b), (e) 25.0 μm, and (c), (f) 51.0 μm.
Fig. 10 The locating results of standard particles: (a) a synthesized image; (b) the result of the center detection of (a); and (c) the extracted center coordinates (x, y) of Fig. 8(d), 8(e), 8(f) and 9(b).
5.2 Particle size measurement
The in-focus images shown as in Figs. 9(a)–9(c) are used to measure the particle size. After extracting the glare points of each particle, and calculating the auto-correlation for each focused image, the glare-point separation of a particle is obtained by Gaussian interpolation, then particle diameter can be obtained by using Eq. (4). Figures. 11(a) and 11(b) show, respectively, the auto-correlation value, its first-order peak and the result of Gaussian interpolation. The interval of 0th and 1st peak of the auto-correlation value corresponds to the glare-point separation ∆l, and the estimated separation ∆l = 18.83 pixel, and then the particle diameter is calculated using Eq. (4) and d = 24.50 μm, the absolute error and relative error being 0.50 μm and 2.0%, respectively. Each particle in Fig. 9(b) is analyzed, and the particle size distribution is shown as in Fig. 11(c). The particle diameter d = 25.25 ± 0.86 μm, the absolute error is 0.25 μm and the relative error is 1.0%. The result shown as in Fig. 11(c) is fitted by Gaussian fitting method, denoted with green lines, and the peak particle diameter d = 25.25 μm, the absolute error is 0.25 μm and the relative error is 1.0%.
Fig. 11 Results of particle size measurement of 25.0 μm: (a) distribution of auto-correlation intensity; (b) fine peak detection and spacing estimation; (c) particle size distribution and peak particle diameter.
Finally, we test the algorithm presented using a lot of experimental images with diameter of 10.7, 14.9, 15.1, 21.4, 25.0, 44.7 and 51.0 μm, the overall result of the glare-point image locating and spacing estimation are similar to the one presented in Fig. 11, and is thus very globally satisfactory, then the particle diameter is calculated using Eq. (4). Figure 12 plots the correlation value between the measured peak particle diameter and the nominal peak particle diameter, and the data points are fitted by linear fitting, and the slope of 0.989, the relative error of the measured particle diameter is 1.1%. The high accuracy of particle size measurement can be obtained.
In addition, we do not take account of sources of noise present (particle non-sphericity, diffraction effect, the laser sheet thickness, etc.), the M parameter error is relatively small due to large M (M = 3.746). The final precision of the particle diameter measurement has been improved in our experiment, and the method presented is very powerful in particle sizing.
6. Conclusion
In this paper, we focus on an algorithm for automatically and simultaneously extracting particle location coordinates and glare-point separation based on the mask matching and autocorrelation method, sub-pixel accuracy of the extracted spacing is obtained with a Gaussian interpolation, and the high accuracy of the position coordinate can be determined by using serial particle mask. The performance of algorithm is estimated by numerical simulations and experimental measurements. The absolute error and relative error of particle location in the simulation experiments are less than 0.052 and 0.27%, respectively. The relative error of the measured particle diameter is less than 1.1%. Additionally, in the present algorithm, the issues of glare point overlapping can be nicely dealt with by mask template selection, which will be reported in other papers. In summary, the method presented in this paper is highly accurate both in particle location detection and particle size measurement. The focused image method of IPI presented in this paper, by which the number density of the measurable particles increases compared with the method of interferometric image, is promising for its application to the high-density spray field.
Funding
National Natural Science Foundation of China (NSFC) (61275019).
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