3D Particle Position Reconstruction Accuracy in Plenoptic PIV

Transcription

1 AIAA SciTech January 2014, National Harbor, Maryland 52nd Aerospace Sciences Meeting AIAA D Particle Position Reconstruction Accuracy in Plenoptic PIV Timothy W. Fahringer and Brian S. Thurow Auburn University, AL, 36849, USA The particle reconstruction capabilities of a novel 3-D, 3-C PIV technique, based on volume illumination and a plenoptic camera to measure a velocity field, are tested. This technique is based on light-field photography, which uses a dense microlens array mounted near a camera sensor to sample the spatial and angular distribution of light entering the camera. Tomographic algorithms (MART) are then used to reconstruct a volumetric intensity field after the image is taken, and cross-correlation algorithms extract the velocity field from the reconstructed volume. This paper provides an introduction to the concepts of light fields and describes the tomographic algorithms used to reconstruct the measurement volume. A preliminary test on the accuracy of single particle reconstructions is presented, showing that laterally we can expect errors to be less than a voxel for most cases. A test on the reconstruction quality is presented for a volume of particles as a function of particle density and is shown to vary little as the particle density is increased. Finally a simulated Gaussian ring vortex is presented showing a full simulation as well as the velocity accuracy. I. Introduction Experimentally quantifying the topology of unsteady coherent flow structures in turbulent flows remains at the forefront of fluid mechanics research. The inability of planar methods, such as particle image velocimetry (PIV), to describe this phenomena can be directly attributed to their 2D nature. An instantaneous fully three-dimensional (3D), three-component (3C) velocity field would be instrumental in the quantification of turbulent flows. The need for 3D techniques has led to numerous efforts over the years with advances such as stereoscopic-piv 1 and dual-plane stereoscopic-piv 2 which allow 3C measurements within a 2D plane. Since these methods only capture 3C data within one or two two-dimensional planes, they are not considered truly three-dimensional. Four techniques that are capable of acquiring fully 3D, 3C velocity fields are defocusing PIV, 3, 4 holographic PIV, 5 tomographic PIV, 6 and synthetic aperture PIV. 7 This effort is a continuation of the work by Lynch, 8 Lynch and Thurow, 9 Lynch et al., 10 and Fahringer and Thurow. 11 In Lynch, 8 a preliminary analysis of particle imaging using a plenoptic camera, with a focus on refocusing was described. Using only synthetic data a plenoptic simulator was designed as well as the first attempt at reconstructing a volume from the light field data. The method used was similar to that in synthetic aperture PIV and is based on refocusing the light field at multiple focal planes, then thresholding the data such that only the bright in-focus particles remain. Building off of these tools Lynch and Thurow 9 describes a home built plenoptic camera, demonstrates the refocusing/thresholding reconstruction and cross-correlation procedures, as well as the first experimentally captured images with the prototype camera. These experimentally captured images were refocused at different focal planes, verifying the plenoptic camera concept. Starting with Lynch et al. 10 the refocusing/thresholding technique was replaced with a direct tomographic approach. It was found that the refocusing/thresholding technique is limited to imaging a sparse number of high intensity particles. This work detailed a preliminary attempt at tomographic reconstruction and provided a synthetic velocity field of an Oseen vortex. Fahringer and Thurow 11 detailed the unique tomographic algorithms used for reconstruction of a volume using a plenoptic camera and tested the algorithms on both synthetic and experimental data. This work will focus on accessing the ability of the tomographic reconstruction algorithm to reconstruct volumes of particles. The first case considered is Graduate Research Assistant. Associate Professor, 211 Davis Hall, thurow@auburn.edu 1 of 10 Copyright 2014 by Brian Thurow. Published by the, Inc., with permission.

2 that of a single particle, representing the best case scenario in terms of accuracy, as well as providing an understanding of the limitations in the method. For this case the accuracy is presented in terms of the absolute error in position accuracy. Subsequently, large numbers of particles are simulated and their accuracy is assessed using the reconstruction quality factor, Q. Finally a synthetic Gaussian ring vortex is generated, reconstructed, and the accuracy of the final velocity measurement is presented. II. Light Field Imaging The term light field is used to describe the complete distribution of light rays in space, and can be described by a 5-D function, sometimes termed the plenoptic function. Each ray in the light field can be parameterized in terms of the plenoptic function as its position (x, y, z) and angle of propagation (θ, φ). This function can be simplified if the light field is in a transparent medium, such as air, where the propagation along one of the spatial coordinates is assumed to be a straight line and therefore redundant, resulting in a 4-D parameterization of the light field denoted as L F (x, y, θ, φ). The modern concept of light field imaging with a plenoptic camera started with Adelson and Wang 12 and continued by Ng. et al. 13 for handheld photography and Levoy et al. 14 for microscopy. In contrast to conventional photography, which only captures the spatial distribution of the 4-D light field, light field photography can capture the full 4-D light field. As described in Levoy, 15 one of several ways to capture the light field is to use a microlens array mounted near a CCD to encode the this information onto a single sensor. This device, termed the plenoptic camera, is the focus of this work. A conventional camera, depicted in Figure 1a, maps a spatial location (x, y, z) on the world focal plane to another spatial location (x p, y p ) at the sensor plane. In reference to Figure 1a the gray area represents the path of light rays emanating from the point (x, y, z) going through the main lens aperture and onto the sensor. Since all rays terminate at the same location on the sensor the angle at which each ray propagated cannot be determined. In contrast, a plenoptic camera maps a spatial location (x, y, z) on the world focal plane to a spatial location (x p, y p ) at the microlens plane which then focuses the light onto the sensor plane. Figure 1b shows how one pixel views the point (x, y, z), where the green shaded area is the pixel s line of sight. Figure 1c shows all pixels associated with the point on the world focal plane and their unique lines of sight on the scene. Each pixel, with its unique viewing angles on the point, is thus representative of the angular distribution (θ, φ) of the light field. This along with the microlens array capturing the spatial information allows the plenoptic camera to capture the entire 4D light field. (x,y,z) World Focal Plane Aperture Plane (u,v) or (θ,ϕ) (x,y,z) World Focal Plane Aperture Plane (u,v) or (θ,ϕ) (x,y,z) World Focal Plane Aperture Plane (x p,y p ) Sensor Plane (x p,y p ) Microlens Plane Sensor Plane (x p,y p ) Microlens Plane Sensor Plane (a) Conventional camera (b) Single pixel s line of sight (c) All pixel s line of sight Figure 1: Difference between a conventional and a plenoptic camera 2 of 10

3 III. Reconstruction Algorithm To reconstruct a volumetric intensity field useful for PIV, tomo-piv principles are used with appropriate modifications. The working principle of tomo-piv as detailed in Elsinga et. al. 6 involves immersing tracer particles into a flow-field, and illuminating the particles within a 3-D region of interest using a pulsed light source. The light scattered from the particles is then recorded onto a plenoptic camera. The 3D particle fields are reconstructed from the images obtained in the recording, then the velocity field is determined from the displacement of the particles calculated using a 3D cross correlation algorithm. The reconstruction of the particle fields is in general both ill-posed and under-determined leading to ambiguity in the solution. A special class of reconstruction algorithms are better suited for these problems and are known as algebraic methods as described by Herman and Lent. 16 These methods rely on iteratively solving a system of linear equations which model the imaging system. As with conventional tomo-piv the 3D volume to be reconstructed is discretized into cubic voxel (volume equivalent of a pixel) elements, with intensity E(x, y, z). The size of the voxel was chosen to be similar to that of a microlens, since they govern the spatial resolution of a plenoptic camera. The problem can be stated as the projection of the volume intensity distribution E(x, y, z) onto a pixel located at (x i, y i ) yields the known intensity of that pixel I(x i, y i ). In equation form this is given by w i,j E(x j, y j, z j ) = I(x i, y i ) j N i where N i represents the number of voxels in the line-of-sight of the ith pixel. The weighting function w i,j describes the relationship between the recorded image (ith pixel) and the 3-D volume of interest (jth voxel), and was detailed in Fahringer. 11 In order to solve this set of equations, techniques have been developed that update the current solution for E based on the previous relation. For additive techniques such as ART 16 (algebraic reconstruction technique) the update is based on the difference between the image intensity data and the projection of the volume such that when they are equal the update added to the solution is zero. For multiplicative techniques such as MART 16 (multiplicative algebraic reconstruction technique) the update is based on the ratio of the image intensity data to the projection of the volume such that when they are equal the update multiplied to the solution is unity. The algorithm used in this work is the standard MART algorithm for its ability to accurately reconstruct particles. Starting from an initial guess of the volume E(x, y, z) 0 = 1 MART is updated via the following relation E(x j, y j, z j ) k+1 = E(x j, y j, z j ) k I(x i, y i ) j N j w i,j E(x j, y j, z j ) k where µ is the relaxation parameter which must be less than or equal to one. The exponent restricts the updates to parts of the volume affected by the ith pixel. IV. Synthetic Particle Reconstructions As an illustration of the reconstruction process a small volume (cube with sides of mm) of 20 particles was generated with random positions, synthetically imaged using the plenoptic simulation tool (Lynch 8 ), and reconstructed using the MART algorithm. For this exercise a small plenoptic camera consisting of a 1000 x 1000 pixel image sensor behind a microlens array of 60 x 60 lenslets was used to cut down on the computational time. The rest of the parameters are the same as the full scale simulations as well as the prototpye camera built at Auburn University and are shown in Table 1. The image simualated from the volume of particles is shown in Figure 2. µw i,j 3 of 10

4 As a means of comparison a volume using the actual particle positions was generated using a 3 x 3 x 3 voxel Gaussian blob fit to the particles position. The final reconstruction of the particles is shown in Figure 3 and the true particle positions are shown in Figure 4. Figure 3b shows a front view of the reconstructed volume. When compared to the actual particle positions (Fig 4b) the reconstructed particles are shown to match the actual particles in both size and location. Alternatively, when the reconstructed particles are compared to the actual particles in depth (Figs 3c & 4c) they are shown to match locations, but the reconstructed particles are elongated in depth. This can be attributed to the limited range of angles that a plenoptic camera has on the scene. Fortunately, the intensity in depth is not a constant. Figure 5 shows a single reconstructed particle iso-surface as well as a slice through the center of the particle on the YZ plane. The particle has a hot center with decreasing intensity at the front and back as shown in Figure 5b. This allows for resolution of the location of the Figure 2: Synthetic Raw Image center of the particle in depth, where a constant intensity would create a large ambiguity. The lateral spatial resolution of this particles reconstruction is limited to a single voxel. For other particles this may be four voxels or larger depending on their location spatially as well as in depth. In particular, the reconstruction of a particle far away from the focal plane is more elongated in depth and blurred spatially. (a) Isometric view (b) Front (XY) (c) Top (YZ) Figure 3: Example tomographic reconstruction of a synthetic particle field generated from synthetic image shown in Figure 2 (a) Isometric view (b) Front (XY) (c) Top (YZ) Figure 4: Actual particle positions used to generate synthetic image shown in Figure 2, represented as a 3D Gaussian blob 4 of 10

5 (a) 3D Iso-metric view of a single particle (b) Slice through center of particle Figure 5: View of single particle reconstruction A. Single Particle Accuracy Using the synthetic image generation technique mentioned previously 40 particles are simulated (generating 40 different images) 1 mm (8 voxels) apart from each other in depth along the optical axis of the camera. The volume for each reconstruction was kept constant, such that the weighting matrix was identical for each reconstruction. The volume of size x x mm was discretized into a grid of 75 x 75 x 402 voxels creating cubic voxel elements of size x x mm. For the reconstruction, a relaxation patameter of 0.75 was used and the MART algorithm was run for 3 iterations. Since the particle locations are known, the error in the reconstructed particles can be calculated. To precisely determine the particle location with sub-voxel accuracy, a 3D Gaussian function was fit to the reconstructed intensity data and the peak location was taken to be the location of the reconstructed particle. The results are shown in Figure 6 with the absolute error (in voxels) on the y-axis and the relative position of the particle to the focal plane of the camera (100 mm away from the lens plane) on the x-axis. The results shown use a nominal magnification of -1, it is noted that the results will vary for other magnifications, however those are not considered in this work. xact xfit (voxels) x y zact zfit (voxels) Particle Location Relative to Focal Plane (mm) (a) X and Y (Lateral) Particle Location Relative to Focal Plane (mm) (b) Z (Depth) Figure 6: Error in reconstruction accuracy via Gaussian fit of 40 particles space 1 mm apart along optical axis Figure 6a reveals the lateral accuracy of the algorithm as a function of depth for this optical configuration. In this case the particle position was perfectly aligned with a voxel, representing the best case scenario. For the region near the focal plane [-10, 10], the mean of the absolute value of the error is 0.49 and 0.4 voxels for 5 of 10

6 x and y respectively, with standard deviations of 0.28 and 0.25 voxels. A notable aberration is the particle located at the focal plane of the camera. This is due to ambiguity in a 1 mm region around the focal plane caused by the nominal depth of field of our camera. More specifically, in this region light emanating from a particle strikes a single microlens whereas in other locations, the light is spread across multiple microlenses. Thus the algorithm does not have the information to interpolate between microlenses. The MART algorithm spreads the intensity throughout this region often leaving two peaks: one before and one after the focal plane. This results in the 1 voxel error shown.further away from the focal plane the algorithm is shown to be less accurate, however the absolute error is only 1.5 voxels. There is some noticeable peak locking occurring causing the solution to be forced into a single voxel. The depth accuracy is shown in Figure 6b as a function of depth. In the region near the focal plane [-10 10] the error in depth was shown on average to be 1 voxel, with a standard deviation of 0.74 voxels. Outside of this region the average error is 5 voxels. It is noted that the depth accuracy is worse than the spatial accuracy as is to be expected. Using the same Gaussian fit for determining the particle positions, some statistics on the shape of the particles can be obtained. The function used for the Gaussian fit is given by ( ( )) (x x o ) 2 f(x, y, z) = A exp 2σ 2 x + (y y o) 2 2σ 2 y + (z z o) 2 2σ 2 z where A is the amplitude, the point (x o, y o, z o ) is the center of the blob (taken to be the center of the particle), and σ is the width of the particle in a particular direction. Figure 7a shows σ x and σ y plotted as a function of depth. Only considering the region of [-10 10], the average width for the lateral positions x and y was calculated to be and voxels respectively. Figure 7b depicts σ z as a function of depth. In the same [-10 10] region the average width of the particle in depth was shown to be voxels. Using these average parameters a representative Gaussian blob is determined and is used as the comparison for further tests. σ Particle Location Relative to Focal Plane (a) X and Y (Lateral) x y σ Particle Location Relative to Focal Plane (b) Z (Depth) Figure 7: Error in reconstruction accuracy via Gaussian fit of 40 particles space 1 mm apart along optical axis An extension to the single particle test is to calculate the reconstruction error in multiple particles simultaneously. For this test a volume of size 30 x 20 x 20 mm discretized into 300 x 200 x 200 voxels was used. The camera parameters for this and subsequent simulations are shown in Table 1. Inside the volume 558 particles (0.01 particles per microlens) were randomly positioned and an image was generated. This is still a relatively small particle density, however the purpose of this test is to obtain the accuracy of individual particles in the presence of additional particles. To determine the error in the reconstruction a sub-volume around the area of a known particle location was extracted (sub-volume was of size 6 x 6 x 30 voxels), and fit with a Gaussian blob yielding the peak location. This is subtracted from the actual known location of the particle resulting in the absolute reconstruction error of the particles. A plot of the absolute X error v. absolute Z error is shown in Figure 8. The absolute error in X has a mean of voxels and a standard deviation of voxels. The absolute error in Z has a mean of voxels and a standard deviation of voxels. This is consistent with the single particle data in the range of depths used. 6 of 10

7 X Error (voxels) Z Error (voxels) Figure 8: X and Z absolute error in reconstruction of 500 simulated particles B. Full Simulation Reconstruction Quality For a full simulation of particles calculating the reconstruction error for each individual particle is inaccurate. Therefore, to determine the accuracy of the reconstruction process, a statistical measure, known as the reconstruction quality factor is used. This work utilizes the zero-mean reconstruction quality factor, Q defined in La Foy and Vlachos. 17 They demonstrated that as the particle density increased the zero-mean reconstruction quality factor became a more accurate measure than the normal reconstruction quality factor defined in Elsinga et al. 6 The zero mean quality factor is defined as: Ẽ(x, y, z) Ẽ 0 (x, y, z) Q x,y,z = Ẽ(x, y, z) 2 Ẽ 0 (x, y, z) 2 x,y,z where Ẽ(x, y, z) and Ẽ0(x, y, z) are the zero mean reconstructed intensity field and the zero mean exact intensity field respectively. Using the aforementioned quality factor the reconstruction quality is used as a measure of the accuracy as a function of particle density. For comparison the exact intensity volume was created using Gaussian blobs consistent with the average single particle reconstruction.conventionally, particle density is defined as the number of particles per pixel, however for a plenoptic camera the spatial resolution is governed by the microlens array, therefore the results are presented as number of particles per microlens (ppm). For completion the results are also presented as particles per pixel (ppp). The focus of this section is to show the change in quality of the reconstruction as a function of particle density, therefore the results are normalized by the single particle reconstruction quality, Q 0 = The results are shown, with µ = 0.75 after 3 iterations, in Figure 9, with the reconstruction quality on the y-axis and the particle density on the x-axis. The results show that their is little variance with respect to particle density at these particle concentrations. As the particle density approaches 3 ppm the image becomes white and the algorithm no longer produces accurate results. x,y,z Table 1: Plenoptic simulation parameters Parameter Symbol Value Microlens Pitch p l mm Microlens Focal Length f l 0.5 mm Number of Mircolenses: X-direction n lx 289 Number of Mircolenses: Y-direction n ly 193 Pixel Pitch p p mm Number of Pixels: X-direction n px 4904 Number of Pixels: Y-direction n py of 10

8 Q Particles per microlens [ppm] Particles per pixel [ppp] x 10 3 Figure 9: Quality factor as a function of particle density C. Simulated Gaussian Ring Vortex The final test for the accuracy of the reconstruction of a 3D particle field is to test the effect of the errors on the final velocity data. In order to test the velocity a synthetic displacement is applied to a randomized particle field (0.5 ppm) and two synthetic images are acquired. The displacement field used is a Gaussian ring vortex. The vortex core s centerline is aligned with the y-axis and is located at the center of the volume forming a ring with an 8 mm diameter. The tangential displacement (in voxel coordinates) is given by V θ = Γ 2πr ( ) 1 e r2 c θ where r is the distance from the particles location to the center of the vortex. circulation Γ and c θ are defined as Γ = 2πr2 c v rc c θ = r2 c γ The other parameters, where r c, the radius of the vortex, is 40 voxels, the tangential displacement at r c, v rc, is 8 voxels and γ is a constant equal to To compare the reconstructed velocity field, a synthetic particle field was generated using the actual positions of the particles with a 3x3x3 Gaussian blob representing each particle. The synthetic volumes were then run through the same cross-correlation algorithms, thus providing an accurate baseline to test the reconstruction algorithm. Both volume pairs were run through a multi-pass FFT based cross-correlation algorithm with final window sizes of 16x16x16 voxels with 75% overlap. The results of the reconstructed velocity field are presented in Figure 10 and the actual field in Figure 11. Figure 10a shows a cross-section of the vortex in the XY plane at Z = 100 voxels (center of the volume) with velocity vectors and contours of the z component of vorticity. Figure 10b shows a cross-section in the ZY plane at X = 150 voxels with a contour showing the axial component of velocity. When compared to the exact solution (Fig. 11) the solution matches well, but has some spurious vectors in the vortex ring (Fig. 10a), and the axial velocity rings are less well defined (Fig. 10b). The overall RMS error of the velocity field is 1.02 voxels with each component having an RMS error of 0.165, 0.23, and for the u, v, and w components respectively. 8 of 10

9 (a) Cross section of velocity in XY plane at Z = 100 voxels. Contours show the vorticity in the z- direction. (b) Cross section of v velocity shown as a contour in the ZY plane at X = 150 voxels. Figure 10: Cross sections of reconstructed Gaussian ring vortex. (a) Cross section of velocity in XY plane at Z = 100 voxels. Contours show the vorticity in the z- direction. (b) Cross section of v velocity shown as a contour in the ZY plane at X = 150 voxels. Figure 11: Cross sections of simualted Gaussian ring vortex. V. Conclusions Using the previously determined unique weighting function, and MART algorithms a preliminary study of the reconstruction accuracy was conducted for both a single particle as well as a volume of particles. Further optimization is needed to determine the optimal grid size, relaxation parameter, number of iterations for MART, as well as the optimal cross-correlation parameters. As such, we expect the results to improve as we learn more about how these parameters affect the reconstruction. Even so it was shown that the accuracy of a single particle reconstruction was 0.5 voxels in the lateral spatial dimensions (x and y) and 1 voxel in depth on average in the range of [-10 10] mm relative to the nominal focal plane of the camera. The particles where shown to be best described using a Gaussian blob with widths of 0.2, 0.2, and 2 voxels in the x, y, and z directions. When testing a volume of particles the reconstruction quality was shown to vary little as the number of particles increased. Finally, a synthetic Gaussian ring vortex was used to demonstrate the velocity accuracy and the total RMS error was shown to be approximately 1 voxel. Acknowledgments This work has been supported through funding provided by the Air Force Office of Scientific Research, specifically grant FA (program manager: Dr. Doug Smith). The authors would like to gratefully acknowledge Marc Levoy from Stanford University for permission to use a template for manufacturing our microlens array, and for a variety of helpful discussions. Additionally the authors thank Stanley Reeves from Auburn University for continued discussions that have led us to consider the direct tomographic approach. 9 of 10

10 References 1 Arroyo, M. P. and Greated, C. A., Stereoscopic particle image velocimetry, Measurement Science and Technology, Vol. 2, pp Kahler, C. J. and Kompenhans, J., Fundamentals of multiple plane stereo particle image velocimetry. Experiments in Fluids, Vol. 29, 2000, pp. S70 S77. 3 Willert, C. E. and Gharib, M., Three-dimensional particle imaging with a single camera, Experiments in Fluids, Vol. 12, 1992, pp Pereira, F., Gharib, M., Dabiri, D., and Madarress, D., Defocusing digital particle image velocimetry: a 3-component 3-dimensional DPIV measurement technique. Application to bubbly flows. Experiments in Fluids, 2000, pp. S78 S84. 5 Hinsch, K. D., Holographic particle image velocimetry. Measurement Science and Technology, Elsinga, G. E., Scarano, F., and Wienke, B., Tomographic particle image velocimetry, Experiments in Fluids, Vol. 41, Belden, J., Truscott, T. T., Axiak, M. C., and Techet, A. M., Three-dimensional synthetic aperture particle image velocimetry, Measurement Science and Technology, Vol. 21, 2010, pp Lynch, K., Development of a 3-D Fluid Velocimetry Technique Based on Light Field Imaging, Master s thesis, Auburn University, Lynch, K. and Thurow, B., Preliminary Development of a 3-D, 3-C PIV Technique using Light Field Imaging, 29th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, AIAA, Honolulu, HI, June Lynch, K., Fahringer, T., and Thurow, B., Three-Dimensional Particle Image Velocimetry Using a Plenoptic Camera, 50th AIAA Aerospace Sciences Meeting, AIAA, Nashville, TN, January Fahringer, T. and Thurow, B., Tomographic Reconstruction of a 3-D Flow Field Using a Plenoptic Camera, 42nd AIAA Fluid Dynamics Conference and Exhibit, AIAA, New Orleans, LA, June Adelson, E. H. and Wang, J. Y., Single Lens Stereo with a Plenoptic Camera, IEEE Transactions on Parrern Analysis and Machine Intelligence, Vol. 14, No. 2, 1992, pp Ng, R., Levoy, M., Bredif, M., Duval, G., Horowitz, M., and Hanrahan, P., Light Field Photography with a Hand-held Plenoptic Camera, Stanford tech report ctsr Levoy, M., Ng, R., Adams, A., Footer, M., and Horowitz, M., Light Field Microscopy, ACM Trasaction on Graphics Proc: SIGGRAPH, Vol. 25, Levoy, M., Light Fields and Computational Imaging, IEEE Computer, Herman, G. T. and Lent, A., Iterative reconstruction algorithms, Computers in Biology and Medicine, Vol. 6, pp Foy, R. R. L. and Vlachos, P., Multi-Camera Plenoptic Particle Image Velocimetry, 10th International Symposium on Particle Image Velocimetry, Delft, The Netherlands, July of 10