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1870 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 H.-E. Hwang and P. Han

Fast algorithm of phase masks for imageencryption in the Fresnel domain

Hone-Ene Hwang

Department of Electronic Engineering, Chung Chou Institute of Technology, Yuan-lin 510, Changhua, Taiwan

Pin Han

Institute of Precision Engineering, National Chung Hsing University, 250, Kuo Kuang Road, Taichung 402, Taiwan

Received November 4, 2005; revised February 7, 2006; accepted February 10, 2006; posted February 17, 2006 (Doc. ID 65799)

A high performance lensless optical security system based on the discrete Fresnel transform is presented. Twophase-only masks are generated with what we believe to be a novel and efficient algorithm. Their positioncoordinates and the wavelength are used as encoding parameters in the encryption process. Compared withprevious studies, the main advantage of this proposed encryption system is that it does not need any iterativealgorithms to produce the masks, and that makes it very efficient and easy to implement without losing theencryption security. © 2006 Optical Society of America

OCIS codes: 070.4560, 100.2000, 200.4740.

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. INTRODUCTIONs a result of the enormous amount of data and images

ransmitted through the Internet, optical information se-urity techniques and encryption algorithms with digitalransmission ability have gained increasing interest.1–21

ome schemes12,13 have been proposed to improve theonventional 4f correlator architecture that is used totore the encrypted data holographically.1–4,16 For ex-mple, a phase-only mask (POM) is used in the Fourierlane of a 4f correlator. However, obtaining those previ-us POMs required complicated iterative algorithms, andhe decrypted image is only an approximation to the tar-et image. The degree of similarity is decided by the cor-elation coefficient, which would consume more iterationime if a higher value were needed.17

An exact image retrieval method with much more effi-ient algorithms without iteration is suggested in thisork. By using the discrete Fresnel transform, two POMs

an be generated straightforwardly, and the recovered im-ge is identical to the target image. Furthermore the se-urity of this scheme is no less than that of other POMystems.

Figure 1 illustrates the optical setup of the suggestedystem. Three planes are defined: the input plane inhich POM2 is located, the transform plane in which theOM1 is located and that is z2 away from the input plane,nd the output plane, which is z1 away from the trans-orm plane. Distance parameters z1 and z2 are decided ac-ording to the size of the POMs to satisfy the Fresnel ap-roximation. The target image can be obtained at theutput plane when the system is directly illuminated withn appropriate light whose wavelength is one of the pa-ameters for encryption.

The encryption process in this proposed system is digi-al while the decryption process can be implemented op-ically or digitally. In the optical implementation case, theesigned phase distribution can be fabricated in the form

1084-7529/06/081870-5/$15.00 © 2

f POMs with micro-optics fabrication techniques.18 How-ver, as shown below in Section 3, the requirements onhe wavelength precision and the POM location precisionre of the order of 10−5 nm and 1.0 nm, respectively,hich is very difficult to achieve for today’s optical engi-eering. Thus this suggested algorithm is more suitableor digital usage. For digital implementation, the encryp-ion process is simply performed with a digital computer;his flexibility gives another advantage: The encryptedata can be directly transmitted over the digital commu-ication lines and then decrypted with the correct keysthe positions of POMs and the wavelength) at the re-eiver.

The noniterative and efficient algorithm is presented inection 2. In Section 3, this system will be simulated anderformed digitally; also the wavelength sensitivity andOM position shift tolerance are studied.

. ENCRYPTION ALGORITHMhe encryption problem is to encode the target image into

he phase functions. We denote the target image g�x0 ,y0�nd the distribution of POM1 and POM2 exp� j�1�x1 ,y1��nd exp� j�2�x2 ,y2��, respectively. A unit plane wave is as-umed to be used in the encryption process. With theresnel approximation, the complex amplitude u�x1 ,y1� in

he transform plane with respect to exp� j�2�x2 ,y2�� is pre-ented as

�x1,y1�

=�� exp� j�2�x2,y2��h�x1,y1;x2,y2;z1,z2;��dx2dy2,

�1�

here

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Fig. 2. Image used as the target image for computer simulation.

Fig. 3. Phase distribution of �2�x2 ,y2�.

H.-E. Hwang and P. Han Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 1871

h�x1,y1;x2,y2;z2;�� =exp� j2�z2/��

j�z2

�exp�−j�

�z2��x1 − x2�2 + �y1 − y2�2��

s the point spread function of the first stage of the sys-em, and � is the wavelength of the plane wave. We canewrite Eq. (1) for simplicity as

u�x1,y1� = FrT��exp� j�2�x2,y2��;z2, �2�

here FrT denotes Fresnel transform.Now we introduce the encryption process. First the

hase function �2�x2 ,y2� is generated arbitrarily with aandom number generator. Then we calculate the inverseresnel transform17 (IFrT) of the target image g�x0 ,y0� atistance z1. It is

u��x1,y1� = IFrT��g�x0,y0�;z1. �3�

hus the phase function �1�x1 ,y1� can be determined byhe relation exp� j�1�x1 ,y1��=u��x1 ,y1� /u�x1 ,y1�; that is,

�1�x1,y1� = arg�u��x1,y1�

u�x1,y1� � , �4�

here arg is the argument.For illustrating how this scheme works, the decryption

f the POMs is explained. Construct the setup as in Fig. 1ith correct parameters z1, z2, and �. At the first stage,

he complex amplitude field u�x1 ,y1� in Eq. (2) will appeart z2. Then u�x1 ,y1� is multiplied by the phase function ofOM1, and again it is Fresnel transformed to the outputlane at z1, where it is denoted as g�x0 ,y0�. Thus g�x0 ,y0�,ith the help of Eqs. (3) and (4), is found through the fol-

owing derivation:

g�x0,y0� = FrT��u�x1,y1�exp� j�1�x1,y1��;z1

= FrT�u�x1,y1�u��x1,y1�

u�x1,y1�;z1� = FrT��u��x1,y1�;z1�

= FrT��IFrT��g�x0,y0�;z1�;z1 = g�x0,y0�. �5�

It is obvious that the decryption image is exactly theame as the target image and no information is lost. Thischeme is very straightforward and efficient, and no itera-ive procedure is needed.

. SIMULATION AND ANALYSISn this section the proposed algorithms are implementedo simulate the results. The wavelength and the axial

Fig. 1. Optical setup of the lensless optical security system.

Fig. 4. Phase distribution of � �x ,y �.

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1872 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 H.-E. Hwang and P. Han

hift sensitivity of the system are also studied. In theimulation, the image shown in Fig. 2 is used as targetmage. It is of size 256�256 with 256 grayscales. Theizes of both of the POMs are the same as that of the tar-et image. We assume that the operation wavelength � ofhe incident unity plane wave for encryption is 632.8 nm;he position parameters z1 and z2 are 60 and 30 mm, re-pectively; and the aperture of the output plane is.2 mm. The algorithm starts with arbitrary �2�x2 ,y2�, ashown in Fig. 3, which is a random-noiselike phase distri-ution. In order to implement this scheme digitally, weeed to perform the discrete Fresnel transform (DFrT).he definitions of DFrT and its inverse transform (IDFrT)

n Ref. 22 are used in the following encryption process.ith Eq. (4) and IDFrT, �1�x1 ,y1� can be obtained as

hown in Fig. 4, which is still a random-noiselike func-ion.

In decryption, the recovered image is generated by us-ng the two POMs in Eqs. (2) and (4) and substituting intohe first line of Eq. (5) by performing the DFrT. The finalecryption image is shown in Fig. 5 and it is exactly theame as the target image. The correlation coefficient be-ween them is 1; thus, no information of the target images lost.

After illustrating the feasibility of this system, we cannalyze the security and the sensitivity of the parameterskeys). Assume that the POMs are designed with the pa-ameters provided in the first paragraph of this section.he correlation coefficient � is used to study the param-ters’ sensitivity and is defined as17:

� =E��g − E�g����g� − E��g���

�E��g − E�g��2E���g� − E��g���2�1/2, �6�

here g�x0 ,y0� and g�x0 ,y0� are target image and recov-red image, respectively.

In order to determine the wavelength sensitivity, theecryption wavelength is shifted from the encryptionavelength by ��. The corresponding correlation coeffi-

0−5 nm, �=0.5, (b) wavelength difference ��=5.0�10−4 nm, �

ig. 5. Decrypted image obtained at the output with correct�x ,y �, � �x ,y � and the corresponding keys.

ig. 6. Correlation coefficient between the target and the recov-red image as a function of the wavelength difference betweenhe encryption and decryption beam.

ig. 7. Retrieved image when (a) wavelength difference ��=2.2�10.215.

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H.-E. Hwang and P. Han Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 1873

ient between the target and the recovered image as theunction of the wavelength difference ��, which rangesrom −50 to 50 nm, is shown in Fig. 6. It indicates thathe quality of the recovered image is quite sensitive to theavelength of the decryption beam. The correlation coef-cient decreases very rapidly as �� increases, and hasropped to 0.5 with a wavelength difference of only ±2.210−5 nm. The corresponding retrieved image is shown

n Fig. 7(a). When the ��=5�10−4 nm, the correlation co-fficient is 0.215, and the recovered image is almost un-ecognizable, as shown in Fig. 7(b). Therefore, the wave-ength can be used as the key to recover the target image.

To evaluate the axial shifting sensitivity, the sameavelength for encryption and decryption is used. The po-

ition parameters z1 and z2 are 30 and 20 mm, respec-ively, and the pixel size in all these planes is therefore.25 �m. For simplicity, the distance between the inputnd the output plane is fixed to be �z1+z2�=50 mm. Weow shift POM1 from its matched position by a distance

ig. 8. Correlation coefficient between the target and the recov-red image as a function of the axial offset of POM1 from itsatched position.

Fig. 9. Retrieved image when (a) �z

z along the axis. Therefore, the distances between thenput POM2 and POM1 and between POM1 and the out-ut plane become �z2+�z� and �z1−�z�, respectively.Figure 8 shows the behavior of the correlation coeffi-

ient between the recovered and the target image versusz, which varies from −10 �m to 100 �m. It indicates

hat the correlation coefficient decreases sharply as �z in-reases. It drops to 0.5 when the �z is only ±1.08 nm. Theorresponding retrieved image is shown in Fig. 9(a).hen �z=18 nm, the correlation coefficient is 0.282, and

he recovered image is nearly unrecognizable, as shown inig. 9(b). Therefore, the position parameters z1 and z2 canlso be used as keys to recover the target image.Now that the recovered image is sensitive to the de-

ryption wavelength and the positions of the POMs, thesearameters as well as the phase codes can be used as keysf the security system. These additional keys introduce aery useful property to the system. It is easily seen thathe generated phase functions, or essentially, the en-rypted data �1�x1 ,y1� and �2�x2 ,y2�, are real and there-ore can be directly transmitted over digital communica-ion lines. All we need to do is transform the encryptedata �1�x1 ,y1� and �2�x2 ,y2� according to the suggested al-orithm to the output with correct keys z1, z2, and �. Itould be quite difficult for any intruder in the Internet toecipher the encrypted data, because decrypting the in-ormation without knowledge of the position and wave-ength keys requires a random search in an infinite keypace.

. CONCLUSIONnovel lensless optical security system based on

omputer-generated POMs is proposed. Its algorithm isery efficient, secure, and straightforward compared withrevious work.16–18 This system has two valuable proper-ies which no prior work can offer: First, it does not needny iterative process to generate POMs. Second, the re-overed image is exactly the same as the target image and

nm, �=0.5; (b) �z=18.0 nm, �=0.282.

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1874 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 H.-E. Hwang and P. Han

o approximation is made. We mathematically analyzehe principle it is based on; subsequently, the simulationesults show that it works very well. Also the sensitivityf the recovered image to the decryption wavelength dif-erence and axial shifting of the POMs is investigated.umerical results indicate that wavelength and positionarameters are very sensitive and can be successfullysed as additional keys besides the phase code of the se-urity system. Higher security can thus be achieved. An-ther advantage is that the encryption data generatedith this proposed system can be directly transmitted via

ommunication lines and then decrypted with the correctddition keys at the receiver. This is very important in to-ay’s high-speed digital transmission environment.

CKNOWLEDGMENTShis study was supported by Chung Chou Institute ofechnology and the National Chung Hsing University. Itas supported also by the National Science Council ofaiwan under contracts NSC 94-2215-E-235-002 andSC 94-2215-E-005-012. The authors also thank the re-iewers for their thoughtful and helpful comments.

Corresponding author H.-E. Hwang may be reached byhone at 886-4-8311498, ext. 2202; fax at 886-4-8314515;-mail at n741@ms26.hinet.net or hikodragon.ccut.edu.tw.

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