## Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display |

Optics Express, Vol. 18, Issue 19, pp. 19504-19509 (2010)

http://dx.doi.org/10.1364/OE.18.019504

Acrobat PDF (602 KB)

### 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 × 10^{4} object points over 10 frames per second.

© 2010 Optical Society of America

## 1. Introduction

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

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]

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]

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]

^{4}object points in real-time.

## 2. Improved wave-recording plane method by LUT method

*u*of all points of a 3D object using the following equation [3

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

*x*) on the WRP, (

_{w},y_{w}*x*) and

_{j},y_{j},z_{j}*A*are the coordinates whose origin is on the CGH and intensity of

_{j}*j*-th object point,

*λ*is the wavelength of the reference light, and

*N*is the total number of 3D object points,

*d*=

_{j}*z*−

_{j}*z*is the perpendicular distance between the WRP and a 3D object point, and

*W*) on the WRP. Therefore, the computational amount of Eq.(1) is very small.

_{j}*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:

*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*), so that the numerical calculation of Eq.(2) is accelerated by using two fast Fourier transform (FFT) operations. In Eq.(2),

*𝓕*[

*u*(

_{w}*x,y*)] is accelerated by the FFT and

*𝓕*

^{−1}[·] is accelerated by the inverse FFT.

*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

*θ*for reconstructing a 3D object light from the CGH is the following equation: sin

_{o}*θ*=

_{o}*λ*/(2

*p*), where,

*p*is the sampling pitch on the CGH. We need to set the spread angle for a 3D object point under the angle

*θ*. The radius

_{o}*W*on the WRP is the following equation:

_{j}*θ*is small.

_{o}*W*. During the calculation, we need to judge either the circular regions or other regions. In this paper, in order to accelerate the judgment, we adopt the rectangular region inscribed in the circular region as shown in Fig.1(b). The length of a side on the rectangular region,

_{j}*L*is,

_{j}*d*into the LUT. During the calculation of the first step, we accumulate the distributions of the complex amplitude by referencing the LUT using the parameter,

_{j}*x*and,

_{j},y_{j}*d*

_{j}*d*, is decided by the following equation:

_{j}*max*{·} and

*min*{·} take a maximum and minimum value in the argument. In Fig.1(c), the areas of rectangular regions increase proportional to

*d*, thus, the LUT has a pyramid-like structure. For example, in case of a 3D object with the depth range 0.5

_{j}*mm*≤

*d*≤10.5

_{j}*mm*, the sampling pitch along the depth 0.5

*mm*,

*p*= 8

*µm*and

*λ*= 532

*nm*, the amount of memory is only about 220 Kbytes.

*L*depends on

_{j}*d*, let us treat the average of

_{j}*L*for all points of a 3D object:

_{j}*L̄*

^{2}. Therefore, the computational amount in the first step is 2

*αNL̄*

^{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).

*βN*

^{2}

_{x}log

*N*, where the CGH size is

_{x}*N*×

_{x}*N*,

_{x}*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.

*αNL̄*

^{2}+ 2

*βN*

^{2}

_{x}log

*N*. And, if

_{x}*N*is very large, the total computational amount is expressed as 2

*αNL̄*

^{2}+ 2

*βN*

^{2}

_{x}log

*N*≈ 2

_{x}*αNL̄*

^{2}. When placing the WRP close to the 3D object,

*L̄*is much smaller than

*N*, therefore, we can see that the computational amount of the proposed method is much smaller than that of conventional ray-tracing algorithms [3

_{x}**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]

*αNN*

^{2}

_{x}.

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]

## 3. Results

**2**, 28–34 (1993). [CrossRef]

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]

*d*is 2

_{j}*mm*≤

*d*≤ 12

_{j}*mm*,

*z*= 300

*mm*,

*λ*= 532

*nm*,

*p*= 8

*µm*and

*N*×

_{x}*N*= 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.

_{y}*µm*and 1,920 × 1,080 pixels. We displayed a portion of 2,048 × 2,048 pixels CGH on the LCD panel. The size of all the reconstructed images is approximately 16

*mm*(

*Width*) × 16

*mm*(

*Height*) × 10

*mm*(

*Depth*).

## 4. Conclusion

## References and links

1. | C. Slinger, C. Cameron, and M. Stanley
M, “Computer-Generated Holography as a Generic Display Technology,” Computer |

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 |

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. |

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. |

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. |

8. | |

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 |

11. | L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express |

12. | H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. |

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 |

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. |

16. | T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express |

17. | |

18. |

**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

Sort: Year | Journal | Reset

### References

- C. Slinger, C. Cameron, and M. Stanley, “Computer-Generated Holography as a Generic Display Technology,” Computer 38, 46–53 (2005). [CrossRef]
- S. A. Benton et al., Holographic Imaging (Wiley-Interscience, 2008).
- M. Lucente, “Interactive Computation of holograms using a Look-up Table,” J. Electron. Imaging 2, 28–34 (1993). [CrossRef]
- 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]
- 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]
- 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).
- 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]
- http://openmp.org/wp/
- M. Lucente and T. A. Galyean, “Rendering Interactive Holographic Images,” Proc. of SIGGRAPH 95 387–394 (1995).
- 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]
- 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]
- H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- http://thegwolibrary.sourceforge.net/
- http://www.fftw.org/

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.