## Channel analysis of the volume holographic correlator for scene matching |

Optics Express, Vol. 19, Issue 5, pp. 3870-3880 (2011)

http://dx.doi.org/10.1364/OE.19.003870

Acrobat PDF (1088 KB)

### Abstract

A channel model of the volume holographic correlator (VHC) is proposed and demonstrated to improve the accuracy in the scene matching application with the multi-sample parallel estimation (MPE) algorithm. A quantity related to the space-bandwidth product is used to describe the recognition ability in the scene matching system by MPE. A curve is given to optimize the number of samples with the required recognition accuracy. The theoretical simulation and the experimental results show the validity of the channel model. The proposed model provides essential theoretical predictions and implementation guidelines for using the multi-sample parallel estimation method to achieve the highest accuracy.

© 2011 OSA

## 1. Introduction

1. S. L. Wang, Q. F. Tan, L. C. Cao, Q. S. He, and G. F. Jin, “Multi-sample parallel estimation in volume holographic correlator for remote sensing image recognition,” Opt. Express **17**(24), 21738–21747 (2009). [CrossRef] [PubMed]

3. Y. Takashima and L. Hesselink, “Media tilt tolerance of bit-based and page-based holographic storage systems,” Opt. Lett. **31**(10), 1513–1515 (2006). [CrossRef] [PubMed]

4. J. Joseph, A. Bhagatji, and K. Singh, “Content-addressable holographic data storage system for invariant pattern recognition of gray-scale images,” Appl. Opt. **49**(3), 471–478 (2010). [CrossRef] [PubMed]

6. A. Heifetz, J. T. Shen, J. K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a superparallel holographic random access memory,” Opt. Eng. **45**(2), 025201 (2006). [CrossRef]

7. J. Capon, “A probabilistic mode for run length coding of picture,” IEEE Trans. Inf. Theory **5**(4), 157–163 (1959). [CrossRef]

1. S. L. Wang, Q. F. Tan, L. C. Cao, Q. S. He, and G. F. Jin, “Multi-sample parallel estimation in volume holographic correlator for remote sensing image recognition,” Opt. Express **17**(24), 21738–21747 (2009). [CrossRef] [PubMed]

*R*of the reference image can be expressed as [9

9. S. D. Wei and S. H. Lai, “Robust and efficient image alignment based on relative gradient matching,” IEEE Trans. Image Process. **15**(10), 2936–2943 (2006). [CrossRef] [PubMed]

*a*,

*b*are constants, determined by the variance and mean of the image grayscale.

*α*,

*β*are also constants, whose reciprocal values are called correlation lengths, determined by the characteristics of the horizontal and vertical spatial grayscale distributions. Finally

*x*and

*y*represent horizontal and vertical coordinate differences between the target image and template images, respectively.

*α*, 1/

*β*) of the image and reduce the redundancy correlation between the target image and the template images. The process of template images preparation is mainly based on the choice of the segmentation interval (

*t*

_{x},

*t*). In this process, the reference image is divided vertically and horizontally into a set of template images: each image has the same pixel numbers as the target image, as well as the same vertical and horizontal segmentation intervals, as shown in Fig. 1 . The segmentation intervals (

_{y}*t*

_{x},

*t*) are chosen to allow overlap between different template images. The number of estimation equations is determined by the sample correlation spots, and more samples can help to improve the recognition accuracy. Thus the correlation length, the segmentation interval and the sample number are the main factors in the MPE method.

_{y}## 2. Correlation space-bandwidth product

*αβ*) have significant correlations among each other. As shown in Fig. 2 , the correlation lengths (1/

*α*, 1/

*β*) describe the characteristics of the remote sensing image and the segmentation intervals (

*t*

_{x},

*t*) are limited by the capacity of the VHC. If we regard (1/

_{y}*t*

_{x}, 1/

*t*) as the sampling frequencies, similar to the SBP in the sampling theorem, we can define the correlation space-bandwidth product (CSBP) as the product between the spatial correlation area and the sampling frequency bandwidth, which is

_{y}## 3. Channel analysis of the VHC for the scene matching

13. P. M. Lundquist, C. Poga, R. G. Devoe, Y. Jia, W. E. Moerner, M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, and G. T. Sincerbox, “Holographic digital data storage in a photorefractive polymer,” Opt. Lett. **21**(12), 890–892 (1996). [CrossRef] [PubMed]

15. M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, G. T. Sincerbox, P. Wimmer, and G. Wittmann, “A precision tester for studies of holographic optical storage materials and recording physics,” Appl. Opt. **35**(14), 2360–2374 (1996). [CrossRef] [PubMed]

*dx*and

*dy*can be calculated from Eq. (1). And then

*dx*,

*dy,*and

*d*(

*f*(

*x*,

*y*)), therefore, the recognition error can be expressed as

### 3.1 One-dimensional two-sample system

*z*

_{1}and

*z*

_{2}is independent of each other, and has the same statistical distribution. Then their standard deviations

*σ*

_{1},

*σ*

_{2}satisfy

*x*=

*t*/2.

*σ*.

_{x}*σ*can be written as:

_{x}### 3.2 Two-dimensional multi-sample system

*t*and

_{x}*t*, respectively. Considering a four-sample correlation spots system, the worst situation for recognition is when the target is in the center of the template images (

_{y}*x = t*2,

_{x}/*y = t*2), as is shown in Fig. 3(b). Therefore, the error in this worst case scenario can be defined as the error of the system. According to Eq. (10), for the two-dimensional four-sample system, we can have

_{y}/*n*×

*n*(

*n*is even),

*t*=

_{x}*t*=

_{y}*t*, and the target image is in the center of the system. According to the conclusion above and the symmetry of the system, the coordinate system shown in Fig. 3(c) shows that the recognition error of the

*n*×

*n*sample spots is half of that of the

*N*= (

*n*/2) × (

*n*/2) sample spots in the first quadrant. Therefore, it is sufficient to analyze the recognition accuracy of the

*N*sample spots in the first quadrant. Because these spots are in the same quadrant, the absolute-value signs in Eq. (1) can be removed. Suppose the sample correlation intensities are

*z*

_{1},

*z*

_{2}, …, z

*and the standard deviations of their errors satisfy*

_{N}*N*sample spots, according to the Least Squares Estimation and the sum of the normal distributions, we have

*n*×

*n*, the error is half of the

*N*sample spots, that is

## 4. Simulations based on the channel model

*n*), decreasing the segmentation interval (

*t*), optimizing the image preprocessing (

*α*,

*β*), and improving the accuracy of the VHC (

*α*≈

*β*. To simplify the expressions, we introduce two new parameters

*w*=

*a*and

*t*to

*w*is simply scaling factors, while the contribution of the parameters

*n*and

*p*to

*w*is more complex, we can fix the parameters

*a*and

*t*(

*a*= 0.5,

*t*= 1) to analyze the

*w*as a function of

*n*and

*p*, as shown in Fig. 4(a) . As shown in Fig. 4(a), for each given

*p*,

*w*will decrease as

*n*increases, but not indefinitely. This agrees with the intuition that using more samples will increase the recognition accuracy. However, for large

*p*values, the impact of the parameter

*n*to

*w*is less significant.

*n*, there exists a

*p*that minimizes the error

*w*. When the parameter

*n*changes from 2 to 20 (

*n*being the square root of the sample number), the minimum error

*w*and the corresponding

*p*are shown in Table 1 .According to Table 1 and Eq. (19), the blue line and the black line are drawn in Fig. 4(b). The blue line is called the optimization curve, where each point indicates the highest achievable accuracy for a given sample number; and the black curve is the limit, when the sample number approaches infinity, that is the upper bound for any achievable accuracy with a given system parameter

*p*. When the value of

*p*decreases, the optimization curve approaches the upper bound curve indicating that the optimized accuracy has nearly achieved the limit of the system. For a given tolerable recognition error

*w*, specified by the user, the parameters on the optimization line represent a system that requires the minimum number of sample points, therefore, the least amount of post-processing. In other words, the optimization curve is the best accuracy we can achieve for a given value of

*n*and is our optimization goal.

*n*and

*p*can also be found as

*w*) and the CSBP (

*M*) is derived. And if the tolerable recognition error (

*w*) is given, the required system CSBP (

*M*) can be calculated using Eq. (20), and the minimum sample number (

*n*) can also be determined using Eq. (21). For example, according to Table 1, to achieve a required accuracy corresponding to a recognition error of 2.28

^{2}*n*= 10-12) and the system parameter

*p*should be about 0.35-0.39 (

*M =*105

*-*131).

*t*) determination. According to storage capacity of the VHC, it is possible to determine the interval. (2) CSBP (

_{x}, t_{y}*M*) determination. According to the recognition accuracy required by the user and the accuracy of the VHC, the error ratio

*w*can be derived. Then CBSP can be derived from Eq. (20). (3) Correlation length (1/

*α*, 1/

*β*) determination. According to the value of the segmentation interval and CSBP, the correlation length can be determined using Eq. (2). And the appropriate preprocessing of the target image should be performed to match the correlation length. (4) Sample number determination. According to Eq. (21) and Fig. 4 (b), choose the minimum number of samples (on the blue line) to achieve the required system performance. The accuracy can be further improved by increasing the sample number, within the limit of the black line.

## 5. Experimental results

*λ*= 532 nm). The holograms are stored in an Fe: LiNbO

_{3}crystal using angle fractal multiplexing, while a CCD camera (MINTRON MTV-1881EX) is used to detect the correlation spots. The thickness of the recording medium is 15mm and the thickness of volume grating in the recording medium is about 6mm.

17. C. Ouyang, L. C. Cao, Q. S. He, Y. Liao, M. X. Wu, and G. F. Jin, “Sidelobe suppression in volume holographic optical correlators by use of speckle modulation,” Opt. Lett. **28**(20), 1972–1974 (2003). [CrossRef] [PubMed]

18. K. Ni, Z. Y. Qu, L. C. Cao, P. Su, Q. S. He, and G. F. Jin, “Improving accuracy of multichannel volume holographic correlators by using a two-dimensional interleaving method,” Opt. Lett. **32**(20), 2973–2974 (2007). [CrossRef]

_{VHC}can be tested prior to the experiment, as follows. Some testing images are stored in the VHC. Another testing images can be input into the VHC for

*N*times, where

*N*should be larger than 1000. The

*N*correlation values are acquired and a Gaussian distribution curve of the

*N*values can be obtained. The parameter σ

_{VHC}can be determined by fitting the Gaussian curve. In this experiment, the normalized error 3σ

_{VHC}of the VHC is equal to 0.08.

*α*= 0.122,

*β*= 0.160,

*a*= 0.453, and

*b*= 0.491. With segmentation intervals

*t*and

_{x}*t*chosen to be 3 pixels, the template images are stored into the VHC. Thus the CBSP(M) is about 91. Then 400 different target images are taken and inputted into the VHC to compute the coordinates of the targets by using the MPE method. When a white image is inputted into the VHC, the correlation spots are detected by the CCD, as shown in Fig. 6(b). When a target image is inputted into the VHC, the correlation spots are detected by the CCD, as shown in Fig. 6(c). The results of the horizontal and vertical errors for using 16 samples correlation spots are shown in Fig. 7 . The ratio between the horizontal error and the vertical error is determined by the ratio between the horizontal and the vertical correlation lengths. According to Eq. (17), the theoretical results are

_{y}*w*= 9.88, 3

*σ*and 3

*σ*is 99.7%. Thus the value 3

*σ*can be regarded as the error of the VHC. Thus, the recognition error is about

*p*increasing), the impact of the sample number to the recognition error will be less significant. In further experiments, we have adjusted

*p*by changing the segmentation intervals to verify this conclusion. The segmentation intervals were set to be 1 pixel, 3 pixels and 5 pixels, respectively. By using the MPE method, the sample numbers 16, 36, 64, 100 were used to estimate the location of the target. To compare the results from the 3 different segmentation intervals, the error was normalized by the corresponding segmentation interval. The results are shown in Table 2 and Fig. 8 .

*n*when the value of

*p*is low (

*p*= 0.12, 1 pixel). When the value of

*p*increases to 0.36 (3 pixels), the accuracy cannot be increased significantly with the increasing

*n*. When the value of

*p*reaches about 0.60 (5 pixels), the accuracy is nearly no longer increasing with the increasing sample number. The results also have good agreement with the theoretical analysis in Fig. 4.

## 6. Conclusion

## Acknowledgment

## References and links

1. | S. L. Wang, Q. F. Tan, L. C. Cao, Q. S. He, and G. F. Jin, “Multi-sample parallel estimation in volume holographic correlator for remote sensing image recognition,” Opt. Express |

2. | G. W. Burr, F. H. Mok, and D. Psaltis, “Large-scale volume holographic storage in the long interaction length architecture,” Proc. SPIE |

3. | Y. Takashima and L. Hesselink, “Media tilt tolerance of bit-based and page-based holographic storage systems,” Opt. Lett. |

4. | J. Joseph, A. Bhagatji, and K. Singh, “Content-addressable holographic data storage system for invariant pattern recognition of gray-scale images,” Appl. Opt. |

5. | E. Watanabe, A. Naito, and K. Kodate, “Ultrahigh-speed compact optical correlation system using holographic disc,” Proc. SPIE |

6. | A. Heifetz, J. T. Shen, J. K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a superparallel holographic random access memory,” Opt. Eng. |

7. | J. Capon, “A probabilistic mode for run length coding of picture,” IEEE Trans. Inf. Theory |

8. | F. Saitoh, “Image template matching based on edge-spin correlation,” Electr. Eng. |

9. | S. D. Wei and S. H. Lai, “Robust and efficient image alignment based on relative gradient matching,” IEEE Trans. Image Process. |

10. | T. S. Huang, “PCM picture transmission,” IEEE Spectr. |

11. | M. A. Neifeld, “Information, resolution, and space-bandwidth product,” Opt. Lett. |

12. | J. W. Goodman, |

13. | P. M. Lundquist, C. Poga, R. G. Devoe, Y. Jia, W. E. Moerner, M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, and G. T. Sincerbox, “Holographic digital data storage in a photorefractive polymer,” Opt. Lett. |

14. | M. R. Vant, R. W. Herring, and E. Shaw, “Digital processing techniques for satellite-borne SAR,” Can. J. Rem. Sens. |

15. | M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, G. T. Sincerbox, P. Wimmer, and G. Wittmann, “A precision tester for studies of holographic optical storage materials and recording physics,” Appl. Opt. |

16. | H. A. Jazwinskl, |

17. | C. Ouyang, L. C. Cao, Q. S. He, Y. Liao, M. X. Wu, and G. F. Jin, “Sidelobe suppression in volume holographic optical correlators by use of speckle modulation,” Opt. Lett. |

18. | K. Ni, Z. Y. Qu, L. C. Cao, P. Su, Q. S. He, and G. F. Jin, “Improving accuracy of multichannel volume holographic correlators by using a two-dimensional interleaving method,” Opt. Lett. |

**OCIS Codes**

(070.4550) Fourier optics and signal processing : Correlators

(090.7330) Holography : Volume gratings

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: November 24, 2010

Revised Manuscript: January 29, 2011

Manuscript Accepted: February 3, 2011

Published: February 14, 2011

**Citation**

Shunli Wang, Liangcai Cao, Huarong Gu, Qingsheng He, Claire Gu, and Guofan Jin, "Channel analysis of the volume holographic correlator for scene matching," Opt. Express **19**, 3870-3880 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3870

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### References

- S. L. Wang, Q. F. Tan, L. C. Cao, Q. S. He, and G. F. Jin, “Multi-sample parallel estimation in volume holographic correlator for remote sensing image recognition,” Opt. Express 17(24), 21738–21747 (2009). [CrossRef] [PubMed]
- G. W. Burr, F. H. Mok, and D. Psaltis, “Large-scale volume holographic storage in the long interaction length architecture,” Proc. SPIE 2297, 402–414 (1994). [CrossRef]
- Y. Takashima and L. Hesselink, “Media tilt tolerance of bit-based and page-based holographic storage systems,” Opt. Lett. 31(10), 1513–1515 (2006). [CrossRef] [PubMed]
- J. Joseph, A. Bhagatji, and K. Singh, “Content-addressable holographic data storage system for invariant pattern recognition of gray-scale images,” Appl. Opt. 49(3), 471–478 (2010). [CrossRef] [PubMed]
- E. Watanabe, A. Naito, and K. Kodate, “Ultrahigh-speed compact optical correlation system using holographic disc,” Proc. SPIE 7442, 1–8 (2010).
- A. Heifetz, J. T. Shen, J. K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a superparallel holographic random access memory,” Opt. Eng. 45(2), 025201 (2006). [CrossRef]
- J. Capon, “A probabilistic mode for run length coding of picture,” IEEE Trans. Inf. Theory 5(4), 157–163 (1959). [CrossRef]
- F. Saitoh, “Image template matching based on edge-spin correlation,” Electr. Eng. 153, 1592–1596 (2005).
- S. D. Wei and S. H. Lai, “Robust and efficient image alignment based on relative gradient matching,” IEEE Trans. Image Process. 15(10), 2936–2943 (2006). [CrossRef] [PubMed]
- T. S. Huang, “PCM picture transmission,” IEEE Spectr. 2, 57–63 (1965).
- M. A. Neifeld, “Information, resolution, and space-bandwidth product,” Opt. Lett. 23(18), 1477–1479 (1998). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1966).
- P. M. Lundquist, C. Poga, R. G. Devoe, Y. Jia, W. E. Moerner, M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, and G. T. Sincerbox, “Holographic digital data storage in a photorefractive polymer,” Opt. Lett. 21(12), 890–892 (1996). [CrossRef] [PubMed]
- M. R. Vant, R. W. Herring, and E. Shaw, “Digital processing techniques for satellite-borne SAR,” Can. J. Rem. Sens. 5, 67 (1979).
- M.-P. Bernal, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, R. M. Macfarlane, R. M. Shelby, G. T. Sincerbox, P. Wimmer, and G. Wittmann, “A precision tester for studies of holographic optical storage materials and recording physics,” Appl. Opt. 35(14), 2360–2374 (1996). [CrossRef] [PubMed]
- H. A. Jazwinskl, Stochastic process and filtering theory (Academic Press, 1970).
- C. Ouyang, L. C. Cao, Q. S. He, Y. Liao, M. X. Wu, and G. F. Jin, “Sidelobe suppression in volume holographic optical correlators by use of speckle modulation,” Opt. Lett. 28(20), 1972–1974 (2003). [CrossRef] [PubMed]
- K. Ni, Z. Y. Qu, L. C. Cao, P. Su, Q. S. He, and G. F. Jin, “Improving accuracy of multichannel volume holographic correlators by using a two-dimensional interleaving method,” Opt. Lett. 32(20), 2973–2974 (2007). [CrossRef]

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