## Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms |

Optics Express, Vol. 20, Issue 11, pp. 12021-12034 (2012)

http://dx.doi.org/10.1364/OE.20.012021

Acrobat PDF (5320 KB)

### Abstract

We propose a novel approach to massively reduce the memory of the novel look-up table (N-LUT) for computer-generated holograms by employing one-dimensional (1-D) sub-principle fringe patterns (sub-PFPs). Two-dimensional (2-D) PFPs used in the conventional N-LUT method are decomposed into a pair of 1-D sub-PFPs through a trigonometric relation. Then, these 1-D sub-PFPs are pre-calculated and stored in the proposed method, which results in a remarkable reduction of the memory of the N-LUT. Experimental results reveal that the memory capacity of the LUT, N-LUT and proposed methods have been calculated to be 149.01 TB, 2.29 GB and 1.51 MB, respectively for the 3-D object having image points of 500 × 500 × 256, which means the memory of the proposed method could be reduced by 103 × 10^{6} fold and 1.55 × 10^{3} fold compared to those of the conventional LUT and N-LUT methods, respectively.

© 2012 OSA

## 1. Introduction

14. S.-C. Kim, W.-Y. Choe, and E.-S. Kim, “Accelerated computation of hologram patterns by use of interline redundancy of 3-D object images,” Opt. Eng. **50**(9), 091305 (2011). [CrossRef]

10. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging **2**(1), 28–34 (1993). [CrossRef]

11. 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]

10. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging **2**(1), 28–34 (1993). [CrossRef]

11. 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]

^{3}fold memory reduction has been obtained in the N-LUT compared to that of the conventional LUT [11

11. 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]

12. S.-C. Kim and E.-S. Kim, “Fast computation of hologram patterns of a 3-D object using run-length encoding and novel look-up table methods,” Appl. Opt. **48**(6), 1030–1041 (2009). [CrossRef]

14. S.-C. Kim, W.-Y. Choe, and E.-S. Kim, “Accelerated computation of hologram patterns by use of interline redundancy of 3-D object images,” Opt. Eng. **50**(9), 091305 (2011). [CrossRef]

7. T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television,” Opt. Express **12**(9), 1788–1793 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-9-1788. [CrossRef] [PubMed]

## 2. Generation of CGH patterns of a 3-D object using the N-LUT method

*p*-th object point is specified by (

*x*,

_{p}*y*,

_{p}*z*) and each object point is assumed to have an associated, real-valued magnitude and phase of

_{p}*a*,

_{p}*φ*, respectively. Additionally, the CGH pattern is assumed to be positioned on the plane of

_{p}*z*= 0 [11

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*z*-direction, and each image plane having a fixed depth is approximated as a collection of self-luminous object points of light. As mentioned above, in the N-LUT method, only the 2-D PFPs representing the fringe patterns of the center-located object points on each depth plane are pre-calculated and stored.

*z*) positioned on the center of an image plane having a depth of

_{p}*z*,

_{p}*T*(

*x*,

*y*;

*z*) can be defined as Eq. (1) [11

_{p}**47**, D55–D62 (2008). [CrossRef] [PubMed]

14. S.-C. Kim, W.-Y. Choe, and E.-S. Kim, “Accelerated computation of hologram patterns by use of interline redundancy of 3-D object images,” Opt. Eng. **50**(9), 091305 (2011). [CrossRef]

*k*is defined as

*k*= 2

*π*/

*λ*, in which

*λ*is the free-space wavelength of the light.

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*I*(

*x*,

*y*) in the N-LUT method can be expressed in terms of the shifted versions of pre-calculated 2-D PFPs of Eq. (1) as shown in Eq. (2)Where

*N*is the number of object points. Here, the phase of the object point is set to be zero for all points in N-LUT method. This phase information of the object point is invisible to the viewers and important only in the interactions among the image points. If the discretization step of the object image gets smaller than the spot size of the display system, overlapping may occur between the image points, which results in cross-talks between the object points. Therefore, the discretization step of the object image must be much larger than the spot size of the display system [10

10. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging **2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

## 3. Proposed method

16. S.-C. Kim, J.-H. Kim, and E.-S. Kim, “Effective reduction of the novel look-up table memory size based on a relationship between the pixel pitch and reconstruction distance of a computer-generated hologram,” Appl. Opt. **50**(19), 3375–3382 (2011). [CrossRef] [PubMed]

16. S.-C. Kim, J.-H. Kim, and E.-S. Kim, “Effective reduction of the novel look-up table memory size based on a relationship between the pixel pitch and reconstruction distance of a computer-generated hologram,” Appl. Opt. **50**(19), 3375–3382 (2011). [CrossRef] [PubMed]

### 3.1 Construction of the N-LUT with a pair of 1-D sub-PFPs

**47**, D55–D62 (2008). [CrossRef] [PubMed]

**50**(9), 091305 (2011). [CrossRef]

*x*=

*x*-

*x*and Δ

_{p}*y*=

*y*-

*y*. As you can see in the first line of Eq. (3), the 2-D PFP is a function of two variables,

_{p}*x*and

*y*. However, in the third line of Eq. (3), each term is a function of only one variable,

*x*or

*y*. That is, a function having two variables can be decomposed into four sub-functions having only one variable. Therefore, in the case where four terms of 1-D sub-PFPs having only one variable are prepared, the 2-D PFP to be used for the generation of the CGH pattern can be restored from these 1-D sub-PFPs.

### 3.2 Extraction of 3-D data from an object to be generated

### 3.3 Outline detection of the object images and restoration of the 2-D PFPs

*A*(

*x*

_{1},

*y*

_{1},

*z*

_{1}),

*B*(

*x*

_{2},

*y*

_{2},

*z*

_{1}),

*C*(

*x*

_{3},

*y*

_{3},

*z*

_{1}), and

*D*(

*x*

_{4},

*y*

_{4},

*z*

_{1}), respectively on the depth plane of

*z*

_{1}as shown in Fig. 6(a) . Figure 6(b) shows the process to generate the hologram pattern for this object image with the proposed N-LUT method.

*A*, area of

*A*'(

*x*

_{1}×

*disc*,

*y*

_{1}×

*disc*,

*z*

_{1}) named here as

*Region I*, is calculated by considering the discretization step (

*disc*) and the viewing-distance [11

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*Region II*,

*III*and

*IV*corresponding to the object points

*B*,

*C*and

*D*are also calculated and extracted. Then, by adding up these four regions together, the CGH pattern for the object points of Fig. 6(a) can be generated as shown in Fig. 6(d).

*h*and

_{x}*h*represent the horizontal and vertical resolution of the hologram pattern, respectively.

_{y}*disc*describes the discretization step, and

*x*,

_{leftmost}*x*,

_{rightmost}*y*and

_{top}*y*represents the leftmost, rightmost, top and bottom object points on a depth plane, respectively. In Fig. 6(a),

_{bottom}*x*

_{1},

*x*

_{3},

*y*

_{1}and

*y*

_{4}may represent

*x*,

_{leftmost}*x*,

_{rightmost}*y*and

_{top}*y*, as well.

_{bottom}### 3.4 Generation of the CGH pattern with the restored 2-D PFPs

*z*

_{1}) and the other is the one for the image point located at the center of the image plane of

*z*

_{2,}as shown in Fig. 8(b). Then, with these 2-D PFPs, fringe patterns for four object points of Fig. 8(a) can be easily calculated through simple shifting and adding operations.

*A*(-

*x*

_{1},

*y*

_{1},

*z*

_{1}), this object point is displaced from the center point

*O*

_{1}(0, 0,

*z*

_{1}) on the image plane of

*z*

_{1,}so that the diffraction pattern for this point can be obtained by simply shifting the restored 2-D PFP for the center point contained in the N-LUT with amounts of –

*x*

_{1}and

*y*

_{1}in the direction of

*x*and

*y*, respectively, which is shown in the upper side of Fig. 8(c). Following the same procedure mentioned above, the diffraction pattern for the object point of

*C*(-

*x*

_{3}, -

*y*

_{3},

*z*

_{2}) located on another image plane of

*z*

_{2}can also be obtained by just shifting the restored 2-D PFP for the center image point

*O*

_{2}(0, 0,

*z*

_{2}) with amounts of –

*x*

_{3}and –

*y*

_{3}in the direction of

*x*and

*y*, respectively as shown in the lower side of Fig. 8(c).

### 3.5 Reconstruction of the 3-D object from the CGH

*μm*× 10

*μm*. In general, the human visual system may see as continuous two points that are separated by 3 milliradians of arc [10

**2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*μm*(100

*mm*× 0.003 = 30

*μm*) are chosen because the viewing-distance is assumed to be 100

*mm*here [10

**2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

## 4. Performance Analysis

### 4.1 Computation time

*I*(

*x*,

*y*) [10

**2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*t*,

_{gen_holo}, t_{multi_amp}*N*and

_{D}*d*

_{max}represent the generation time for one object point, the multiplication time of the amplitudes of the object points, the number of object points in a depth plane, and the number of depth planes of the input 3-D object, respectively [11

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*e*required for each CGH pattern

*I*(

*x*,

*y*) of an object point using the loaded 2-D EFP. That is, the total computation time in the LUT method can be given by Eq. (6).Where

*t*represents the loading-time of the 2-D EFP [10

_{load_EFP}**2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

*t*represents the loading-time of the 2-D PFP [10

_{load_PFP}**2**(1), 28–34 (1993). [CrossRef]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

**50**(9), 091305 (2011). [CrossRef]

*t*and

_{load_sub-PFP}*t*represent the loading-time of the 1-D sub-PFPs and the restoring-time of the 2-D PFP from the loaded 1-D sub-PFPs, respectively.

_{restore_PFP}### 4.2 Memory capacity dependence on the object and hologram parameters

*μm*and 500

*mm*, respectively. In Fig. 10, the red, green and blue colors represent the results of the LUT, N-LUT and proposed methods, respectively.

^{6}EFPs should be pre-calculated and stored in the conventional LUT method. Moreover, the file size of each EFP is given by 1,000 × 1,000 × 8 bit = 976.6 KB, so that the total memory capacity of the conventional LUT becomes 976.6 KB × 10

^{6}= 931.32 GB [11

**47**, D55–D62 (2008). [CrossRef] [PubMed]

**47**, D55–D62 (2008). [CrossRef] [PubMed]

## 5. Experiments and results

*μm*× 10

*μm*. The horizontal and vertical discretization steps of less than 30

*μm*(100

*mm*× 0.003 = 30

*μm*) are chosen since the viewing-distance is assumed to be 100

*mm*here.

^{6}fold and 1.25 × 10

^{3}fold smaller than those of the conventional LUT and N-LUT methods.

^{6}fold and 1.40 × 10

^{3}fold for the input image with the resolution of 400 × 400 pixels and 103 × 10

^{6}fold, 1.55 × 10

^{3}fold for the input image with the resolution of 500 × 500 pixels compared to those required in the conventional LUT and N-LUT methods, respectively.

7. T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television,” Opt. Express **12**(9), 1788–1793 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-9-1788. [CrossRef] [PubMed]

8. T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express **18**(10), 9955–9960 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-9955. [CrossRef] [PubMed]

12. S.-C. Kim and E.-S. Kim, “Fast computation of hologram patterns of a 3-D object using run-length encoding and novel look-up table methods,” Appl. Opt. **48**(6), 1030–1041 (2009). [CrossRef]

**50**(9), 091305 (2011). [CrossRef]

^{6}fold and 1.55 × 10

^{3}fold compared to the conventional LUT and N-LUT methods, respectively for the case of the 3-D object having the resolution of 500 × 500 × 256 pixels even though its computation time has been slightly increased compared to the conventional N-LUT method. Furthermore, as we can see in Table 1, the memory capacity of the proposed method may not sharply increase as the resolution of the object image enhances, so that the proposed method could be regarded as the most effective approach for generation of the high-resolution hologram images with a moderate memory size of megabytes (MB).

## 6. Conclusions

^{6}fold and 1.55 × 10

^{3}fold smaller than those of the conventional LUT and N-LUT methods. Good experimental results confirm the feasibility of the proposed method in the field of CGHs.

## Acknowledgment

## References and links

1. | C. J. Kuo and M. H. Tsai, “ |

2. | U. Schnars and W. Jueptner, “ |

3. | T.-C. Poon, “ |

4. | G. L. Rogers, “Gabor diffraction microscopy: the hologram as a generalized zone-plate,” Nature |

5. | T. Kurihara and Y. Takaki, “Shading of a computer-generated hologram by zone plate modulation,” Opt. Express |

6. | P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T.-C. Poon, “Modern Methods for fast generation of digital holograms,” 3D Research |

7. | T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television,” Opt. Express |

8. | T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express |

9. | Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. A. Tanjung, C. Tan, and T.-C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express |

10. | M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging |

11. | 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. |

12. | S.-C. Kim and E.-S. Kim, “Fast computation of hologram patterns of a 3-D object using run-length encoding and novel look-up table methods,” Appl. Opt. |

13. | S.-C. Kim, J.-H. Yoon, and E.-S. Kim, “Fast generation of three-dimensional video holograms by combined use of data compression and lookup table techniques,” Appl. Opt. |

14. | S.-C. Kim, W.-Y. Choe, and E.-S. Kim, “Accelerated computation of hologram patterns by use of interline redundancy of 3-D object images,” Opt. Eng. |

15. | P. Hariharan, “ |

16. | S.-C. Kim, J.-H. Kim, and E.-S. Kim, “Effective reduction of the novel look-up table memory size based on a relationship between the pixel pitch and reconstruction distance of a computer-generated hologram,” Appl. Opt. |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.6890) Image processing : Three-dimensional image processing

(090.5694) Holography : Real-time holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 7, 2012

Revised Manuscript: May 7, 2012

Manuscript Accepted: May 9, 2012

Published: May 11, 2012

**Citation**

Seung-Cheol Kim, Jae-Man Kim, and Eun-Soo Kim, "Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms," Opt. Express **20**, 12021-12034 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12021

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### References

- C. J. Kuo and M. H. Tsai, “Three-Dimensional Holographic Imaging,” (John Wiley & Sons, 2002).
- U. Schnars and W. Jueptner, “Digital Holography -Digital Hologram Recording, Numerical Reconstruction, and Related Techniques,” (Springer Verlag, 2004).
- T.-C. Poon, “Digital Holography and Three-dimensional Display,” (Springer Verlag, 2007).
- G. L. Rogers, “Gabor diffraction microscopy: the hologram as a generalized zone-plate,” Nature166(4214), 237 (1950). [CrossRef] [PubMed]
- T. Kurihara and Y. Takaki, “Shading of a computer-generated hologram by zone plate modulation,” Opt. Express20(4), 3529–3540 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3529 . [CrossRef] [PubMed]
- P. W. M. Tsang, J. P. Liu, K. W. K. Cheung, and T.-C. Poon, “Modern Methods for fast generation of digital holograms,” 3D Research1(2), 11–18 (2010). [CrossRef]
- T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television,” Opt. Express12(9), 1788–1793 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-9-1788 . [CrossRef] [PubMed]
- T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express18(10), 9955–9960 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-9955 . [CrossRef] [PubMed]
- Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. A. Tanjung, C. Tan, and T.-C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express17(21), 18543–18555 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-21-18543 . [CrossRef] [PubMed]
- M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging2(1), 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 3-D object using run-length encoding and novel look-up table methods,” Appl. Opt.48(6), 1030–1041 (2009). [CrossRef]
- S.-C. Kim, J.-H. Yoon, and E.-S. Kim, “Fast generation of three-dimensional video holograms by combined use of data compression and lookup table techniques,” Appl. Opt.47, 5986–5995 (2008). [CrossRef] [PubMed]
- S.-C. Kim, W.-Y. Choe, and E.-S. Kim, “Accelerated computation of hologram patterns by use of interline redundancy of 3-D object images,” Opt. Eng.50(9), 091305 (2011). [CrossRef]
- P. Hariharan, “Optical Holography; Principles, techniques, and applications,” (Cambridge Studies in Modern Optics, 1996).
- S.-C. Kim, J.-H. Kim, and E.-S. Kim, “Effective reduction of the novel look-up table memory size based on a relationship between the pixel pitch and reconstruction distance of a computer-generated hologram,” Appl. Opt.50(19), 3375–3382 (2011). [CrossRef] [PubMed]

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