## Abstract

In this paper, we propose a new method for color image coding and synthesis based on fractional Fourier transforms and wavelength multiplexing with digital holography. A color image is divided into three channels and each channel, in which the information is encrypted with different wavelength, fractional orders and random phase masks, is independently encrypted or synthesized. The system parameters are additional keys and this method would improve the security of information encryption. The images are fused or subtracted by phase shifting technique. The possible optical implementations for color image encryption and synthesis are also proposed with some simulation results that show the possibility of the proposed idea.

©2007 Optical Society of America

## 1. Introduction

In recent years, optical information processing techniques have attracted a great deal of interest. In order to protect people’s privacy, information security problem has been paid more and more attention with the development of science and technology. Since Refregier and Javidi [1] proposed the double random phase encryption method for the first time in 1995, people have presented a number of optical image encryption techniques for their interesting and meaningful in practical applications [2–15]. An image is encrypted to a white noise picture in a double random encoding system with two random phase plates, which are displayed respectively at the input and Fourier planes. After that, the encryption systems based on fractional Fourier transforms (FRFT) and Fresnel transforms were presented [3, 4]. Besides double random phase masks, the system parameters are also their important keys in decryption in the two methods above. And there are many other good techniques for optical image encryption, such as encryption with digital holography [9, 11, 12] or joint transform correlators, etc. Optical image subtraction is also an important processing operation used extensively in earth resource studies, photography, inspection, etc. Meanwhile, we often want to add different images together, and it will be more beautiful or interesting than the original pictures. Several optical image addition and subtraction techniques have been proposed in the past four decades [16–24]. The optical image synthesis was described by Gabor *et al* firstly in 1965, by superposing the complex diffraction patterns of two or more images at the same holographic plate [16]. A comprehensive review paper on image subtraction was published in 1975 [17]. Recently, we have described a new image addition and subtraction technique with phase-shifting interferometry [24]. However, in most of these methods, the images are illuminated by monochromatic light, and the reconstructed images will lose their color information which is useful in information processing and practical applications in many cases. To our knowledge, only two methods that one is color image encryption based on Fourier transform introduced by Zhang and Karim [25] in 1999, while the other is an optical color image encryption by lensless Fresnel transform holograms [26] proposed by us have been published. Wavelength multiplexing method was proposed in 2004 for multiple image encryption [15], and a real color FRFT has been presented recently [27]. Phase-shifting and wavelength-multiplexing techniques have been applied to color information reconstruction by digital holography [28]. In this paper, we will introduce a color information processing technique by wavelength multiplexing and FRFT. Color images are encrypted with double random phase keys and fractional orders, therefore the system parameters are additional keys which indicates that our method will improve the security and have potential applications both in optical and wireless color image coding. The images can also be fused or subtracted by phase shifting techniques. Image addition and subtraction of encrypted data has many potential applications in content distribution and multiple authentication. A color image is divided into three channels and each channel is to be independently encrypted with different wavelength or even different phase masks. The keys can be kept in several places or assigned to a group of authorized people. All the keys in the three parts are important in decryption for that the image can be recovered only when all of them are correct. The method of color information coding and synthesis which is realized with digital simulations may have potential applications in digital color information protection and authentication.

This paper is organized as follows. Section 2 gives the basic theory of color image decomposition and processing with wavelength multiplexing. In section 3, the technique of double random image encryption and fractional Fourier holography is presented and some numerical simulations are given based on both on-axis and off-axis illuminations. While in section 4, image addition and subtraction by two-step phase shifting technique and digital fractional Fourier holography are introduced. Then color image processing (encryption and synthesis) is analyzed, and its possibility is demonstrated by computer simulations in part 5. Finally, some conclusions are outlined in section 6.

## 2. Color image decomposition and processing with wavelength multiplexing

In a color image, each pixel is represented by three RGB values of its color, as shown in Fig. 1. The real color is the composition of three RGB values with certain proportions. Therefore, color image *f*(*x,y*) can be decomposed into three components R(*x,y*), G(*x,y*) and B(*x,y*), each of which corresponds to one color (red, green or blue).

Figure 2 is computer simulation result of real color image decomposition. Figure 2(a) is an original RGB JPEG image with 800×600 pixels and Figs. 2(b), (c) and (d) are its corresponding red, green and blue parts. To incorporate the color information, it is suggested by using a wavelength multiplexing method to synthesize the image. The information is separated into red, green and blue three channels and each channel is independently encrypted or synthesized. If the incident wavelength of each channel is close to the wavelength of basic color (*λ*
_{1}=700nm, *λ*
_{2}=546.1nm, *λ*
_{3}=435.8nm), the real color information can be recovered at the output plane.

## 3. Double random image encryption and fractional Fourier holography

Here we first discuss the basic theory of the proposed idea. The original image *f*(*x,y*) is encoded for the first time at the input plane with random distributed phases *P*(*x,y*), which is expressed as exp[*i2πp*(*x, y*)] and *p*(*x,y*) is independent white sequences uniformly distributed on the interval [0, 1], and fractional Fourier transformed with an input wavelength *λ*. It can be described as

$$\times \mathrm{exp}\left(i\pi \frac{{y}^{2}+{{y}_{1}}^{2}}{\lambda {f}_{s2}\mathrm{tan}{\varphi}_{2}}-2\pi i\frac{{\mathrm{yy}}_{1}}{\lambda {f}_{s2}\mathrm{sin}{\varphi}_{2}}\right)\text{d}x\text{d}y,$$

where *ϕ*
_{1}=*p _{1}π*/2 and

*ϕ*/2,

_{2}=p_{2}π*p*

_{1}and

*p*

_{2}are the fractional orders of FRFT,

*f*

_{s1}and

*f*

_{s2}are the standard focal lengths of FRFT in the

*x*and

*y*directions, respectively. After recording in a hologram halide or CCD camera, it becomes [9, 11, 23]

Where *Q*(*u,v*) is diffractive field of phase distribution *Q*(*x, y*) which is expressed as exp[*i2πq*(*x, y*)] and *q*(*x,y*) is another independent white sequences uniformly distributed on the interval [0, 1]. The corresponding optical encryption implementation is illustrated in Fig. 3, which can also be found in Ref. [7]. RPM1 is associated with *p*(*x,y*) and RPM_{2} is associated with *q*(*x,y*). The relationships of fractional orders, focal lengths and propagations distances can be obtained in Ref. [29].

Therefore the original image is encoded for the second time, the information is randomly distributed at the fractional plane, and people cannot obtain the correct information easily without the correct keys. By removing the input image, random phases RPM1 and lens *f*
_{1} and illuminating with a plane wave of uniform unitary amplitude, the holographic data of the fractional Fourier phase mask is [9, 11, 23]

When the information is recorded by on-axis hologram, all the information would be mixed together and people cannot retrieve the part they want by this way, so the recovered image would have some noise of the other terms. However, if the reference beam is a slightly inclined planar beam, the last term on the right-side of Eq. (2) and the first term on the right-side of Eq. (3) can be extracted. But in this way, an important factor that is the space-bandwidth product of the random phase mask must be considered. In order to ensure the lossless transmission of the information, we must keep the space-bandwidth product of the information not to exceed that of the system. It should be carefully dealt with in the optical experiments including the choice of numerical aperture of lens and the dimensions of the CCD array, etc [9, 11].

When the retrieved fractional Fourier phase mask is placed before the recorded intensity plane and illuminated by the same wavelength, the last term of Eq. (2) can be used to recover the original image, and it is expressed as

$$\times \mathrm{exp}\left(i\pi \frac{{y}^{2}+{{y}_{1}}^{2}}{\lambda {f}_{s2}\mathrm{tan}\left({-\varphi}_{2}\right)}-2\pi i\frac{{\mathrm{yy}}_{1}}{\lambda {f}_{s2}\mathrm{sin}\left({-\varphi}_{2}\right)}\right)d{x}_{1}d{y}_{1}$$

$$=f(x,y)P(x,y).$$

Here, *P*(*x,y*) would not influence the reconstruction of original images.

In a color image encryption system, the information is encrypted independently in three channels and the keys in all these channels should be correct in decryption, otherwise the color information cannot be correctly recovered. Fractional orders and the double random phases in all the three channels are the encryption keys, so it would strengthen the information protection.

In the following part, we give some computer simulation results based on the concept of digital holography and “virtual optics”. The concept is based on optics but we realize the whole process in the computer simulations. In the actual optical realizations, the focal lengths of the lens and the distances from input planes to the lens should be carefully chosen in each channel, to make sure the three parts of a color image are recovered at the same plane and with the same size. Algorithm for reconstruction of digital holograms with adjustable magnification was proposed in 2004 [30]. As a result of “virtual optics”, there are no costly optical elements, no physical limitations and no difficulties in optical alignment. Both the encryption and decryption processes are performed digitally. It avoids image re-scaling and pixel-mismatch problems in the computer simulations, because we use the fast Fourier transform and convolution method in processing. And it enables us to store and transmit data easily in the computer.

At first, we simulate the result with on-axis illumination. A landscape picture is used as an original input image with 1024×768 pixels as shown in Fig. 4(a). The actual size of the image and the phase masks are 10 mm in the simulations. Figure 4(b) is its color encrypted fractional hologram, which is expressed as Eq. (2). In the first channel encryption, we use red wavelength *λ*
_{1}=700nm, *p*=0.40 in both x and y directions. And the parameters for other two channels are *λ*
_{2}=546.1nm, *p*=0.50 ; *λ*
_{3}=435.8nm, *p*=0.667, respectively. *λ*
_{1}, *λ*
_{2} and *λ*
_{3} are the wavelengths of basic RGB color in the computer. The image is encrypted in three channels, and we can find the encrypted picture contains three noises (Red, Green and Blue). Figure 4(c) is the key hologram as expressed in Eq. (3). Retrieved from the above two holograms, Fig. 4(d) is the recovered color image with some background noise. We can notice that the reconstructed image is obscure to recognize with on-axis illumination method.

In the following simulations, we reconstruct the image with the corresponding useful terms that can be retrieved by off-axis illumination. Figure 5(a) is the distribution of the last term on the right side of Eq. (2), which is retrieved from Fig. 4(b). Figure 5(b) is the distribution of the first term on the right of Eq. (3), which is retrieved from Fig. 4(c). Figure 5(c) shows one of the incorrectly decrypted images when all the fractional orders are incorrect in decryption, with *p*
^{–}=-0.2,-0.25,-0.33, respectively, in the three channels. Figure 5(d) is the decryption image when all the random phase masks are incorrect. Figure 5(e) shows the result with both fractional orders and random phases are incorrectly selected at the same time. The recovered pictures are obscure to recognize if the keys are incorrect in decryption. When only one or two channels are correctly decrypted, the correct color information also cannot be obtained. In Fig. 5(f) the blue component is incorrectly decrypted, and the other two parts are correct. In Fig. 5(g) the red component is incorrectly decrypted, and in Fig. 5(h) the green color is wrong recovered. We can notice that not only the color of trees but also the flowers, clouds and hills are distorted. For example, the color of clouds changes to yellow in Fig. 5(f) and purple in Fig. 5(h), and this is terrible because it transfers the pseudo information to people. However, in Figs. 5(i)–5(k), the results are worse, as only one color component is correctly decoded, the reconstructed images are single color images with some noise of other color information. In Fig. 5(i), green and blue components are incorrectly decrypted. Fig. 5(j) is the result when keys in red and blue segments are wrong selected. While Fig. 5(k) shows the result when red and green components are incorrectly decrypted. Finally, Fig. 5(l) is the correctly decrypted image.

## 4. Image addition and subtraction by two-step phase shifting technique and digital holography with FRFT

People often use Fourier transform to implement image addition or subtraction. As we know, FRFT is the generalization of Fourier transform and it is widely used in information processing, therefore we can use FRFT to realize image synthesis. Two images *f ^{′}*

_{1}(

*x,y*) and

*f*

^{′}_{2}(

*x,y*) are fractional Fourier transformed before being superposed and can be expressed as

$$\times \mathrm{exp}\left(i\pi \frac{{y}^{2}+{{y}_{1}}^{2}}{\lambda {f}_{s2}\mathrm{tan}{\varphi}_{2}}-2\pi i\frac{{\mathrm{yy}}_{1}}{\lambda {f}_{s2}\mathrm{sin}{\varphi}_{2}}\right)dxdy.$$

The similar optical implementation to Fig. 3 can be used in image addition and subtraction with the replacement of RPM2 by a *π* phase delay (PD). In recording the first image, the phase delay is omitted in the reference beam. But the other image to be subtracted should be multiplied by *π* phase delay. The intensities of the two images recorded at the output plane can be described as

The reference beam is supposed to be a slightly inclined planar beam, therefore the last term on the right-side of Eq. (6) and Eq. (7) can be extracted. And *Q*
_{1}(*u, v*) denotes its field distribution after diffraction. When the same reference beam *Q*
_{1}(*u, v*) illuminates at the input plane of the reconstruction implementation, the last terms on the right-side of Eq. (6) and Eq. (7) are used to recover the subtracted images. Combined the corresponding two terms, it can be written as

$$={g}_{1}^{\prime}({x}_{1},{y}_{1})+{g}_{2}^{\prime}({x}_{1},{y}_{1}),$$

There would be a constant phase factor in the above formula. But it does not influence the image recovery. And we omit it. The subtracted image is recovered after an inverse FRFT, just as

$$\times \mathrm{exp}\left(i\pi \frac{{y}^{2}+{{y}_{1}}^{2}}{\lambda {f}_{s2}\mathrm{tan}\left({-\varphi}_{2}\right)}-2\pi i\frac{{\mathrm{yy}}_{1}}{\lambda {f}_{s2}\mathrm{sin}\left({-\varphi}_{2}\right)}\right){dx}_{1}d{y}_{1}$$

$$=\mathrm{IFrFT}\left[{g}_{1}^{\prime}({x}_{1},{y}_{1})\right]-\mathrm{IFrFT}\left[{g}_{2}^{\prime}({x}_{1},{y}_{1})\right]$$

$$={f}_{1}^{\prime}(x,y)-{f}_{2}^{\prime}(x,y).$$

When the phase delay is omitted in recording both two images, it becomes

then the fused image would be reconstructed at the output plane of inverse FRFT,

$$={f}_{1}^{\prime}(x,y)+{f}_{2}^{\prime}(x,y).$$

When the three wavelengths are illuminated at the input plane one by one in the recording process and the information is recorded at the same CCD camera, the addition or subtraction of color images can be realized. The added or subtracted color image would be recovered at the output plane of inverse FRFT in reconstruction process. Some computer simulations are demonstrated in the following pictures. Figure 6 is color image addition and subtraction results. After transformations and synthesis, some of the pixel values of the final result are much greater than 255, so we make them unitary and then normalization to make sure that they are distributed on the interval [0, 255]. Figure 6(a) and Fig. 6(b) are two original images with 1024×768 pixels, and Fig. 6(c) is their color addition image. The intensity of addition image is increased after synthesis and the image would be somewhat bright. It is a kind of color distortion and Fig. 6(d) is the result after some post-processing. Figure 6(e) and Fig. 6(f) are other two original images with 257×255 pixels, while Fig. 6(g) is their color subtraction result. We notice that some color information of the candle has been lost. Some post-processing can be made, which is shown in Fig. 6(h). It can also be realized in optical realization, as in the input plane we can mask the corresponding area that we won’t want to be subtracted in the first image in recording the second image. Then the corresponding intensity of the first image would be remained and the information would be recovered after subtraction and inverse FRFT. In this simulation, we also use wavelength *λ*
_{1}=700nm, *p*=0.40 in both *x* and *y* directions in red channel; *λ*
_{2}=546.1nm, *p*=0.50 ; and *λ*
_{3}=435.8nm, *p*=0.667, respectively, in the other two channels.

## 5. Color image processing (encryption and synthesis) with double random phase encryption, two-step phase shifting technique and wavelength multiplexing in FRFT domain

In this section, we give the computer simulation results of color image processing. The images are encoded by double random phase encryption method and added or subtracted with twostep phase shifting technique. Then the added or subtracted images would be reconstructed in the decryption with correct keys. The same keys should be used in the same channel for every color image. The whole process is also realized with digital simulations and based on the fast Fourier transforms and convolution algorithms. In order to keep the image to be seen clearly, we extract the useful information based on the off-axis illumination.

Figure 7(a) is the encryption result of addition images, and the three original images are shown in Fig. 4(a), Fig. 6(a) and Fig. 6(b). In the first channel encryption, we use red wavelength *λ*
_{1}=700nm, *p*=0.90 in both *x* and *y* axes. And the parameters of other two channels are *λ*
_{2}=546.1nm, *p*=0.70 ; *λ*
_{3}=435.8nm, *p*=0.30, respectively. Figure 7(b) shows the decryption image when fractional order keys are incorrect with *p*
^{′}=-0.72,-0.35,-0.15, respectively, in the three channels. Figure 7(c) is the decryption result with incorrect random phase masks in all the three channels. Figure 7(d) is only the red component correctly decrypted, and the other parts are incorrect. In Fig. 7(e) the red component is wrong, the green and blue parts are correct in recovery. At last, we can notice that the correct addition image appears in Fig. 7(f).

Figure 8(a) is the corresponding encryption result of subtraction images, and the two original images are shown in Fig. 6(e) and Fig. 6(f). The same parameters are used in encryption and decryption as simulations in Fig. 7. Figure 8(b) is an incorrect decryption image with wrong fractional orders in the three channels. Figure 8(c) is the result when random phase masks are incorrect in decryption. Figure 8(d) is only the blue component correctly decrypted and Fig. 8(e) is the red color and the blue color correctly recovered. Finally, Fig. 8(f) is the correct decryption image, and the result can also be post-processed to recover the color information of the candle.

From Fig. 7 and Fig. 8, we may find that the color image encryption systems are more difficult in decryption. When only one or two channels are correctly decrypted, the correct color information also cannot be obtained, which means that people need more keys to obtain the correct information and that it increases the difficulty in attacking the system by an unauthorized people. The fused or subtracted images are more beautiful and interesting than the original ones and people can get more information in the synthesized pictures. Therefore we believe our method would have great potential applications in optical and digital information processing.

## 6. Conclusions

In this paper, we have proposed a new method for color image processing including image encoding and synthesis. A color image is decomposed into three channels (Red, Green and Blue), each of which is encrypted, decrypted, added or subtracted respectively, and finally combined together in reconstruction. The encryption method is based on double random encryption technique and fractional Fourier holography, and the image addition or subtraction is realized by two-step phase shifting technique. The simulations are on the concept of digital holography and virtual optics, which is based on optics but we realize the whole process in the computer simulations. Therefore our method would be useful both in optics and digital color information processing.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under grants 60478041, and the program of the Ministry of Education of China for New Century Excellent Talents in University. One of the authors (D. Zhao as a visiting scholar at the University of Rochester) was grateful to the support by Pao Yu-Kong and Pao Zhao-Long Scholarship.

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