A Hybrid Digital Image Watermarking based on Discrete Wavelet Transform, Discrete Cosine Transform, and General Regression Neural Network

Transcription

1 A Hybrd Dgtal Image Watermarkng based on Dscrete Wavelet Transform, Dscrete Cosne Transform, and General Regresson Neural Network Ayoub Taher ; ABSTRACT In ths paper, a new hybrd dgtal watermarkng technque for gray mage based on combnaton of dscrete wavelet transform (DWT), dscrete cosne transform (DCT), and general regresson neural network (GRNN) s proposed. The watermarkng sgnal s embedded n hgher DCT frequency, whch s n the lower and mddle frequences of orgnal mage by DWT joned wth DCT. The ablty of attractng s mproved by pretreatment and re-treatment of mage scramblng and GRNN. Features of lower and mddle frequences sub-band of wavelet coeffcent lmt the postons for watermarkng embeddng. The postons are taken as ndexes of ponts for embeddng watermarkng and map these postons to low and medum frequency spaces to carry watermarkng. Detecton of watermarkng s free of orgnal mage and the balance between transparency and robustness are realzed. The mplementaton results show that the watermarkng algorthm has very good robustness to all knd of attacks. KEYWORDS Hybrd Dgtal Image Watermarkng, Dscrete Wavelet Transform, Dscrete Cosne Transform, General Regresson Neural Network. 1. INTRODUCTION Dgtal watermarkng plays a sgnfcant role to handle the problems of copyrght protecton, owner s dentfcaton and nformaton securty. Dgtal watermarkng [1] s the process of embeddng the watermark bts nto the dgtal protected meda such as mage, audo and vdeo. Imperceptblty, robustness, capacty and securty are the man features of any watermarkng algorthm []-[4]. The watermarkng can be done ether n spatal doman [] or n frequency doman [4], [6]-[8]. The study of watermarkng schemes ether n spatal doman or n frequency doman have shown that transform doman technques are more mperceptble and more robust to common mage processng operatons lke addton of nose, JPEG compresson, blurrng n spatal doman, sharpenng and geometrc operatons such as scalng, translaton and rotaton as comparson to spatal doman schemes. Neural network technology n the applcaton of dgtal watermarkng s proposed only n recent years. The lterature [3]-[6],[9] desgned dfferent watermarkng algorthm, usng neural networks to classfy or generatng adaptve watermark on the mage when watermarkng embedded, ts purpose s to mprove the strength of watermark embedded and fdelty of the mage. The lterature [11] s the watermark detecton by neural network. Through tranng to learn and adjust the weghts of neural networks approxmate the relatonshp of the orgnal sgnal and the watermark sgnal, the watermark s extracted by usng the traned neural network n the recever, whch ams to mprove the accuracy of watermark detecton rate, and realze the blnd watermark detecton. Gven a network archtecture, a set of tranng nput and the expected output, the network can learn from the tranng set and then can be used to classfy or predct the unseen data [10], [5], and [1]. In the feld of sgnal and mage processng, efforts have been made to take the advantage of transform algorthms lke fast Fourer transform (FFT) [7], dscreet cosne transform [1],[13], wavelet transform [14] are used n watermark embeddng and extractng procedure. The problems assocated wth conventonal. Another knd of algorthm realzes watermarkng postonng by means of recordng the postons n a secret table durng watermarkng embeddng [15]. Ths makes the poston of watermarkng very precse and not affected by watermarkng embeddng, mage processng and compresson, even malcous attacks. The dsadvantage of ths knd of algorthm s that the owner A. Taher s wth the Group of IT Engneerng, Payam Noor Unversty, Broujen, Iran (e-mal:ayoob.taher@gmal.com). 1 Amrkabr/ Vol. / No. -Group (Subject ) Month Year

2 of dgtal mages must pay more attentons to storng the poston tables one by one. Usng secret keys and specal algorthms to create postons of watermarkng [16], [17] s a good choce by whch the cost on storage s decreased. In fact, the postons produced by the secret keys and specal algorthms do not take the transparency of watermarkng nto account whch wll result obvous trace of watermarkng embeddng. The tradeoff between transparency and robustness cannot be realzed.. SCHEME DESIGN Scheme desgn s organzed as follows; Scramblng Watermark s gven n secton A. DWT coeffcents and embeddng postons are dscussed n secton B. Dscrete cosne transform (DCT) after DWT s presented n secton C. The tranng of GRNN s descrbed n secton D. The Watermark embeddng, and extractng processes are shown n secton E, and F. A. Watermark Scramblng Orgnal watermark s the logo of company or nsttute where s a black-whte mage wth sze 64 64; the entres of ths mage are zero and one values. Scramblng process can be mplemented n both spatal doman such as color space, poston space, and frequency doman of a dgtal mage, whch s regarded as a cryptographc method to an mage, allows rghtful users to choose proper algorthm and parameters easly. As a result, the llegal decrypton becomes more dffcult, and securty of the watermark more strengthened. Scramblng mage n spatal doman s to change correlaton between pxels, leadng to the mage beyond recognton, but mantan the same hstogram. In a practcal applcaton, the scramblng algorthm wth small computaton and hgh scramblng degree s needed. Ths paper apples the famous toral Automorphsm mappng, Arnold transformaton [0], whch was put forward by V.I.Arnold when he was researchng rng endomorphsm, a specal case of toral Automorphsm. Arnold transformaton s descrbed as the followng formula: x 1 1 y 1x mod y 64 Where x,y s the coordnates of a pont n the plane, and x,y s the ones after beng transformed. The constant, 64 s relevant to orgnal watermark mage sze. Arnold transformaton changes the layout of an mage by changng the coordnates of the mage, so as to scramble the mage. Furthermore, the transformaton wth a perodcty lke T, the watermark mage goes back to ts orgnal state after T transformatons. In the recoverng process, the transformaton can scatter damaged pxel bts to reduce the vsual mpact and mprove the vsual effect, whch s often used to scramble (1) the watermark mage. In ths paper, the perodcty T s for 4, scramblng process s dsplayed as the followng Fgure 1(a) ~ (d), whch are orgnal watermark mage, 6, 1, and 4 Arnold transformng effect. For T, here s for 4, the 4 transformng s equvalent to the recoverng effect. Let T=k 1 +k, Scramblng the watermark mage k 1 tmes before embeddng t, then after extractng scrambled watermark form watermark mage, k tmes of transformaton can recover the orgnal extracted watermark, where k 1, and k are secret keys. After scramblng watermark mage, t s arranged to one dmensonal array W (n), where n=1, Fgure 1: Image effect after beng Arnold transformed B. DWT coeffcents and embeddng postons The wavelet transform s based on small waves. It was n 1987 when the wavelets became the base of the multresoluton analyss. In two-dmensonal DWT, each level of decomposton produces four bands of data, one correspondng to the low pass band (LL), and three other correspondng to horzontal (HL), and vertcal (LH) mddle pass bands, and dagonal (HH) hgh pass band. The decomposed mage shows an approxmaton mage n the lowest resoluton low pass band, and three detal mages n hgher bands. The low pass band can further be decomposed to obtan another level of decomposton. Fgure, shows three level decompostons of DWT coeffcents. Fgure : Three level decompostons of -D DWT coeffcents Watermarkng postonng s a crtcal technque for watermarkng. For detectng watermarkng precsely, the embeddng ponts must be stable n varety of condtons. It requres the embeddng ponts have followng features: Amrkabr/ Vol. / No. -Group (Subject ) Month Year

3 The embeddng poston shall have deal poston features; Watermarkng embeddng shall have nsgnfcant affecton to embeddng postons; The embeddng postons shall have good nose proof propertes; The embeddng postons shall be stable after mage processng or deformaton. To avod the dsadvantages n algorthm [14]-[18], the watermarkng algorthm based on two-dmensonal Dscrete Wavelet Transform (D-DWT) and D-DCT, for watermarkng postonng s proposed n ths paper. The algorthm utlzes the good local spatal propertes of DWT and establshes wavelet coeffcent tree to ndex the postons for watermarkng. Expermental results show that the postons are stable aganst nterferences and mage compresson. Determnaton of embeddng postons conssts of two steps. The frst step s mult-resoluton decomposton of mage and the second step s the algorthm for determnng the root node of wavelet tree. In the frst step, orgnal mage wll subject to 3-level multple resoluton decomposton by usng D-DWT to form 10 sub-bands wth dfferent resolutons n dfferent drectons. Among these sub-bands, the one n lower frequency labeled LL3 wth sze of concentrates 95 percent of energy of orgnal mage and has property of ant-reference. For establsh the relatonshp between coeffcents n dfferent sub-bands and dfferent drectons. A wavelet tree s created by taken any pont n sub-band LL3 as root node combnng wth ponts n horzontal, vertcal and dagonal drectons n dfferent resolutons accordng to mult-resoluton features of wavelet transform. An establshed 3-level wavelet tree s shown n Fgure 3. In the wavelet coeffcent tree shown n fgure 3, the route n any drecton ponted by arrows conssts of several coeffcent blocks, for example the route LL3 HH3 HH. It can be shown that f the root node s determned, the whole wavelet coeffcent tree s unquely determned. The second step s to determne the root node. As ndex of watermarkng poston, the root node must be stable n any condtons. So, t shall satsfy the condtons mentoned above. The steps for determnng the root node are as follows: (1) Carry out 3-level wavelet decomposton to the orgnal mage and extract sub bands LL3, HL3, HL, HL1, LH3,LH,LH1,HH3, and HH,; the sub band HH1 s not consdered, because of hgh frequency features. () Carry out mage flter to the mentoned sub bands order sequentally, and the output mage s local varance of each pont n 3 3 neghborhood. The calculaton formula s below: ( x, y) 1 j 1 1 j 1 ( f ( x, y) f ( x, y j)) 8 () Fgure 3: 3-level wavelet coeffcent tree Where f(x,y) s the DWT coeffcent, L x, y H, and L, and H are the low, and hgh DWT sub band coordnates, for example: L (LL3) =0, H (LL3) =63. (3) Dvde the matrx constructed by local varances n to several data blocks wth sze of 8 8 and calculate the average of each data block: 1 E k k ( x, y) 64 x y Where E k s the average value of kth block, k ( x, y) s the varance entry n the kth block. (4) Set threshold Th= 1 max {Ek }. Select the data blocks that the average values are less than Th as reference blocks. The reference blocks n varance matrx have the relatve blocks n mentoned DWT sub bands, because the sze of varance matrx s wth the same sze as the mentoned sub bands. The selected data blocks n 3-level DWT coeffcent wth sze 8 8 are consdered as nput matrxes for DCT and GRNN processes that are descrbed n watermark embeddng and extractng algorthms. The postons of selected blocks are saved n the key matrx K for future use n watermark embeddng and extractng processes. C. Dscrete cosne transform (DCT) after DWT The mage transformed by wavelet, most of ts energy s concentrated n low-frequency sub-band, f watermark s embedded drectly n low frequency sub-band, and the transparency of mage contanng watermark wll declne. If watermark s embedded drectly n hgh frequency subband, a lot of hgh frequency nformaton s lossed when contanng watermark mage after flterng, and the algorthm robustness wll declne. The correlaton between every coeffcent s larger n low frequency subband, and then t separated further by dscrete cosne transform (DCT). After DCT, most of the energy n low (3) 3 Amrkabr/ Vol. / No. -Group (Subject ) Month Year

4 frequency sub-band focused on few low frequency coeffcents, so most of the energy s concentrated about the whole mage. The mage changed bgger f these coeffcents amended arbtrary; therefore, t should guarantee that these coeffcents dd not amend. Because the hgh coeffcents s not senstve n the eye, the hgh frequency DCT components n low and mddle DWT frequency sub- bands are the deal regonals of embedded watermark [18], the contradctons about watermarkng robustness and transparency can be solved. After a selected 8 8 DWT coeffcents block s transformed wth DCT, the coeffcents n low-frequency doman contan very low energy of the mage. If the watermark s embedded n ths range, the robustness of the watermark wll not be guaranteed whle the nonvsblty of the watermark wll be poor, and also authentcaton to the carrer content. On the contrary, the coeffcents n mddle frequency doman contan less energy of the mage, whch usually stand for the edge and texture part of the mage, f the watermark s embedded n ths range, where the nonvsblty of the watermark wll be ensured whle the ablty to resst aganst normal processes lke data compresson and format converson, all knds of attacks weakens greatly,whch makes the watermark dsappear and beng destroyed easly.so we select 36 hgh-frequency coeffcents n each 8 by 8 block as the destnaton area where the watermark s embedded, whch are labeled as the followng Fgure 4. Fgure 4: DCT hgher frequency coeffcents As a result, each watermark bt s embedded nto the 36 gven locatons, whch mplements redundant watermark embeddng, beng equvalent to an applcaton of spread spectrum technology, mproves the robustness and securty of dgtal watermark [17]. D. General regresson neural network The GRNN, proposed by Donald F. Specht n [10], s specal network n the category of probablstc neural networks (PNN). GRNN s a one-pass learnng algorthm wth a hghly parallel structure. Ths makes GRNN a powerful tool to do predctons and comparsons of large data sets. A block dagram of GRNN s llustrated n Fgure 5 Fgure 5: GRNN structure The nput unts are the dstrbuton unts. There s no calculaton at ths layer. It just dstrbutes the entre measurement varable X to all of the neurons n the pattern unts layer. The pattern unts frst calculate the cluster center of the nput vector, X. When a new vector X s entered the network, t s subtracted from the correspondng stored cluster center. The square D dfferences are summed and fed nto the actvaton functon f(x), and are gven by D D f ( X ) exp( ) (5) The sgnal of the pattern neuron gong to the numerator neuron s weghted wth the correspondng values of the observed values (target values), Y, to obtan the output value of the numerator neuron Y N ( X ). The weghts on the sgnals gong to the denumerator neuron are one, and the output value of the denumerator neuron s Y D ( X ) The output of the GRNN s gven by relaton (8). Y Y ( X N D ( X ) ( X ) Y Y ( X ) Y T X ).( X X ) p 1 p 1 N D Y f ( X ) f ( X ) (4) (6) (7) (8) Amrkabr/ Vol. / No. -Group (Subject ) Month Year 4

5 In GRNN, only the standard devaton or smooth parameter, σ, the kernel wdth of Gaussan functon s subject for a search [10]. In our work, we have two GRNN structures; each GRNN has 17 nput neurons, 17 pattern neurons, summaton neurons for a numerator neuron and a denumerator neuron, and 1 output neuron. The detal how ths GRNN works s descrbed n the next secton. E. Watermark embeddng process Each hgh frequency coeffcent of 8 8 DCT block s scanned n a zgzag manner, and arranged as shown n Fgure 6. Let Y 1 HF ( k ), Y HF ( k 1) be the desred outputs for two GRNN structures, whch are the central values of nput hgh frequency coeffcents and nput vectors are: X { HF ( k 17 ), HF ( k 16 ),..., HF ( k 1)}, 1 X { HF ( k ), HF ( k 3),..., HF ( k 18 )} Then the watermark bts are embedded nto output obtaned by traned GRNNs, the central coeffcents are replaced by the nserton of watermark bts accordng to the followng rules: HF ( k ) HF ( k ) ( W ( n ) 1), HF ( k 1) HF ( k 1) ( W ( n 1) 1) (9). Where, HF ( k ), HF ( k 1) are the outputs obtaned by two GRNNs. The constant α s the watermarkng strength and W (n), and W (n+1) are the even, and odd bts of watermark, n other words, n each DCT block two watermark bts are nserted, t s possble to have desred watermarkng capacty, f only at least half of DWT data blocks select for embeddng, that the value of threshold Th guaranteed t. Fgure 6: Coeffcent selecton from 8 8 DCT block usng zgzag scannng and arrangement of hgh frequency DCT coeffcents of each block for tranng values for two GRNNs After embeddng all of the watermark bts, gettng IDCT for each DCT block, performng IDWT, and makng the watermarked mage fnally. The total embeddng process s shown n Fgure 7. Fgure 7: Watermark embeddng dagram F. Watermark extractng process Extractng he watermark from the watermarked mage s the reverse process of watermark embeddng whch ncludes the followng steps: Perform 3-level DWT for watermarked mage and select postons of watermark nserton data blocks based on key matrx K as descrbed n secton B. Transform each data block usng DCT. Scan each selected DCT block n a zgzag order as shown n Fgure 6, organze nput vectors, and desred outputs for two GRNN structures smlar to embeddng process, then the watermark bts are extracted to the followng rules: 1 HF ( k ) HF ( k ) W ( n ), 0 otherwse 1 HF ( k 1) HF ( k 1) W ( n 1) 0 otherwse (10). Where, HF ( k ), HF ( k 1) are the outputs obtaned by two GRNNs for watermarked mage. After obtanng all of the watermark bts, the descramblng s performed 5 Amrkabr/ Vol. / No. -Group (Subject ) Month Year

6 for watermark sequence and extracted watermark mage can be obtaned. The dagram of extractng algorthm s shown n Fgure 8. of Barbara and Baboon, and results have been ntegrated n table 1, and table. In relaton (11) W s the orgnal watermark and W s W. W SIM ( W, W ) W. W PSNR 10 log(, j 55 I(, j) I ) (, j) (11) (1) the Extracted logo watermark mage. Dot operaton n ths relaton s explanatory sum of product of respectve entres between matrx W and W. Square operaton s explanatory sum of product of each entry of matrx W wth tself. w Fgure 8: Watermark extractng dagram 3. IMPLEMENTATION RESULTS Two orgnal and watermarked mages wth sze have been shown n Fgure 9, Fgure 10, Fgure 11, and Fgure 1, Barbara, and Baboon mages have been used to mplement the watermarkng algorthm. Orgnal Watermark s a bnary mage and ts sze s The orgnal watermark mage s shown n Fgure 13. Extracted watermarks after some knd of attack on mentoned watermarked mages for Barbara and Baboon have been shown n Fgure 14, and Fgure 15. The performed attacks on the watermarked mages are as follows: Gaussan nose; medan flterng 3*3; low pass flterng; and reszng 1/5 the mage; jpeg compresson wth qualty factors of 10, 5, 50, 75 and fnally jpeg 000 compresson wth bt rate 3. The estmate of smlarty between the extracted watermark mage and the orgnal watermark mage accordng to relaton (11), along the peak sgnal to nose rato (PSNR) of watermarked mage and Orgnal mage, to relaton (1), were calculated havng performed each one of the mentoned attacks on the watermarked mage Fgure 9: Orgnal Barbara mage Fgure 10: Watermarked Barbara mage. Amrkabr/ Vol. / No. -Group (Subject ) Month Year 6

7 Fgure 11: Orgnal Baboon mage Fgure 14: Extracted watermark mage after some knds of watermarkng attacks for Barbara mage Fgure 1 Watermarked Baboon mage Fgure 13: Orgnal watermark mage Fgure 15 Extracted watermark mage after some knds of watermarkng attacks for Baboon mage 7 Amrkabr/ Vol. / No. -Group (Subject ) Month Year

8 TABLE 1: IMPLEMENTATION RESULTS AND COMPARISONS FOR BARBARA IMAGE Knd of attack Our method Method n [] SIM PSNR SIM PSNR Gaussan Nose Low Pass Flter Medan Pass Flter Scalng 1/ JPEG 75% JPEG 50% JPEG 5% JPEG 10% JPEG 000 wth bt rate 3 TABLE : IMPLEMENTATION RESULTS AND COMPARISONS FOR BABOON IMAGE Knd of attack Our method Method n [19] SIM PSNR SIM PSNR Gaussan Nose Low Pass Flter Medan Pass Flter Scalng 1/ JPEG 75% JPEG 50% JPEG 5% JPEG 10% JPEG 000 wth bt rate 3 4. REFERENCES Perodcals: [1] Ingemar J. Cox, Joe Klan, F. Thomson Leghton, and Talal Shamoon: Secure Spread Spectrum Watermarkng for Multmeda. IEEE Transactons on Image Processng.6 (1): , 1997 [] M. Kutter and F.A.P. Pettcolas: A far benchmark for mage watermarkng systems. In Proc. Electronc Imagng 99, Securty and Watermarkng of Multmeda Contents, vol SanJose, CA, pp. 6 39(1999). [3] F.A.P. Pettcolas: Watermarkng Schemes Evaluaton. IEEE Sgnal Processng Magazne pp , (000). [4] Chu, W.C.: DCT based mage watermarkng usng sub samplng. IEEE Transactonon Multmeda 5:34-38(003). [5] Pao-Ta Yu, Hung-Hsu Tsa, and Jyh-Shyan Ln, Dgtal watermarkng based on neural networks for color mages, ELSEVIER Sgnal Processng Journal, vol. 81, pp , October 001. [6] Cheng-R Pa, We-zhong Fan, Woo and Seung-Soo Han: Robust Dgtal Image watermarkng Algorthm Usng BPN Neural Network. In J Wang et al(eds.)isnn 006.LNCS 3973, 85-9 Sprnger, Hedelberg(006). [7] GUI Guo-fu, JIANG Lng-ge, and HE Chen: A new asymmetrc watermarkng scheme based on real fractonal DCT-1 transforms. Journal of Zhejang Unversty SCIENCE A 7(3): (006). [8] Hung-Hsu Tsa: Decson-Based Hybrd Image watermarkng n wavelet Doman Usng HVS and Neural Networks. In D Lu et al (Eds.) ISNN 007.LNCS 4493, , sprnger, Hedelberg (007). [9] MEI Sh-chun, LI Ren-hou1,FANG Ha-jan. An adaptve mage watermarkng algorthm based on neural-networks. Journal of Chna Insttute of Communcatons. 00,3(1): [10] Donald F. Specht, A general regresson neural network, IEEE Trans. Neural Networks, vol., no. 6, November [11] Hu Jnyan,Zhang Tay,Lu Congde, Zhang Chunme. Audo Watermarkng wth Neural Networks n the Wavelet Doman. Journal of Xan Jaotong Unversty. 003,37(4): [1] S.C.Pe and M.H.Yeh: The dscrete fractonal cosne and sne transforms. IEEE transactons on Sgnal Processng.6 (49), (001). [13] G. Carolaro, T. Erseghe and P. Kranauskas: The Fractonal Dscrete Cosne Transform. IEEE Transactons on Sgnal Processng.50 (4): Aprl 00 [14] Gaurav Bhatnagar, Balasubramanyam Raman: encrypton Based Robust watermarkng n fractonal wavelet doman. M grgc et al. (Eds.): Rec. advance n Mult sg. process. and commun., SCI 31, pp , Sprnger Hedelberg, 009. [15] M. Kutter, S. K. Bhattacharjee, T. Ebrahm. Towards Second Generaton Watermarkng Scheme. Image Processng Proceedngs, ICIP 99. Kobe, Japan; vol.1: [16] I. Hong, I. Km, S. S. Han. A Blnd Watermarkng Technque usng wavelet Transform. ISIE 001 Proceedngs. Pusan, Korea Vol.1: 1946~1950. [17] Chu, W.C.: DCT based mage watermarkng usng sub samplng. IEEE Transactonon Multmeda 5:34-38(003) [18] Zhou Yaxun, YeQngwe, XuTefeng. A knd of watermarkng algorthm based on wavelet transform and cosne.chnese Jouranl of Electroncs. 001,9(1): [19] Quan Lu, Jang Xueme. Desgn and Realzaton of a Meanngful Dgtal Watermarkng Algorthm Based on RBF Neural Network, Proceedngs of The 006 Sxth World Congress on Intellgent Control and Automaton, WCICA 006. vol. 1, pp , 006. Books: [0] Ed. Pearson, 1999, p. 83.D.K. Arrowsmth and C.M.Place, An Introducton to Dynamcal systems, Cambrdge Unv. Press [1] [1] Smon Haykn, Neural Networks: A Comprehensve Foundaton, Second Ed. Pearson, 1999, p. 83. Papers from Conference Proceedngs (Publshed): [] Qun-tng Yang, Te-gang Gao, L Fan, A Novel Robust Watermarkng Based on Neural Network, Intellgent Computng and Integrated Systems, Internatonal Conference On page(s): 71-75, 010. [3] Davs K J, Najaran K. Maxmzng strength of dgtal watermarks usng neural networks. Proceedngs of the Internatonal Jont Conference on Neural Networks, 001,4: [4] K. Murakam, Y. Ueno. A Robust Dgtal Watermark usng Multresoluton Analyss of Image. Sgnal Processng Proceedngs, WCCC-ICSP th Internatonal Conference; Bejng, Chna vol.: Papers Presented at Conferences (Unpublshed): [5] Yu Pao-Ta, Tsa Hung-Hsu, Ln Jyh-Shyan. Dgtal watermarkng based on neural networks for color mages. Sgnal Processng,001,81: [6] Xu Jun,Ye Chengqng, Xang Hu. An Algorthm Of Image Dgtal Watermarkng Based On Neural Network Classfyng. Pattern Recognton and Artfcal Intellgence, 001,14(3): [7] Ozaktas, H.M., Zalevsky, Z. Kutay, and M.A.: The fractonal fourer transform wth applcatons n optcs and sgnal processng. Wley, New York (000) Amrkabr/ Vol. / No. -Group (Subject ) Month Year 8