## Fully-phase asymmetric-image verification system based on joint transform correlator

Optics Express, Vol. 14, Issue 4, pp. 1458-1467 (2006)

http://dx.doi.org/10.1364/OE.14.001458

Acrobat PDF (449 KB)

### Abstract

A fully-phase optical asymmetric-image verification system based on a joint transform correlator (JTC) is proposed in this study. Conventional joint power spectra in JTCs are used as the amplitude information for reconstructing only the symmetric target images at the output plane. A previous method, in which an additional phase mask is used at the frequency domain as the phase information, proposed by Chang and Chen [9] was proposed to enable the reconstruction of asymmetric images at the output plane. However, the dominating effect arose from the additional phase makes the wrongly reconstructed image recognizable when the phase key at the input plane is incorrect. In the proposed method, the joint power spectra is nonlinearly transformed into the phase information for reconstructing both symmetric and asymmetric images at the output plane, while the dominating effect in the previous method can be released as well. Simulation results of using two different nonlinear transformations with different parameters are provided to verify the proposed method.

© 2006 Optical Society of America

## 1 Introduction

1. C.J. Weaver and J.W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. **5**, 1248–1249 (1966) [CrossRef] [PubMed]

*h*

_{1}(

*x*,

*y*) and

*h*

_{2}(

*x*,

*y*) and their joint Fourier transform

*S*(

_{I}*u*,

*v*) is detected by the use of a charge-coupled device (CCD) in the intensity form (i.e., the joint power spectrum). By passing the joint power spectrum to the spatial light modulator (SLM) as the transmittance signal

*O*(

*u*,

*v*), the cross-correlation and convolution signals of two input images can be obtained at the output plane.

2. B. Javidi and J.L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. **33**, 1752–1756 (1994) [CrossRef]

3. B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. **39**, 2439–2443 (2000) [CrossRef]

2. B. Javidi and J.L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. **33**, 1752–1756 (1994) [CrossRef]

4. T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. **39**, 2031–2035 (2000) [CrossRef]

7. J. Rosen, “Learning in correlators based on projection onto constraint sets,” Opt. Lett. **18**, 1183–1185 (1993) [CrossRef] [PubMed]

8. H.T. Chang and C.T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. **11**, 165–169 (2004) [CrossRef]

*h*

_{1}(

*x*,

*y*) and

*h*

_{2}(

*x*,

*y*) at the input plane are phase-only That is,

*h*

_{1}(

*x*,

*y*) = exp[

*i*2

*πp*

_{1}(

*x*,

*y*)] and

*h*

_{2}(

*x*,

*y*) = exp[

*i*2

*πp*

_{2}(

*x*,

*y*)], where both functions

*p*

_{1}(

*x*,

*y*) and

*p*

_{2}(

*x*,

*y*) are the random numbers within the range [0, 1]. Given a specific target image

*o*(

*x*,

*y*) at the output plane, both the phase functions in the input plane can be determined by use of some phase-retrieval algorithms [6

6. J.R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. **22**, 2758–2769 (1982) [CrossRef]

*h*

_{2}(

*x*,

*y*) is fixed and refers to as a lock and the other phase function

*h*

_{1}(

*x*,

*y*) refers to as a key. To obtain a target image

*o*(

*x*,

*y*) at the output plane for the fixed phase function

*h*

_{2}(

*x*,

*y*), the other one

*h*

_{1}(

*x*,

*y*) is iteratively retrieved using the phase retrieval algorithms such as the projection onto constraint set (POCS) algorithm [7

7. J. Rosen, “Learning in correlators based on projection onto constraint sets,” Opt. Lett. **18**, 1183–1185 (1993) [CrossRef] [PubMed]

5. D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. **40**, 1584–1589 (2001) [CrossRef]

8. H.T. Chang and C.T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. **11**, 165–169 (2004) [CrossRef]

9. H.T. Chang and Ching T. Chen, “Asymmetric-image verification for security optical systems based on joint transform correlator architecture,” Opt. Commun. **239**, 43–54 (2004) [CrossRef]

*H*

_{3}(

*u*,

*v*) = exp[

*i*2

*πP*

_{3}(

*u*,

*v*)] to the SLM that displays the joint power spectrum, which is shown in the amplitude form. The first phase key

*h*

_{1}(

*x*,

*y*) at the input plane and this attached phase mask, as the second phase key

*H*

_{3}(

*u*,

*v*), are pairwised and iteratively retrieved by the use of the POCS and the multiple phase retrieval algorithms [10

10. H.T. Chang, W. C. Lu, and C. J. Kuo, “Multiple-phase retrieval for optical security systems using random phase encoding,” Appl. Opt. **41**, 4825–4834 (2002) [CrossRef] [PubMed]

*H*

_{3}(

*u*,

*v*) attached at the SLM dominates the reconstructed image quality [9

9. H.T. Chang and Ching T. Chen, “Asymmetric-image verification for security optical systems based on joint transform correlator architecture,” Opt. Commun. **239**, 43–54 (2004) [CrossRef]

*H*

_{3}(

*u*,

*v*) is much larger than that of the phase function

*h*

_{1}(

*x*,

*y*) at the input plane. Therefore, even two phase keys belong to different iterations, the resulting images that are obtained from using the mismatched phase keys still show significant meaning with the target image, especially for binary images. That is, any of the retrieved phase attached at the SLM dominates the quality of the reconstructed image. It could be a disadvantage of this method because the recovered image may be visible when mismatched keys are used. On the other hand, to current techniques, SLMs usually can only display either the amplitude or phase information with a limited resolution. Moreover, additional phase mask increases the extra cost and the alignment requirement in the optical system. Therefore, it is desirable to perform asymmetric-image verification using only a single SLM in the reconstruction stage, which is the case of traditional JTC architecture.

*h*

_{1}(

*x*,

*y*) at the input plane is wrong, the reconstructed image is noise-like even the other phase key in the frequency domain is correct.

## 2 The Proposed Architecture

*S*(

_{P}*u*,

*v*) is converted to the phase information exp[

*iS*(

_{P}*u*,

*v*)] for further processing. (2) A nonlinear transformation is applied to the converted phase information such that a more broaden histogram of the transformed phase information exp[

*iS*(

_{I}*u*,

*v*)] can be obtained. On the other hand, the converted phase information will vary a lot when the input phase key

*h*

_{1}(

*x*,

*y*) is changed. (3) An extra phase key exp[

*iH*

_{3}(

*u*,

*v*)], which is required in the proposed method and can further increase the system security level, is employed. The first two operations shown above can be performed by the use of digital signal processing techniques in a computer. On the other hand, the extra phase key exp[

*iH*

_{3}(

*u*,

*v*)] is determined by the use of the retrieved phase exp[

*iO*(

*u*,

*v*)] and the transformed phase exp[

*iS*(

_{I}*u*,

*v*)]. The detailed steps for retrieving the second phase key exp[

*iH*

_{3}(

*u*,

*v*)] based on a given target image

*o*(

*x*,

*y*) and the first phase key

*h*

_{1}(

*x*,

*y*) will be provided.

*h*

_{1}(

*x*,

*y*) and

*h*

_{2}(

*x*,

*y*) at the input plane are fourier transformed and then detected by a square-law detector such as a CCD to obtain the joint power spectrum exp[

*iS*(

_{P}*u*,

*v*)]. That is,

*o*(

*x*,

*y*) at the output plane, the phase required to be appeared at the SLM can be retrieved by applying the POCS algorithm to the optical 2-

*f*architecture [7

7. J. Rosen, “Learning in correlators based on projection onto constraint sets,” Opt. Lett. **18**, 1183–1185 (1993) [CrossRef] [PubMed]

*k*iterations. The retrieved phase function is denoted as

*O*(

_{k}*u*,

*v*) and the corresponding reconstruction image becomes |

*FT*{

*O*(

_{k}*u*,

*v*)}| = |

*o*(

_{k}*x*,

*y*)| ≈

*o*(

*x*,

*y*). The quality of reconstructed image can be shown much better than that of the conventional image verification system based on the JTC architecture.

*π*], which is similar to that of the phase retrieved from the 2-

*f*optical architecture using POCS algorithm. Two representative nonlinear functions are employed in the proposed method: the power-law and log-sigmoid functions. For a given input signal

*ξ*, the output

*η*of the nonlinear function

*g*is denoted as

*η*=

*g*(

*ξ*). The power-law function is defined as

*a*and

*b*are constants. Figures 4(a) and 4(b) show the power-law and log-sigmoid functions under different parameters, respectively. Note that the ranges of the power spectrum and phase information should be normalized, respectively, to the input and output of the nonlinear functions.

*iH*

_{3}(

*u*,

*v*)] is thus required to make the retrieved phase information

*O*(

_{k}*u*,

*v*) appear at the SLM. Thus the target image can be reconstructed at the output plane. The second phase key exp[

*iH*

_{3}(

*u*,

*v*)] is determined as

9. H.T. Chang and Ching T. Chen, “Asymmetric-image verification for security optical systems based on joint transform correlator architecture,” Opt. Commun. **239**, 43–54 (2004) [CrossRef]

10. H.T. Chang, W. C. Lu, and C. J. Kuo, “Multiple-phase retrieval for optical security systems using random phase encoding,” Appl. Opt. **41**, 4825–4834 (2002) [CrossRef] [PubMed]

*f*architecture. In the proposed method, the first phase key

*h*

_{1}(

*x*,

*y*) can be randomly generated rather than retrieved from the iterative POCS algorithm. The second phase key is determined by the use of transformed phase and the retrieved phase at the SLM, which is much easier to be obtained than using the iterative algorithm through the whole JTC architecture, as shown in the previous study. Note that the second phase key is one of the inputs of the computer. Therefore, only one SLM is required and only the phase information will appear at the SLM for asymmetric image reconstruction. Compared with the previous method, the proposed architecture is more practical for system implementation in addition to releasing the dominating effect caused from the second phase key.

## 3 Simulation Results

*B*×

*B*pixels are represented by the mean squared error (MSE), which is defined as

*h*

_{1}(

*x*,

*y*) and the phase lock

*h*

_{2}(

*x*,

*y*) at the input plane. Note that the phase information has been transformed to the grayscale values for visualization. Figures 6(b) and 6(c) show the visualized phase information exp[

*iS*(

_{P}*u*,

*v*)] after using the power-law and log-sigmoid functions, respectively. The histograms of Figs. 6(b)–6(d) are shown in Fig. 7(a)–7(c), respectively. Obviously, the histograms of the phase information shown in Figs. 6(a) and 6(b) are more nonuniform and not enough uniformly distributed, respectively. Figure 6(c) shows the much better equalization effect is obtained by the use of the log-sigmoid function with the parameter

*a*= 9. On the other side, the iteration algorithm is used to retrieve the phase information exp[

*iO*(

_{k}*u*,

*v*)] at the SLM for a given target image. Then the second phase key exp[

*iH*(

*u*,

*v*)] can be computed according to Eqs. (4) and (5). Note that the reconstructed image quality depends only on this stage and will not be affected by any randomly generated first phase key

*h*

_{1}(

*x*,

*y*) and the phase lock

*h*

_{2}(

*x*,

*y*).

**239**, 43–54 (2004) [CrossRef]

*h*

_{1}(

*x*,

*y*), the reconstructed symmetric and asymmetric images shown in Figs. 8(c) and 8(d) are still recognizable. Thus the attached second phase key dominates the reconstructed image quality in this method. Figures 9(a) and 9(b) show the reconstructed symmetric and asymmetric target images, respectively, with the correct phase keys and using the power-law function (

*b*= 0.3) in the nonlinear transformation. Given a wrong (randomly generated) first phase key

*h*

_{1}(

*x*,

*y*) and the correct second phase key exp[

*iH*

_{3}(

*u*,

*v*)], the dominating effects of the second phase key are obvious in Figs. 9(c) and 9(d), in which the reconstructed images are recognizable, although they are much less than that in the previous method. On the other hand, Figs. 10(a) and 10(b) show the reconstructed symmetric and asymmetric target images, respectively, with the correct phase keys and using the log-sigmoid function (

*a*= 9) in the nonlinear transform. The dominating effects of the second phase key exp[

*iH*

_{3}(

*u*,

*v*)] are eliminated in Figs. 10(c) and 10(d), in which the reconstructed images are totally unrecognizable. It is verified that the proposed method, while using the log-sigmoid function in the nonlinear transform, can release the dominating effect of the second phase key.

*h*

_{1}(

*x*,

*y*) and

*H*

_{3}(

*u*.

*v*), are correct and when one of the phase keys,

*h*

_{1}(

*x*,

*y*), is wrong are provided for comparison. Table 1 summarizes the MSE results of the proposed method using the power-law function with different values of the parameter

*b*. As shown in this table, the parameter value

*b*= 0.3 can lead almost the best performance for both the symmetric and asymmetric images. On the other hand, Table 2 summarizes the MSE results of the proposed method using the log-sigmoid function with different values of the parameter

*a*. As shown in this table, the parameter value

*a*= 9 can lead almost the best performance for both the symmetric and asymmetric images. From the MSE results shown in both tables, the reconstructed image quality is about the same when both keys are correct. On the other hand, the log-sigmoid function can make the MSE of the reconstructed image much larger than that of using power-law function when the first phase key

*h*

_{1}(

*x*,

*y*) at the input plane is wrong. Therefore, the log-sigmoid function is preferred to as the nonlinear function for transforming the joint power spectrum in the proposed method. Since there are many nonlinear transform functions, different transforms and parameters will lead to different performances on the security level and reconstructed image quality. Furthermore, it will be worthy to explore or investigate other nonlinear transforms for obtaining better performance.

## 4 Conclusion

## References and links

1. | C.J. Weaver and J.W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. |

2. | B. Javidi and J.L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. |

3. | B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. |

4. | T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. |

5. | D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. |

6. | J.R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. |

7. | J. Rosen, “Learning in correlators based on projection onto constraint sets,” Opt. Lett. |

8. | H.T. Chang and C.T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. |

9. | H.T. Chang and Ching T. Chen, “Asymmetric-image verification for security optical systems based on joint transform correlator architecture,” Opt. Commun. |

10. | H.T. Chang, W. C. Lu, and C. J. Kuo, “Multiple-phase retrieval for optical security systems using random phase encoding,” Appl. Opt. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.4550) Fourier optics and signal processing : Correlators

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: December 20, 2005

Revised Manuscript: February 13, 2006

Manuscript Accepted: February 14, 2006

Published: February 20, 2006

**Citation**

Hsuan Chang and Chao-C. Chen, "Fully-phase asymmetric-image verification system based on joint transform correlator," Opt. Express **14**, 1458-1467 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1458

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### References

- C.J. Weaver and J.W. Goodman, "A technique for optically convolving two functions," Appl. Opt. 5, 1248-1249 (1966). [CrossRef] [PubMed]
- B. Javidi and J.L. Horner, "Optical pattern recognition for validation and security verification," Opt. Eng. 33, 1752-1756 (1994). [CrossRef]
- B. Javidi and T. Nomura, "Polarization encoding for optical security systems," Opt. Eng. 39, 2439-2443 (2000). [CrossRef]
- T. Nomura and B. Javidi, "Optical encryption using a joint transform correlator architecture," Opt. Eng. 39, 2031-2035 (2000). [CrossRef]
- D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001). [CrossRef]
- J.R. Fienup, "Phase retrieval algorithm: a comparison," Appl. Opt. 22, 2758-2769 (1982). [CrossRef]
- J. Rosen, "Learning in correlators based on projection onto constraint sets," Opt. Lett. 18, 1183-1185 (1993). [CrossRef] [PubMed]
- H.T. Chang and C.T. Chen, "Enhanced optical image verification based on joint transform correlator adopting Fourier hologram," Opt. Rev. 11, 165-169 (2004). [CrossRef]
- H.T. Chang and ChingT. Chen, "Asymmetric-image verification for security optical systems based on joint transform correlator architecture," Opt. Commun. 239, 43-54 (2004). [CrossRef]
- H.T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems using random phase encoding," Appl. Opt. 41, 4825-4834 (2002). [CrossRef] [PubMed]

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