Volumetric Velocimetry via Scanning Back-Projection and Least-Squares-Matching Algorithms of a Vortex Ring

Transcription

1 Volumetric Velocimetry via Scanning Back-Projection and Least-Squares-Matching Algorithms of a Vortex Ring Benjamin Ponitz 1,*, Mark Sastuba 1, Christoph Brücker 1 and Jens Kitzhofer 2 1: Institute of Mechanics and Fluid Dynamics, University of Freiberg, Germany 2: Dantec Dynamics A/S, Skovlunde, Denmark * correspondent author: Benjamin.Ponitz@imfd.tu-freiberg.de Abstract In the following paper a volumetric reconstruction method is applied on a vortex ring. In the future the evaluation effort of 3D flow studies becomes more widespread due to higher pixel resolutions of the camera set-ups. Most reconstruction approaches use iterative methods which need a huge processing overhead. This work presents the advantages of combining a scanning technique with a fast reconstruction procedure. A back-projection algorithm and a Least-Squares-Matching method are applied to gain the threedimensional velocity distribution with a multiple camera configuration. Thereby the scanning backprojection method is used to reconstruct the entire measuring volume from each captured camera image. Besides the required calibration an additional computation of a disparity error correction improves the reconstruction accuracy significantly. The velocity field evaluation of the measuring volume is performed by a Least-Squares-Matching Algorithm. The results of the volumetric velocimetry method illustrate the timeresolved behavior of an unsteady vortex ring in a 7x5x5 cm³ area. 1. Introduction For the time-resolved and three-dimensional evaluation of flow fields the amount of information increases continuously due to higher pixel resolutions of the camera set-ups. Hence the requirements of the analysis procedure become more and more challenging, especially the processing time. An extensive overview of volumetric velocimetry methods based on PIV is given in [4]. Therein the working principle of Tomographic PIV is presented in a well detailed way by implementing a common Multiplicative Algebraic Reconstruction Technique (MART). In [7] the advantages and disadvantages of this method are discussed. To reduce reconstruction ambiguities a camera set-up of four cameras is used. Furthermore this approach is an iterative procedure which needs a huge processing overhead. This paper introduces the advantages of combining a scanning technique with a fast reconstruction procedure. It shows that a time-resolved and three-dimensional velocity distribution is gained with a multiple camera configuration of three cameras by applying an analytical back-projection algorithm [2,3] and a Least-Squares-Matching (LSM) method [6,8]. 2. Experimental Set-up The experimental set-up is shown in Figure 1. The laser beam of a continuous Argon-Ion laser (Figure 1a) Coherent Innova 70 (3 W) passes an optical lens system (Figure 1b) to adjust the desired thickness of the light sheets. A rotating mirror drum (Figure 1c) generates successively 10 parallel light sheet planes with a thickness of 10 mm and a plane overlap of 2 mm (Figure 1f). This series of consecutive light sheets leads to a voluminous illumination. The particle images are captured with a synchronized three camera system consisting of digital high speed cameras with a resolution of 1280 x 800 Pixel² and an angular displacement of roughly 45, 90 and 135. The cameras are equipped with telecentric lenses by Sill Optics. The aperture is fully closed to obtain a depth of focus of approximately 10 cm. The observed flow is a vortex ring travelling in an - 1 -

2 octagonal glass tank filled with water (Figure 1e). The vortex is generated at the exit of a piston tube with a diameter of 30 mm (Figure 1g). The neutrally buoyant seeding particles (100 microns) are injected into the center of the vortex generator. To eliminate the travelling speed of the vortex ring related to the camera, the complete glass octagon is moved with the same travelling speed (5 cm/s) but in the opposite direction of the vortex ring. Hence the vortex ring stays in focal depth of the cameras while capturing. The images are taken with a recording rate of 123 frames/s, resulting in a mean particle displacement in the image planes of 8 pixel. The image sizes of the cameras and the thickness of the scan define the measured volume. The recorded volume dimension is defined by the captured image size and the thickness of the light sheets. This leads to a reconstructed volume extension of circa 7x5x5 cm³. [5] Figure 1 Experimental set-up: (a) Ar-Ion-laser, (b) lens system, (c) rotating mirror drum, (d) high speed camera, (e) moving glass octagon, (f) light sheets, (g) vortex generator. 3. Scanning Back-Projection Figure 2 illustrates the tomographic reconstruction procedure. At first the measuring volume is divided in smaller planes by using a scanning illumination. Secondly each illumination sheet has to be discretized into voxels. Next, the voxel elements obtain the grey value information from the captured images by back-projection. As the final reconstruction step the previously subdivided illumination sheets are merged to form an entire continuous volume. Figure 2 Tomographic reconstruction procedure 3.1 Scanning Illumination The measuring volume is gradually subdivided into 10 parallel illumination planes (light sheets) with a certain thickness of 10 mm and with an overlap of 2 mm. This leads to a faster processing speed for the back-projection method on the one hand and to a reduction of ghost phenomena while reconstructing on the other hand

3 3.2 Discretization of the Measuring Volume In this step the measuring volume or more specifically each illumination sheet is discretized into a certain three-dimensional equidistant grid of voxel elements with a resolution of 1000 voxel per mm³. A voxel is the smallest describable grey value filled element of a volume, equivalent to the pixel of an image but with an additional spatial extension. Accordingly a particle is defined by a certain amount of adjoining voxel-elements. 3.3 Back-Projection Algorithm This section describes the main step of the reconstruction process. The reconstruction method relies on the reverse case of a projection. Thus it is called back-projection (Figure 3). The information from a pixel within the image plane is transferred back into a voxel element of the measuring volume. Figure 3 The so-called back-projection method relies on the reverse case of a projection from a scene into an image plane. The relation between pixel and voxel information is determined within the projection matrix which allows the back-projection of pixel information from an image back into voxel information of the measuring volume. The entire back-projection process fills each voxel with the pixel information from each camera chip. For the correct allocation of the voxel grey values a calibration is crucial. For this calibration the three-dimensional coordinates of a certain number of reference points W (x o,y o,z o ) is necessary. These points are given by a reference object. With the coordinates of the projection into the image point M (x I,y I ) the calculation process is done by M = P * W. (1) Due to calibration the relation between pixel location in the image plane and voxel location in the volume is determined within the projection matrix P. Hence the transformation from 2D to 3D which is given by P is important for the reconstruction process via scanning back-projection. Thus the accuracy of the projection matrix determines the accuracy of the reconstruction process. The current maximum deviation of the projection matrix after the calibration procedure amounts 2 to 3 pixel. This displacement error of the reconstructed voxel position leads immediately to a further error of the calculated three dimensionally velocity field. For that reason the accuracy of the coordinates is improved by using a disparity error map to gain a deviation of 0.1 pixel. The disparity error is the displacement dl (dx, dy) of the calculated coordinates N (x P,y P ) and the coordinates of the image point M (x I,y I ). The value of the disparity error depends on the location of the reference point W in the voxel volume. Hence it is essential to have disparity information of - 3 -

4 every position. Therefore the calibration is done for various positions in the whole voxel volume. The area between the individual calibration layers, so-called disparity error map, is interpolated to achieve additional disparity information. For that reason the calibration procedure is done for several selected layers (Figure 4). Figure 4 Whole measuring volume with colored visualization of the disparity error distribution for the calibration layers. The space between these layers is interpolated to achieve additional disparity information. Figure 5 One exemplary disparity error map for all three cameras shown in vector-view (left) and in colored 3Dvisualisation (right)

5 Figure 5 shows one exemplary disparity error map for all three cameras with vector visualization on the left-hand side and with color visualization on the right-hand side. Due to this accuracy enhancement the reconstruction method Scanning Back-Projection leads to a high resolved spatial particle distribution. This reconstruction method basically requires at least two cameras which capture the scene from different perspective. However, in a configuration of two cameras the effect of ghost particles will appear during the reconstruction process (Figure 6). A three-camera configuration is used to eliminate these ghost phenomena. Figure 6 Ghost particle reduction by using a three camera configuration. The grey value of the position in a third camera (Cam2) is negligible low. Hence the sum of the grey value intensities of the real particle I voxel (red colored dot) and ghost particle I voxel,ghost (white colored dot) will be significantly distinguished from each other. Thus a simple threshold criterion allows eliminating the detected ghost particle. An additional third camera assesses if the reconstructed particle is a real or a ghost one by using a grey value comparison. In the second case the particle will not be considered for the final reconstruction process. The selective function works with comparison of the sum of the logarithmic grey values of each camera: I voxel = ( log b I cam,i ). (2) This ensures a maximum voxel grey value I voxel after the logarithmical summation which is equivalent the maximum pixel value I cam in the individually captured image of a particle. In the case of a 8-bit system of grey value distribution (0 255) and a three camera configuration the voxel grey value I voxel is set to a maximum value of 255 with the basis of b = The sum of the grey values for the real and ghost particle will be the same in the case of a two camera configuration (Cam1 and Cam3). This is because both particles belong to the same optical line of the camera sensor. The grey value of the position in a third camera (in this case Cam2) where the ghost particle is not pictured is consequentially negligible low (comparable to background noise). Due to the application of a third camera the sum of the grey value intensities of the real particle I voxel (red colored dot) and ghost particle I voxel,ghost (white colored dot) will be significantly distinguished from each other. Furthermore the logarithmic grey value function increases the gap between real particle intensity and ghost particle intensity significantly which allows to use a simple threshold criterion to eliminates the detected ghost particle. Therefore the combination of a third camera and logarithmic grey value function enables the reduction of ghost particles

6 Due to the implementation of the disparity error map and the resulting enhanced accuracy each voxel in the discretized volume can be identified in the image planes of each camera. In contrast to other iterative applied algorithm like MART (Multiplicative Algebraic Reconstruction Technique), e.g. in [1,4], each voxel is grey value charged only once. This ensures a significant reduction of processing time. The limit for this reconstruction method is determined by the particle density in the measuring volume. Although the correction of the volume coordinates is improved by using a disparity error map it could be possible to get additional noise in the reconstructed area caused by very high particle density (ρ p > 0.02 particles per pixel [ppp]). For that reason the investigated volume has to be subdivided in small light sheet planes which are realized with a scanning technique [5]. Hence the limiting particle density is now related to the smaller light sheet volume. In the case of dividing into 10 light sheets a 10 times higher amount of particles for the whole measuring volume can be achieved: ρ p,volume = n lightsheets * ρ particle,lightsheet = 0.2 ppp. (3) 3.4. Merging Illumination Sheets The previously applied scanning technique divided the measuring volume in 10 parallel illumination sheets. In the last reconstruction step these light sheets have to be merged to form an entire reconstruction volume (Figure 7). For the merging function the overlapping section between two adjacent light sheets are necessary to obtain a continuous transition region. Figure 7 Merging illumination sheets is necessary to obtain a continuous transition region in the overlapping section. The final reconstructed and merged volume is shown in Figure 8 for two time steps. Due to the higher particle density within the vortex the particle distribution shows the outer shape of the vortex ring. This is caused by the injection of a higher particle seeded vortex ring in a lower seeded surrounding fluid of the tank

7 16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Figure 8 Reconstructed particles of two time steps. The voxel volume has a size of 700x500x519 vox³ by a voxel resolution of 0.1 mm/vox. The analyses are performed on a machine with 2 Six-Core AMD Opteron processors at 2.6 GHz, and 32 GB RAM. The reconstruction of each time step with a voxel volume size of 700x500x519 vox3 can be realized by a processing time of 35 minutes. The resolution of the reconstruction is 1000 Voxel per mm3. 4. Velocity Field Evaluation by Least-Squares-Matching The further evaluation of two subsequent reconstructed grey value filled voxel volumes with a Least-Squares-Matching (LSM) algorithm [6,8] leads to a three-dimensional velocity field. The resulting velocity field consists of approximately vectors. Figure 9 shows the threedimensional velocity distribution of a vortex ring for three time steps. The temporal development shows the transition from a steady to and unsteady vortex flow (left to right). The iso-surfaces illustrate the vorticity ωabs = 0.47 s-1 (black colored), the velocity magnitude Vmag=0.014 m/s (yellow colored) and the velocity in y-direction v1=0.06 m/s (light red colored) and v2=0.07 m/s (red colored). The essential elements of the vortex ring (stagnation points, vortex cores, evolution of instabilities) are visualized. Figure 9 Visualisation of the three-dimensional velocity distribution of a vortex ring for three time steps. The temporal development shows the transition from a steady to and unsteady vortex flow (left to right). The iso-surfaces illustrate the vorticity ωabs = 0.47 s-1 (black colored), the velocity magnitude Vmag=0.014 m/s (yellow colored), and the velocity in y-direction v1=0.06 m/s (light red colored) and v2=0.07 m/s (red colored). The essential elements of the vortex ring (stagnation points, vortex cores, evolution of instabilities) are visualized. -7-

8 5. Conclusion The evaluation effort via iterative methods of the increasing investigation volumes becomes more and more time-consuming. This work presents that a scanning technique enables to use a faster particles reconstruction procedure, so-called scanning back-projection. Due to an enhanced calibration process including a disparity error map it was shown that the reconstruction accuracy could be improved which resulted in a reduction of ghost particles. The results of the volumetric velocimetry method illustrate the time-resolved behavior of an unsteady vortex ring in a 7x5x5 cm³ area. 6. References [1] Atkinson CH, Soria J (2007) Algebraic Reconstruction Techniques for Tomographic Particle Image Velocimetry, 16 Austrailian Fluid Mechanics Conference. [2] Calluaud D, David L (2002) Backward Projection Algorithm And Stereoscopic Particle Image Velocimetry Measurements Of The Flow Around A Square Section Cylinder. International Symposium on Applications of Laser Techniques to Fluid Mechanic. [3] Coudert JM, Schon JP (2001) Back-projection algorithm with misalignment corrections for 2D3C. Meas. Sci. Technol. 12: [4] Elsinga GE, Scarano F, Wieneke B, van Oudheusden BW (2006) Tomographic particle image velocimetry. Experiments in Fluids: 41: [5] Kitzhofer J, Kirmse C, Brücker Ch (2009) High Density, Long-Term 3D PTV Using 3D Scanning Illumination and Telecentric Imaging. Imaging Measuremnt Methods: NNFM 106: [6] Kitzhofer J, et al. (2010) Estimation of 3D Deformation and Rotation rate Tensor from volumetric particle data via 3D Least Squares Matching. 15th Int Symp on Applications of Laser Techniques to Fluid Mechanics. [7] Petra S, Schnörr Ch, Schröder A, Wieneke B (2007) Tomographic Image Reconstruction in Experimental Fluid Dynamics: Synopsis and Problems. [8] Westfeld P, Maas HG, Pust O, Kitzhofer J, Brücker Ch (2010) 3D least square matching for volumetric velocity data processing; Proc. 15th Int Symp on Applications of Laser Techniques to Fluid Mechanics