## 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

Optical joint transform correlators (JTCs) [1] play an important role in optical signal processing techniques such as the image correlation framework in pattern recognition. Figure 1 shows the optical setup of a conventional JTC architecture. There are two input images *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.

Recently JTCs have shown the wide applications on optical security [2–5]. Two major categories are the optics-based image encryption [3] and security image verification [2,4,7,8] systems. For the applications on image verification, there is no input plain image and both the input functions *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]. Initially, both phase functions are randomly generated. The phase function *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].

One of the major limitations of the conventional image verification systems based on JTCs is that only the symmetric images can be reconstructed at the output plane [5, 8]. In order to reconstruct asymmetric images at the output plane, a complex or, at least, a phase function is required such that the magnitude of its Fourier transform can be asymmetric. Chang and Chen [9] proposed an optical asymmetric-image verification system based on the joint transform correlator architecture. A complex function at the frequency domain can be obtained by attaching an additional phase mask *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]. The reconstruction of asymmetric images is thus possible and the system security is also improved because two phase keys are required in reconstructing the target image.

However, another drawback is that the second phase key *H*
_{3}(*u*, *v*) attached at the SLM dominates the reconstructed image quality [9] because (1) the phase information is usually more important than the amplitude information in signal reconstruction and (2) the data amount of the phase function *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.

In this paper, a fully phase algorithm, which (1) converts the joint power spectrum to phase function with a nonlinear transformation, and (2) does not need an attached phase mask for the SLM, is proposed. The dominating problem in the previous asymmetric-image verification system is solved, while the system security is well maintained. On the other hand, two nonlinear transformations are discussed. A better selection on the nonlinear transformation can more release the dominating effects. Simulation results have verified that once the first phase key *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

Figure 2 shows the schematic diagram of the proposed asymmetric image verification system based on the fully-phase JTC architecture. Compared with the conventional JTC architecture for image verification, there are three major differences: (1) The detected real and non-negative joint power spectrum *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.

Figure 3 shows the block diagram of the proposed method. The two phase functions *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,

The joint power spectrum will be further processed in the computer. Given a target image *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]. Thus the Fourier phase that can reconstruct the target image in the spatial domain can be retrieved. Suppose that the reconstructed image converges to the target image after *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.

The nonlinear transformation is a critical element in the proposed method. It converts the joint power spectrum into phase information and makes the histogram of the converted phase information more uniformly distributed over the whole range [0,2*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

while the log-sigmoid function is defined as

where both the parameters *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.

Here the second phase key exp[*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

where

In the previous method [9], the second phase key is attached at the SLM, which shows the amplitude information, to obtain a complex signal to reconstruct asymmetric images. However, it is not easy to display both the phase and amplitude information in one SLM. On the other hand, a gap will exist even two SLMs are placed side by side. The phase distortion at the Fourier plane will be introduced and the reconstructed image will degrade as well. Furthermore, both the first phase key used at the input plane and the second phase key attached at the SLM are retrieved from the multiple phase retrieval algorithm [10], which is much more cumbersome than the POCS algorithm used for 2-*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

In computer simulation, the quality of the reconstructed images of size *B* × *B* pixels are represented by the mean squared error (MSE), which is defined as

Figures 5(a) and 5(b) show the symmetric and asymmetric target images, respectively, used for testing the proposed method. Figure 6(a) shows the grayscale image of two randomly-generated phase key *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*).

Figures 8(a) and 8(b) show the reconstructed symmetric and asymmetric target images, respectively, with the correct phase keys and based on the previous optical architecture shown in Ref. 9. Given the correct attached phase key but a wrong phase key *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.

The MSE results of reconstructed symmetric and asymmetric images in the proposed method when two phase keys, *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

In conclusion, a fully-phase implementation for asymmetric image verification system based on the JTC architecture is proposed. The phase retrieval algorithm is much simpler than that in the conventional image verification systems. With a proper selection of the nonlinear transformation and the corresponding parameter, the proposed method successfully solves the dominating problem of the attached phase key in the previous optical asymmetric-image verification system. Moreover, the second phase key can be replaced by an input code to the computer rather than being attached to the SLM. Given any mismatched key, the reconstructed image will be very different from the target image. Therefore, the security level of the asymmetric-image verification system is greatly enhanced, while the optical architecture is as simple as that in conventional JTCs as well.

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