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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19504–19509
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Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display

Tomoyoshi Shimobaba, Hirotaka Nakayama, Nobuyuki Masuda, and Tomoyoshi Ito  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19504-19509 (2010)
http://dx.doi.org/10.1364/OE.18.019504


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Abstract

A rapid calculation method of Fresnel computer-generated-hologram (CGH) using look-up table and wavefront-recording plane (WRP) methods toward three-dimensional (3D) display is presented. The method consists of two steps: the first step is the calculation of a WRP that is placed between a 3D object and a CGH. In the second step, we obtain an amplitude-type or phase-type CGH to execute diffraction calculation from the WRP to the CGH. The first step of the previous WRP method was difficult to calculate in real-time due to the calculation cost. In this paper, in order to obtain greater acceleration, we apply a look-up table method to the first step. In addition, we use a graphics processing unit in the second step. The total computational complexity is dramatically reduced in comparison with conventional CGH calculations. We show optical reconstructions from a 2,048 × 2,048 phase-type CGH generated by about 3 × 104 object points over 10 frames per second.

© 2010 Optical Society of America

1. Introduction

Computer-generated-holograms (CGH) are widely used in the field of optics, especially as a three-dimensional (3D) display technique [1

1. C. Slinger, C. Cameron, and M. Stanley M, “Computer-Generated Holography as a Generic Display Technology,” Computer 38, 46–53 (2005). [CrossRef]

, 2

2. S. A. Benton et al., Holographic Imaging, (Wiley-Interscience, 2008).

]. However, the computational time involved in CGH obstructs the realization of a practical 3D display system using the CGH. Many methods have been proposed for accelerating the computational time toward real-time calculation. Current CGH calculations are mainly categorized into two approaches: polygon-based and point cloud approaches. In this paper, we focus on the acceleration of the point cloud approach, which treats a 3D object as a composite of multiple point light sources, and generates a CGH from a 3D object more flexibly than polygon-based approaches.

Look-up table (LUT) methods [3

3. M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]

, 4

4. S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008). [CrossRef] [PubMed]

, 5

5. S. C. Kim and E. S Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030–1041 (2009). [CrossRef]

] pre-calculate light intensity distributions on a CGH of point light sources, then, we reference the LUT during the CGH calculation. An image hologram approach carried out real-time 3D color reconstruction because the approach placing a 3D object close to a CGH calculates a small region on the CGH in place of calculating the whole region on the CGH [6

6. H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-Time Generation of Full color Image Hologram with Compact Distance Look-up Table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009).

]. In addition, the method accomplishes greater acceleration using a LUT. LUT methods [3

3. M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]

, 4

4. S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008). [CrossRef] [PubMed]

, 5

5. S. C. Kim and E. S Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030–1041 (2009). [CrossRef]

] calculate a CGH faster than without using LUT methods; unfortunately, they required large memory for the light intensity distributions, and cannot decrease the computational complexity, which is proportional to the total number of point light sources and the total number of CGH pixels. The image hologram approach reduces the computational complexity and memory amount for the light intensity distributions; unfortunately, the method cannot reconstruct a 3D object in the Fresnel region.

2. Improved wave-recording plane method by LUT method

The WRP method inspired by Ref.[6

6. H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-Time Generation of Full color Image Hologram with Compact Distance Look-up Table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009).

] reduces the computational complexity of CGH by two-step calculation. Figure 1(a) shows an outline of the algorithm.

Fig. 1. (a) Outline of the calculation with WRP. (b) Rectangular region on WRP (c) LUT with pyramid structure

In the first step, we record the complex amplitude on the WRP of a 3D object using a raytracing between the 3D object and the WRP. We record the complex amplitude uw of all points of a 3D object using the following equation [3

3. M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]

]:

uw(xw,yw)=jNAjRwjexp(i2πλRwj)
(1)

where, Rwj=(xwxj)2+(ywyj)2+dj2dj+((xwxj)2+(ywyj)2)(2dj) is the distance between a 3D object point and a coordinate (xw,yw) on the WRP, (xj,yj,zj) and Aj are the coordinates whose origin is on the CGH and intensity of j-th object point, λ is the wavelength of the reference light, and N is the total number of 3D object points, dj = zjz is the perpendicular distance between the WRP and a 3D object point, and i=1 . When placing the WRP close to the 3D object, the object light traverses small regions (the radius is Wj) on the WRP. Therefore, the computational amount of Eq.(1) is very small.

The second step calculates the complex amplitude u(x,y) on the CGH using the diffraction calculation between the WRP and CGH. Since the WRP has amplitude and phase information for a 3D object light, the diffraction calculation is equivalent to the case of directly calculating the complex amplitude on the CGH from the 3D object. We use the Fresnel diffraction:

u(x,y)=exp(i2πλz)iλzuw(xw,yw)exp(iπλz((xxw)2+(yyw)2))dxwdyw
=exp(i2πλz)iλz1[[uw(x,y)]·[h(x,y)]]
(2)

where, 𝓕[·] and 𝓕 −1[·] are the Fourier and inverse Fourier operators, z is the perpendicular distance between the WRP and the CGH, and the impulse response h(x,y)=exp(iπλz(x2+y2)). .

We analytically obtain the Fourier transform of h(x,y), so that the numerical calculation of Eq.(2) is accelerated by using two fast Fourier transform (FFT) operations. In Eq.(2), 𝓕[uw(x,y)] is accelerated by the FFT and 𝓕 −1[·] is accelerated by the inverse FFT.

Lastly, we calculate the CGH. If we calculate an amplitude-type CGH, we calculate the light intensity I(x,y) = ∣u(x,y)+R(x,y)∣2, where, R(x,y) is a reference light. On the other hand, if we calculate a phase-type CGH, we calculate θ(x,y)=arg{u(x,y)}, where, the operator arg{·} calculates the argument of u(x,y).

2.1. Improvement of the WRP method

As shown in Fig.1(a), the maximum diffraction angle θo for reconstructing a 3D object light from the CGH is the following equation: sin θo = λ/(2p), where, p is the sampling pitch on the CGH. We need to set the spread angle for a 3D object point under the angle θo. The radius Wj on the WRP is the following equation:

Wj=djtan(sin1(λ2p))djλ2p
(3)

where, the approximation is good when θo is small.

Lj=2Wj2djλ1.4p.
(4)

It is possible to omit a condition branch for the judgment of the circular regions in the programming. We could not confirm the deterioration on optical or numerical reconstructions using the rectangular region instead of the circular one.

The amount of memory of the LUT, which is proportional to dj, is decided by the following equation:

Mem=min{dj}max{dj}Lj2=min{dj}max{dj}(djλ1.4p)2.
(5)

where, the operators max{·} and min{·} take a maximum and minimum value in the argument. In Fig.1(c), the areas of rectangular regions increase proportional to dj, thus, the LUT has a pyramid-like structure. For example, in case of a 3D object with the depth range 0.5mmdj≤10.5mm, the sampling pitch along the depth 0.5mm, p = 8µm and λ = 532nm, the amount of memory is only about 220 Kbytes.

We consider the computational amount of the first step. Because Lj depends on dj, let us treat the average of Lj for all points of a 3D object: L¯=1NΣjNLj . The average of the rectangular area on the WRP is expressed as 2. Therefore, the computational amount in the first step is 2αNL̄ 2, where, “2” means the calculations of the real and imaginary parts, and α is the number of arithmetic operations (e.g. additions and multiplications) in the summation of Eq.(1).

Next, the computational amount in the second step, which includes two FFT operations, is about 2βN 2 xlogNx, where the CGH size is Nx × Nx, β is the number of arithmetic operations (e.g. additions and multiplications) in the FFT. Note that this computational amount does not depend on the number of 3D object points.

Therefore, the total computational amount is about 2αNL̄ 2 + 2βN 2 xlogNx. And, if N is very large, the total computational amount is expressed as 2αNL̄ 2 + 2βN 2 xlogNx ≈ 2αNL̄ 2. When placing the WRP close to the 3D object, is much smaller than Nx, therefore, we can see that the computational amount of the proposed method is much smaller than that of conventional ray-tracing algorithms [3

3. M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]

, 4

4. S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008). [CrossRef] [PubMed]

], which is αNN 2 x.

In addition, in order to obtain more acceleration, we use a multi-thread programming by OpenMP [8] in the first step, and GPU computing in the second step. In recent years, several research have been reported on GPU-based CGH calculation [9

9. M. Lucente and T. A. Galyean, “Rendering Interactive Holographic Images,” Proc. of SIGGRAPH 95 387–394 (1995).

, 10

10. N. Masuda, T. Ito, T. Tanaka, A. Shiraki, and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14, 587–592 (2006). [CrossRef] [PubMed]

, 11

11. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14, 7636–7641 (2006). [CrossRef] [PubMed]

, 12

12. H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009). [CrossRef] [PubMed]

, 13

13. Y. Pan, X. Xu, S. Solanki, X. Liang, R. Bin, A. Tanjung, C. Tan, and T. C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express 17, 18543–18555 (2009). [CrossRef]

, 14

14. T. Shimobaba, T. Ito, N. Y. Ichihashi, and N. Takada “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express bf 18, 9955–9960 (2010). [CrossRef]

]. In this paper, for the Fresnel diffraction, we used our GPU-basedWave Optics (GWO) library [15

15. T. Shimobaba, et.al., “Numerical calculation library for diffraction integrals using the graphic processing unit : the GWO library,” J. Opt. A: Pure Appl. Opt. 10, 075308 (5pp) (2008). [CrossRef]

, 16

16. T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express 16, 11776–11781 (2008). [CrossRef] [PubMed]

]. The library is a numerical calculation library for the diffraction calculations using a GPU. The library and a sample code can be downloaded from Ref.[17].

As shown in Eq.(2), the Fresnel diffraction can accelerate the computational time using FFT and inverse FFT; however, recent central processing units (CPUs) do not have sufficient computational power for real-time calculation. Therefore, we use GPU instead of CPU.

3. Results

We used Microsoft Windows XP Professional Service Pack 3 as the operating system, a computer with an Intel Core i7 920 processor (2.8 GHz), memory of 3G bytes, and Microsoft Visual C++ 2005 with Intel C++ compiler version 11.1. We used FFTW3.2.1 library [18] for the FFT operations on the CPU. In addition, we used one Nvidia GeForce GTX260 GPU board, and CUDA 2.3 for GPU programming.

The calculation conditions are as follows: the range of dj is 2mmdj ≤ 12mm, z = 300mm, λ = 532nm, p = 8µm and Nx × Ny = 2,048 × 2,048. We calculated CGHs from two 3D objects: “Nanami”, which is NHK (Japan Broadcasting Corp.) character, and “Earth”, consisting of 4,596 and 30,492 points, respectively.

Table 1. Calculation times of conventional algorithm on the CPU and GPU, and the previous and improved WRP methods.

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Table 2. Calculation times of the step 1 and 2 in the previous and improved WRP methods.

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Fig. 2. (Media 1) (Media 2) Reconstructed 3D movies from 2,048 × 2,048 phase-type CGH by the improved WRP method. (a) Nanami (4,596 points) (b) Earth (30,492 points).

Comparing numerical reconstruction from a CGH generated by the conventional algorithm with that from a CGH generated by the improved WRP algorithm, we find a blur in micrometerorder in the improved WRP algorithm; however, we do not almost observe the blur.

4. Conclusion

References and links

1.

C. Slinger, C. Cameron, and M. Stanley M, “Computer-Generated Holography as a Generic Display Technology,” Computer 38, 46–53 (2005). [CrossRef]

2.

S. A. Benton et al., Holographic Imaging, (Wiley-Interscience, 2008).

3.

M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]

4.

S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008). [CrossRef] [PubMed]

5.

S. C. Kim and E. S Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030–1041 (2009). [CrossRef]

6.

H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-Time Generation of Full color Image Hologram with Compact Distance Look-up Table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009).

7.

T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calclulation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. 34, 3133–3135 (2009). [CrossRef] [PubMed]

8.

http://openmp.org/wp/

9.

M. Lucente and T. A. Galyean, “Rendering Interactive Holographic Images,” Proc. of SIGGRAPH 95 387–394 (1995).

10.

N. Masuda, T. Ito, T. Tanaka, A. Shiraki, and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14, 587–592 (2006). [CrossRef] [PubMed]

11.

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14, 7636–7641 (2006). [CrossRef] [PubMed]

12.

H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009). [CrossRef] [PubMed]

13.

Y. Pan, X. Xu, S. Solanki, X. Liang, R. Bin, A. Tanjung, C. Tan, and T. C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express 17, 18543–18555 (2009). [CrossRef]

14.

T. Shimobaba, T. Ito, N. Y. Ichihashi, and N. Takada “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express bf 18, 9955–9960 (2010). [CrossRef]

15.

T. Shimobaba, et.al., “Numerical calculation library for diffraction integrals using the graphic processing unit : the GWO library,” J. Opt. A: Pure Appl. Opt. 10, 075308 (5pp) (2008). [CrossRef]

16.

T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express 16, 11776–11781 (2008). [CrossRef] [PubMed]

17.

http://thegwolibrary.sourceforge.net/

18.

http://www.fftw.org/

OCIS Codes
(090.1760) Holography : Computer holography
(090.2870) Holography : Holographic display
(090.1995) Holography : Digital holography
(090.5694) Holography : Real-time holography

ToC Category:
Holography

History
Original Manuscript: July 1, 2010
Revised Manuscript: August 23, 2010
Manuscript Accepted: August 26, 2010
Published: August 30, 2010

Citation
Tomoyoshi Shimobaba, Hirotaka Nakayama, Nobuyuki Masuda, and Tomoyoshi Ito, "Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display," Opt. Express 18, 19504-19509 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19504


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References

  1. C. Slinger, C. Cameron, and M. Stanley, “Computer-Generated Holography as a Generic Display Technology,” Computer 38, 46–53 (2005). [CrossRef]
  2. S. A. Benton et al., Holographic Imaging (Wiley-Interscience, 2008).
  3. M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]
  4. S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008). [CrossRef] [PubMed]
  5. S. C. Kim and E. S Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030–1041 (2009). [CrossRef]
  6. H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-Time Generation of Full color Image Hologram with Compact Distance Look-up Table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009).
  7. T. Shimobaba, N. Masuda and T. Ito, “Simple and fast calclulation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. 34, 3133–3135 (2009). [CrossRef] [PubMed]
  8. http://openmp.org/wp/
  9. M. Lucente and T. A. Galyean, “Rendering Interactive Holographic Images,” Proc. of SIGGRAPH 95 387–394 (1995).
  10. N. Masuda, T. Ito, T. Tanaka, A. Shiraki and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14,587–592 (2006). [CrossRef] [PubMed]
  11. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14,7636–7641 (2006). [CrossRef] [PubMed]
  12. H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009). [CrossRef] [PubMed]
  13. Y. Pan, X. Xu, S. Solanki, X. Liang, R. Bin A. Tanjung, C. Tan, and T. C. Chong,” Fast CGH computation using S-LUT on GPU,” Opt. Express 17, 18543–18555 (2009). [CrossRef]
  14. T. Shimobaba, T. Ito, N. Y. Ichihashi, and N. Takada “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express 18, 9955–9960 (2010). [CrossRef]
  15. T. Shimobaba, et.al., “Numerical calculation library for diffraction integrals using the graphic processing unit : the GWO library,” J. Opt. A: Pure Appl. Opt. 10, 075308 (5pp) (2008). [CrossRef]
  16. T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express 16, 11776–11781 (2008). [CrossRef] [PubMed]
  17. http://thegwolibrary.sourceforge.net/
  18. http://www.fftw.org/

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