Reduction of reconstructed particle elongation using iterative min-max filtering in holographic particle image velocimetry

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1 Reduction of reconstructed particle elongation using iterative min-max filtering in holographic particle image velocimetry Yohsuke Tanaka 1, *, Shigeru Murata 1 1: Department of Mechanical System Engineering, Kyoto Institute of Technology, Kyoto, Japan * correspondent author: tyohsuke@kit.ac.jp Abstract This paper proposes a method that reduces reconstructed particle elongation using iterative minmax filtering in holographic particle image velocimetry. As an elementary step before applying this method to HPIV, the method was numerically verified at two particles located on the same line in the direction of elongation. We also applied the method to 100 elongated particles distributed at random. The proposed method enables an iterative filtered light intensity to be completely divided into background and particle intensity. 1. Introduction Holographic particle image velocimetry (HPIV) is one of the powerful tools for measuring threedimensional velocimetry field (Meng et al., 2004). However, the error of velocity in depth becomes much larger than that of the in-plane velocity in the case of in-line holography. This problem is due to small parallax which results from optical devices and the tracer particles installed in the line. As a result, reconstructed particles are elongated in the depth direction. This is the so-called depth-of-focus (DOF) problem (Katz et al., 2010). In order to suppress the elongation, various experimental, theoretical and numerical approaches have been proposed by many researchers. Sheng et al. successfully reduced the elongation using an objective lens in digital holographic microscopy (Sheng et al., 2006). Multiple digital CCD cameras were used to overcome small parallax in recording in-line digital holograms of the same volume of seed particles from multiple orientations (Soria and Atkinson, 2008). Methods of instant and iterative 3d-deconvolutions were applied to reduce the elongation (Latychevskaia et al, 2010). New reconstruction methods based on WDIA (Within Depth Intensity Averaging) and morphological technique are proposed (Singh and Panigrahi, 2010, Nigam and Panigrahi, 2013). This paper proposes a method that reduces reconstructed particle elongation using iterative minmax filtering in a reconstructed intensity volume. The method was numerically verified at two particles located on the same line in the direction of elongation. Finally, we applied the method to 100 elongated particles distributed at random. 2. Iterative min-max filtering Assume that reconstructed the intensity of a small particle is approximated as a Gaussian distribution normalized to amplitude of 1.0, and the intensity distribution can be asymptotic to a delta function as follows:! δ x!!! {exp x! /2(σ! ) }! = lim!! exp ( x! /2(σ! /n)) (1) where, σ and x are deviation and positional vector, respectively. However, intensities of a particle at focused plane are not equal to each other and just one particle can be remained using Eq. (1). Therefore, we propose this equation with a min-max filtering in subvolume generated at iterative step. The proposed method is applied to the reconstructed intensity volume of tracer particles to reduce the particle elongation at each iterative step as shown in Fig. 1. The method consists of several steps that are as follows: - 1 -

2 1. Apply min-max filtering (Westerweel, 1993) to a full reconstructed intensity volume which has greater intensity than a threshold value I th. The volume is normalized by minimum and maximum value of intensity: I! (x, y, z) =!!,!,!!!!"#!!"#!!!"# (1) 2. Multiply the filtered intensity volume I i(x, y, z) by the following equation to reduce the particle elongation. I(x, y, z) =!!!! I! (x, y, z) (2) 3. Divide the intensity volume into eight equal subvolumes. 4. Repeat these steps until minimum subvolume (i.g.: minimum subvolume: voxel, full volume: voxel) at each subvolume. Iterative step = 1 Iterative step = 2 Iterative step = 3 Iterative step = 4 Reconstructed particle Ith> I(x,y,z) Fig. 1 Iterative steps in proposed method with min-max filtering and subvolume In order to verify the principle of the method a synthetic holographic pattern is introduced. The pattern of two particles with 10 m diameters located at 600 m and 1900 m on the same line from holographic plane in the direction of elongation is numerically recorded by illuminating wave (λ: nm, x-y plane: 256 pixel 256 pixel, spatial resolutions Δx and Δy of a holographic plane: 10 m). The intensity volume is reconstructed from holographic pattern using a convolution integral (Kreis et al., 1997) as x-y-z volume: voxel and spatial depth resolution Δz: 10 m. The elongation z e proportional to πd /λ (Coëtmellec, 2001) is almost 1600 m shown as the greensolid line of the centerline intensity profile in Fig.2. It is too long to detect the particle in the depth 2 direction. Red and blue solid lines show the first and final step of the profile, respectively. The red line profile shows the maximum intensity peak and minimum intensity is normalized by min-max filtering, and the elongation is reduced by multiplying the filtered intensity volume. Here, the number of multiple times m and threshold intensity I th are 4 times and 100. The blue line profile shows two peaks are sharply detected ranging between 1 and 3 particle diameters. If the step of division of the intensity volume is not applied, the smaller intensity peak decreases with increasing number of multiplying times. The intensity goes to zero value at the final step due to multiply 4 times. Thus, this method is capable of reducing the elongation of particles using 8 iterative min-max filtering

3 250 Light intensity (8bit) Result Original profile z (µm) voxel sub-volume (First step) voxel sub-volume (Final step) Fig. 2 Comparison of centerline intensity profile between original profile, first and final steps A reduction of multi-particle elongation is examined by the present method before the method is applied to HPIV. A synthetic holographic pattern is numerically recorded by illuminating wave (λ: nm) as 100 particles with 10 m diameters located at random. The pattern is reconstructed as the intensity volume (x-y-z volume: voxel and spatial resolutions Δx, Δy and Δz: 10 m). Figure 3 shows reconstructed intensity distributions of 100 particles with an iterative min-max filtering at each step. As can be seen from the steps, particle elongations decrease with an increasing number of divisions of a subvolume. Step 1 of Fig.3 shows that it is hard to find particles due to elongations. On the other hand, the elongation is almost converged as shown in Step 7. A synthetic particle volume is compared with the volume of Step 9 as shown in Fig. 4. It is observed that some particles are removed until Step3 of Fig. 3, probably because a particle which has a higher intensity in subvolume exists around these particles. It can be improved by decreasing the number of multiple times. Moreover, some particles are observed in greater than a particle diameter. This is because these peaks have the form such as a flat top and the form is enhanced by iterative min-max filtering. Deeper bit depth is expected to avoid a flat top of particle intensity. Frequency distributions of synthetic, reconstructed and iterative filtered particle intensities are compared in Fig. 5. The reconstructed particle intensity is widely distributed and the light intensity around 128 and the light ranging between 200 and 255 correspond to a reference wave and particles, respectively. The proposed method enables an iterative filtered light intensity to be completely divided into background and particle intensity

4 (Step 1) voxel (Step 2) voxel (Step 3) voxel (Step 4) voxel (Step 5) voxel (Step 6) voxel (Step 7) voxel (Step 8) voxel Fig. 3 Reconstructed intensity distributions of 100 particles with iterative min-max filtering at each step - 4 -

5 Step 9 (1 1 1 voxel) Synthetic 100 particles Fig. 4 Comparison between an intensity volume with step 9 and synthetic particles 1.E+08 1.E+07 1.E+06 Frequency (#) 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 Fig. 5 Comparison of intensity frequency among reconstructed, synthetic and iterative filtered particle intensity (Step 9, subvolume: voxel) 4. Conclusion This paper has presented the method that reduces reconstructed particle elongation using iterative min-max filtering in holographic particle image velocimetry. The method was numerically verified in the case of two particles located on the same line in the direction of elongation. It is found that this method is capable of reducing the elongation of particles using iterative min-max filtering. We also applied the method to 100 elongated particles distributed at random. Iterative filtered light intensities of particle were completely divided into background and particle intensity. A larger number of particles and experimental verifications would be considered in future work. Acknowledgements 1.E Light intensity (8bit) Reconstructed particle intensity Synthetic particle intensity Iterative filtered particle intensity - 5 -

6 The authors greatly acknowledge that this work was financially supported by Grant-in-Aid for Young Scientists (B) Grant Number Reference Meng H, Pan G, Pu Y, Woodward SH (2004) Holographic particle image velocimetry: from film to digital recording. Measurement Science and Technology 15: Katz J, Sheng J, (2010) Applications of Holography in Fluid Mechanics and Particle Dynamics. Annual reviews of Fluid Mechanics, 42: Sheng J, Malkiel E, Katz J, (2006) Digital Holographic Microscope for Measuring Threedimensional Particle Distributions and Motions. Applied Optics 45: Soria J, Atkinson C, (2008) Towards 3C-3D digital holographic fluid velocity vector field measurement tomographic digital holographic PIV (Tomo-HPIV). Measurement Science and Technology, 19: (12pp) Latychevskaia T, Gehri F, Fink H.W, (2010) Depth-resolved Holographic Reconstructions by Three-dimensional Deconvolution. Optics Express 18 : Singh DK, Panigrahi PK, (2010) Improved digital holographic reconstruction algorithm for depth error reduction and elimination of out-of-focus particles. Optics Express 18 : Nigam A, Panigrahi PK, (2013) Increase in effectiveness of holographic particle field reconstruction using superposition procedure. Applied Optics 52 : A377-A387 Westerweel J, (1993) Digital Particle Image Velocimetry. Theory and Practice. PhD Thesis, Delft University of Technology, The Netherlands The Netherlands Kreis T, Adams M, Jueptner W.P.O, (2004) Methods of digital holography: comparison. Proc. SPIE3098 Coëtmellec S, Buraga-Lefebvre C, Lebrun D, Özkul C, (2001) Application of In-line Digital Holography to Multiple Plane Velocimetry. Measurement Science and Technology, 12: