## Abstract

We have designed a special purpose computer system for digital holographic particle tracking velocimetry (DHPTV). We present the pipeline for calculating the intensity of an object from a hologram by fast Fourier transform in an FPGA chip. This system uses four FPGA chips and can make 100 reconstructed images from a 256×256-grid hologram in 266 msec. It is expected that this system will improve the efficiency of analysis in DHPTV.

© 2006 Optical Society of America

## 1. Introduction

Holographic particle image velocimetry systems use high-quality information and allow the recording of an instantaneous Three-dimensional velocity field illuminated by only one beam line[1, 2, 3, 4, 5]. However this method takes up most of the process time in image reconstruction. It is also difficult to use it to capture the time evolution of a particle image by single frame recording of instantaneous particles dispersed in a flow field. In contrast, digital holographic techniques can easily capture the time evolution of particles using a digital camera. Members of our group have developed a complete digital holographic particle tracking velocimetry (DHPTV) system that does not use photographic films and instead takes 3D velocity vectors by a high speed digital camera[6]. The high-speed reconstruction of a particle field was successfully performed using a fast Fourier transform (FFT)[4, 5]. However this method requires a high-speed computer to reconstruct particle location by a computer hologram algorithm. To overcome the computational cost, we designed a special purpose computer system using FFT for image reconstruction. Members of our group have also developed a special purpose computer for computer generated holography[7, 8, 9, 10, 11], called HORN (HOlographic ReconstructioN). We call our special purpose computer for DHPTV, FFT-HORN.

## 2. Algorithm of DHPTV

The equation for reconstruction is given by

where ϕ(*x _{i}*,

*y*,

_{i}*Z*) is the amplitude of the object light at point (

_{i}*x*,

_{i}*y*,

_{i}*z*), λ is the wavelength of the reference light,

_{i}*I*is the intensity of the hologram at point (

_{α}*x*,

_{α}*y*, 0),

_{α}*k*is the wavenumber of the reference light, and

*N*is the number of division of the hologram grid in the x or y directions. Note that in Eq. (1), the hologram is on the

*z*= 0 plane. Furthermore, using the Fresnel approximation we obtain

We introduce a new function defined by the following equation,

Substituting Eq. (4) into Eq. (3) we obtain

Equation (5) is the convolution integral and is transformed by a two-dimensional Fourier transform,

where Ψ(*n*,*m*) is the two-dimensional Fourier transform of ϕ(*x _{i}*,

*y*,

_{i}*z*),

_{i}*I*̂(

*n*,

*m*) is the transform of

*I*, and

_{α}*G*(

*n*,

*m*) is the transform of

*g*(

*x*-

_{i}*x*,

_{α}*y*-

_{i}*y*).

_{α}where Δ*x _{α}*, Δ

*y*is the grid size of the hologram.

_{α}The calculation of three-dimensional particle reconstruction is carried out from reproductions in several divided sections in the *z*-direction. Each plane is reconstructed keeping *z* constant. To calculate Eq. (3), the method described below was contrived:

- calculate
*I*̂(*n*,*m*) from*I*_{α} - calculate Ψ(
*n*,*m*) =*I*̂(*n*,*m*) ×*G*(*n*,*m*) - calculate ϕ(
*x*,_{i}*y*,_{i}*Z*) by using a two-dimensional inverse Fourier transform._{i}By this method, the speed of reconstruction is greatly increased.

## 3. Hardware design

Figure 1 shows a block diagram of FFT-HORN. This pipeline can calculate the intensity of an object by the above method, with all calculations being fixed-point operations. The numbers under the slashes are the wordlengths of the data over that line. The unit FFT CORE calculates the FFT and the inverse FFT (IFFT). Here we use a Xilinx LogiCORE as FFT CORE, which can perform a one-dimensional FFT. Hence, in this pipeline, one-dimensional FFT is performed twice to calculate the two-dimensional FFT. There are three RAM units in this pipeline. RAM Hologram is used for the hologram intensity data. RAM Real ande RAM Imaginary are used to save the FFT data temporarily. The G-pipeline unit calculates Eq. (7). The Sel unit controls the data flow in the pipeline to calculate the intensity of the object according to the above method.

Figure 2 shows the FPGA board (HORN-5 board) used in FFT-HORN. HORN-5 board which is developed by Chiba university and the Institute of Physical and Chemical Research (RIKEN) has four FPGA chips, Xilinx XC2VP70-5FF1517C [11]. The chip is equivalent to about 7,000,000 gates. FFT-HORN connects to the host computer through a peripheral component interconnect (PCI) bus. The host computer controls FFT-HORN, writes the intensity data of the hologram to FFT-HORN and sends the start signal. After FFT-HORN finishes calculating the intensity data of the object, the host computer reads the data.

## 4. Performance

We used a Xilinx ISE for computer aided design of the FFT-HORN chip. The clock frequency of the special purpose computer is 133 MHz. There is one pipeline in a FPGA chip and a FPGA board can calculate four planes at a time.

We compared the calculation time of FFT-HORN with a personal computer. The specifications of the personal computer are as follows: Intel Pentium 4 3.2-GHz CPU, 2.0 GB of memory, Microsoft Windows XP operating system and Microsoft Visual C++ C compiler. Table 1 shows the calculation time for 100 reconstructed images from a hologram with a 256 × 256 grid. The calculation time of FFT-HORN is 266 msec, while the calculation time of the PC is 10, 145 msec. Hence, the calculation speed of FFT-HORN is about 40 times faster than that of the personal computer.

We also compared reconstructed images made with FFT-HORN and the PC. For this simulation, holograms were generated by the system described in detail in Ref. [6]. Figure 3(a) shows a 1024 × 1024-grid hologram of the system and Fig.3(b) shows a 256 × 256-grid hologram of a portion of the system. The working fluid was water and the injected particles were 50-micron nylon spheres. Water flowed from top to bottom in Fig. 3(a), with a square with 4-mm sides placed in the flow as an obstacle, shown as a black shadow in Fig. 3(a).

Figure 4(a) shows the reconstructed image made by the PC and Fig. 4(b) shows the reconstructed image made by FFT-HORN. The FFT-HORN image agrees well with that of the PC. The reconstructions of the particles by the computer hologram algorithm are 33.78 mm from the CCD surface. In Fig.4(a) and Fig.4(b), the black small objects are the reconstructed particles.

## 5. Conclusion

We have designed and built a special purpose computer for DHPTV, FFT-HORN. Its clock frequency is 133 MHz and it can make 100 reconstructed images from a 256×256-grid hologram in 266 ms. The FFT-HORN reconstructed images agree well with those of a PC. Hence, this special purpose computer system can be used for particle tracking velocimetry and it is expected that this system will improve the efficiency of analysis in DHPTV. In the future, we plan to expand FFT-HORN to be able to calculate the intensity of an object from a 1024 × 1024-grid hologram.

## Acknowledgments

We would like to thank Dr. T. Shimobaba and Dr. T. Sugie for their useful advice. This research was partly supported by a Grant-in-Aid for Scientific Research (C) (17560031).

## References and links

**01. **D.H. Barnhart, R.J. Adrian, and G.C. Papen,“Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. **33**, 7159–7170 (1994). [CrossRef] [PubMed]

**02. **H. Memg and F. Hussain, “In-line recording and off-axis viewing technique for holographic particle velocimetry,” Appl. Opt. **34**, 1827–1840 (1995). [CrossRef]

**03. **J. Sheng, E. Malkiel, and J. Katz,“Single beam two-views holographic particle image velocimetry,” Appl. Opt. **42**, 235–250 (2003). [CrossRef] [PubMed]

**04. **U. Schnars, T. Kreis, and W. Juptner, “Direct recording of holograms by CCD target and numerical reconstruction,” Appl. Opt. **33**, 179–181 (1994). [CrossRef] [PubMed]

**05. **S. Murata and N. Yasuda, “Potential of digitalholography in particle measurement,” Opt. Laser Technol. **32**, 567–574 (2000). [CrossRef]

**06. **S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev. **11**, 162–164 (2004).

**07. **T. Ito, T. Yabe, M. Ozaki, and M. Yanagi,“Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,”Comp. Phys. Commun. **82**104–110 (1994). [CrossRef]

**08. **T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. **93**13–20(1996). [CrossRef]

**09. **T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. **130**, 75–82(2002). [CrossRef]

**10. **T. Shimobaba and T. Ito, “Special-purpose computer for holography HORN-4 with recurrence algorithm,” Comp. Phys. Commun. **148**, 160–170(2002). [CrossRef]

**11. **T. Ito, K.N. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie “A special-purpose computer for electroholography HORN-5 to realize a real-time reconstruction,” Opt. Express , **13**, 1923–1932(2005),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1923 [CrossRef] [PubMed]