外文文獻翻譯-一類新的置亂變換及其在圖像信息隱蔽中的應用

1、 畢業(yè)設(shè)計（論文） 畢業(yè)論文題目: 外文文獻原文及譯文 基于DXF格式CAD文檔保護系統(tǒng)置亂模塊設(shè)計與實現(xiàn) 文獻中文題目: 一類新的置亂變換及其在圖像信息隱蔽中的應用 文獻英文題目: A new class of scrambling transformation and its application in the image information covering 軟件工程 1043710122 陳愈堅 2008-07-04 哈爾濱工業(yè)大學 哈爾濱工業(yè)大學本科畢業(yè)設(shè)計（論文）（外文文獻） 外文文獻譯文 一類新的置亂變換及其在圖像信息隱蔽中的應用 本文研究了兩種非線性變換，即高維Arnol

2、d變換和高維 Fibonacci_Q 變換；分析了變換的周期性，給出了高維變換具有周期性的充分必要條件； 針對數(shù)字圖像的灰度空間，討論了兩種變換的置亂作用。結(jié)果表明：在圖像 信息隱蔽存儲與傳輸中，這類圖像變換是有應用價值的。 隨著網(wǎng)絡(luò)技術(shù)的發(fā)展，大量個人和公眾信息在網(wǎng)絡(luò)上傳播信息的安全 問題成為人們關(guān)注的熱點，而信息安全中圖像安全是眾所關(guān)心的。對于圖像 信息。傳統(tǒng)的保密學尚缺少足夠的研究。隨著計算機技術(shù)與數(shù)字圖像處理技 術(shù)的發(fā)展，對此已有一些成果。近年來，相繼召開了關(guān)于數(shù)據(jù)加密的國際學 術(shù)會議，圖像信息隱蔽問題為其重要議題之一，且有關(guān)的論文以數(shù)字水印技 術(shù)為主。針對大幅圖像的信息隱蔽問題，置亂

3、技術(shù)是基礎(chǔ)性的工作。值得強 調(diào)指出的是Samile給出的方法，它是基于填滿空間的所謂FASS曲線，這 種方法的應用見文獻5。 我們注意到Arnold變換的特性，將它引入圖像的置亂處理有良好的效 果。由于Arnold變換有周期性，這在編碼與解碼中是有方便之處的。在文 獻5-8中，討論了 Arnold變換在圖像信息隱蔽中的應用，但經(jīng)典的Arnold 變換中的參數(shù)僅有 4個，用于數(shù)據(jù)加密尚嫌太少。文獻9把平面Arnold變 換推廣到空間，從數(shù)學上推廣Arnold變換是有意義的。受 Arnold變換思想 的啟發(fā)，我們一般地研究了什么樣的矩陣變換（模運算）具有周期性的問題， 發(fā)現(xiàn)很廣的一類變換都可用于圖

4、像信息置亂處理，本文的目的是建立任意n 階的矩陣模變換，并且作為本文的主要理論結(jié)果，給出了該新型變換具有周 期性的充分必要條件，為其在圖像置亂編碼的應用打下必要的理論基礎(chǔ)。 1矩陣變換有周期性的條件 數(shù)字圖像可以看作是一個矩陣，矩陣的元素所在的行與列，就是圖像顯 示在計算機屏幕上諸像素點的坐標。元素的數(shù)值就是像素的灰度。對于一幅 圖像，如果把它數(shù)字化就得到一個矩陣，改變矩陣元素的位置或RGB數(shù) 值，圖像就會變成另外一幅圖像。本節(jié)討論的是什么樣的矩陣變換可以把圖 像復原，即周期性的問題。 -5 - 定義1對給定的N階數(shù)字圖像P,我們說變換 aii a22 a23 a2i ai n a2n X2

5、. (mod N) an2 an3 ann （aj為整數(shù)，Xi,Xn 0,1,N1）關(guān)于P的周期為mN，指m”是使得圖像 P經(jīng)一系列變換后回復到 P的最少次數(shù)。 定理I以上變換有周期性的充分必要條件是|A|與N互素。此處 A是 變換的矩陣，|A|是矩陣A的行列式。 2 n維Arnold變換 Arnold變換是Arnold在研究環(huán)面上的自同態(tài)時所提出的。設(shè)M是光滑 流形環(huán)面 （x,y） （modi） , M上的一個自同態(tài)定義如下： （x, y） = （x y,x 2y）（mod i） 顯然 映射導出覆蓋平面（x, y）上的一個線性映射 i i、 = 。 i 2 k丿 定義2設(shè)有單位正方形上的點（

6、x,y）,將點（x, y）變到另一點（x；y）的變 換為 /i (modi), x _ 0=4 2 k Arnold變換，簡稱 其中，（mod i）表示模i運算。此變換稱作二維 Arnold 變換。 將Arnold變換應用在數(shù)字圖像上，可以通過像素坐標的改變而改變圖 像灰度值的布局，把數(shù)字圖像看做一個矩陣，則經(jīng)Arnold變換后的圖像會 變得混亂不堪”但繼續(xù)使用 Arnold變換，一定會出現(xiàn)一幅與原圖相同的 圖像。如果把這類變換應用到數(shù)字圖像的存儲與傳輸，特別是用到圖像信息 交換方面，則可以取得圖像隱蔽的效果?？紤]到數(shù)字圖像的需要，我們把以 上的Arnold變換改寫為 1 2 (mod N)

7、以后 其中x,y 0,1,2,N-1，而N是數(shù)字圖像矩陣的階數(shù)令A= 我們說Arnold變換即指 式。 例1 設(shè)N=2，數(shù)字圖像矩陣為 廣N C、 P = U T I 丿 則經(jīng)過3次Arnold變換后，P恢復了原圖。見下所示 廣N C、 廣N T、 ZN U U T C U T C 丿 丿 2,下列變換稱為Fibonacci變換: 1 1、 (mod N) 其中 x,y 0,1,2,N1, 定義5對于給定的自然數(shù)，下列變換稱為Fibonacci_Q變換: X1 F X =Q p f 、 X1 X2 (mod N) 其中Qp為廣義Fibonacci_Q矩陣， 兀必,Xp 0,1,2,，-N 引理

8、1如果變換 (mod N) (x, y 0,1,2,N)的周 期為mN，則下列變換有周期，且周期也為mN : (mod N) (x, y 0,1,2,，-N) 這個引理的證明較簡單，這里省略.由引理1,很容易得出下列 定理2 對于給定的自然數(shù) N 2,如果二維Arnold變換的周期為 mN，貝U Fibonacci變換的周期為2 m” 由于|Q p|= 1,所以由定理1,我們有 推論2 Fibonacci_Q變換具有周期性 計算機編程結(jié)果如表2和3. 表2 當P=2時廣義Fibonacci_Q變換在不同階數(shù)N下的變換周期 N 2 3 456 7 8 9 10 11 12 25 50 60 10

9、0 120 125 mN 7 8 143156 57 28 24 217 6056 155 1085 1736 2170 1736 775 表3 當P=3時廣義 Fib on acci_Q 變換在不同階數(shù) N下的變換周期 N 23 45 6 7 8 9 10 11 12 25 50 60 100 mN 1580 30312 240 342 60 2401 5601 330 240 1560 1560 3120 1560 4基于相空間的圖像置舌L 從現(xiàn)在開始，我們討論 mxn數(shù)字圖像矩陣P=(Pij)mKn。 APS變換 定義6下列變換稱為APS變換(基于相空間的廣義 Arnold變換): P

10、= AP(modT)(9) 其中A是m維Arnold變換中的變換矩陣。 容易看出，圖像矩陣P中的每一列可看作是 m維空間的一個點，所以 根據(jù)定理1，APS變換是有周期性的。其周期小于或等于m維Arnold變換 的周期口丁，當然對不同的數(shù)字圖像 APS變換可能有不同的變換周期。這與 基于像素點位置改變的圖像變換是不同的。 FPS變換 定義7 下列變換稱為FPS變換（基于相空間的Fibonacci_Q變換）: P =QmP（modT）， 其中Qm是Fib on acci_Q矩陣。 類似于APS變換，根據(jù)定理1，F(xiàn)PS變換具有周期性。 5兩個圖像變換例子 （i ）三維Arnold變換例子：圖版I _

11、1（附本刊后，下同）是利用 式中的 變換對原始圖像（左圖）作兩次變換得到的結(jié)果。原始圖像尺寸為 256X 380; 在PC586用C+完成。 （ii ）APS變換例子：圖版I _2是利用（9）式中的變換對原始圖像（左圖）作 兩次變換得到的結(jié)果。原始圖像尺寸為256X 380 ;在PC586用C+完成。 對數(shù)字圖像實施 APS和FPS變換，則圖像的每一像素點的值依賴于該 點所在的列的所有點的像素值。但我們可以通過改變變換而使每點的像素值 的改變只依賴于它所在的行，甚至依賴于整幅圖像。定義1中指出的變換， 可選擇的參數(shù)有n2個，且n與N互相獨立，這就使得對于圖像隱藏目的編 碼應用中，有很寬的加密容

12、量，無論采用哪種變換（包括這些變換的變體與 推廣），我們大量計算表明，理論分析與實驗結(jié)果一致。此外，一般說來變 換前后的圖像之間差別很大，這對于圖像信息的隱蔽目的來說在應用中是可 資利用的。 哈爾濱工業(yè)大學本科畢業(yè)設(shè)計（論文）（外文文獻） 外文文獻原文 A new class of scrambling transformation and its application in the image information covering Abstract This paper studies two kinds o4 nonlinear transformaiiors, namely the

13、 h巾her-dimensional Arnold transfonnaton 且nd the higher-dimensional Abortacci Q-(ransfomiation and discusses the scram* bling action of the I wo traneformalions focusing on the phase space of the digital knaves. A suffidenl and necftasary condrtlon that a matrix transformation of digital image has pe

14、rioddty is given. The resuits show that the two transfoimatiore have potential application in the amrege and transportation of image information for the purpose of infomiatian securrly. More and ithjit privaJe and public information spreads on the Internet network wilh lhe rapid progresfi in neiwork

15、 lechndogy. So theof information securiry attract the public interest. Fhfi image informatioft Riirity is certainly an important prohlem, The imag? information has rarely Ixien Muditd by the tradiLianal cryptography. Perhap* the reason a that the image 創(chuàng)ores huge infurmaiion. But in recent years peo

16、ple have obtained some results in this field mJ th the rapid develojjrnenl of computer tehiwlogy and digital image prtx：朋r technology1 h Some mtema- tiuna cimferrnces about data-cryptography have ben held* in which the hiding of image information was one of the main copies &nd the paper% contribille

17、d 10 the conferences arc mainly about 山電 digital walfirmarkingThe scrambling technology is the basic mcanft for covering huge infnnnation It should be points out that the method given by Sami Ifon thp FASS curve whose appJicaticn can be found in ivf+ & ” We observed the properties of the Arnold tran

18、sformation and f()und tbit ils properties can effect!vdy urd in lhe imae scrambling. Because Arnold tran#&)rmatian ha periodicity t it U very (xnvenjent fw code-making and code-solvingr In 陽直一右9r Arnold transfojmalion is used for covering of irne in&nnAfion. But the chsaieal Arnold Iransformtition h

19、a only ftiur punimeLehj which arc loo ftbw for dutu-cry pinguard. Tlic two-dimensional Arnold iranfomwtion is genera iKcd ic lhe fipace u . Inspired by the of Arnold traruifonnadon t we study the problem of what matrix Iransformation (imodule operation ) has periodicity - We found that a l4 of trans

20、formaiinns can be applied to Lhe scrambling of image information + The purpose of this paper is to discuss arbi- tnuy malrix module UansFbrmatLQn of order ft. One of the nuiin results in thb work is the sufficient and neceBRary condition for the new tranAformahon to have periHjicity. Ihe result lays

21、 the neecsiarv 加is for lhe ce-making ot the scrambLng of imag已 infomiMian. In the (allowing we give the definition the mutriK module transfurmalion of arbitrary order n and prt)%c the sufFicienl arid neemwury nacei Q-1riinsformation inLn two-dimninnaJtraiifbrniaLion . 1 The condition for a matrix tr

22、anfermution having periodicity A diiLal imagt! cun h* remarried u matrix . Vhe position of an rlement of the mutrixju和the ooardinntes of the image pixel un the computer and lhe element of the matrix is lhe grsy levol of the pixel 1 So we can get another image if we change the positions of rlemrutfi

23、of miitrix nr ihp H(；H - Here* wh disijiss whai mnirix. triinforniatitjnan pr originul iiTiap. Definition 1. anil arbitrary digihd image P, die Lrunsfiimia- 吟 W 2 * x,* x 10 J，/V - I -13 - (1) has period m v with re spent to the imar P r 11 m v be iht1 smallest positive integer n such that the image

24、 P can come back aftrr n lim總& transformHliorm. W t dndte by A the matrix of the right hand of ( I). Theorem For givn pcsitivp integer N、ihp Aiifljcifnt and n=( rssary (nndilion rhiit transfomrahon (1 ) ha periedicily is that del4 and N are piimn eath iher - Hptp del 4 is iht3 leterminanT of the mat

25、m A . 2 The 7i-diineiisioal Arnold transformation Th 2-dimeiisionaJ Arnold transtform田Hon * usudly simply called Arnold transformation was intixxiuceil by Arnold when he studied HTgixlic proLlenis of elasmeal mechanics1 . Lftt M be lhe torus J (Jr ty )mod 11 Mith the usual measure. An automorphism 爭

26、 也 defined 甲匕、j) = (* + y i J + 2y) (mod 1). Thr iTKipping 中 imluces a linear mapping in the covring plan旺(r , y)： =11 / is measure-pre serving. Definition 2 Let (裊.y) be a point in th凸 unit Hquarr:. Hie fnl losing transformation is called the 2D A mol J Iranlurmation (Arnold irartBlunnatiun for sho

27、rt): (mod 1, whtre f rnixl 1) reprnents the mod ration. Using (he Arnold Iran formal ion wr can cViangc thr distribLiLion nf the gray of a digital imagr and make the image mess*. But will gel the ohgind image if rte use Lhe Arnold iranuRirma- lion again and again. So we can hide uur image informatio

28、n in the pnx避少 of storing and Lrune* porting image infnrma(imt especiaJlj thr imagf irdorrnntinn on the Intemrl by ust- of thf Arnold irkinyfurmauun. To meet the need of digiial images h we change the above Amuld irariKR)rmatj(m into the Gdlnwing 2 Hit N x N matrix of digital image is a 2- A - L i ,

29、 So the abovt Arnold tranfiformatiot can Hr rcwriiri 1/ where x y 10 1 P 2 * /V-1 ” From now “n we 陰y thnt the 2-dimenuiuna Arnold transformHlion is (5). After three limes of Example 1 * Set .V 2. The digil卻I ima matrix is /* Amuld Uansfomation the image returns back NTIN UIN q U T! C u) T Ci u T) S

30、o when (V 2 the period of the A mold tninsfbrnuition m j - 3 (table 1) * Table L Tlie perioeb tjf the Amvld trBnafmations for some / IV 2 3 4 5 7 8 9 11 12 3 4 3 JO 12 8 6 12 3Q 3 12 N 23 50 GO LOO 130 J 25 23& 480 31Z 50 150 60 150 60 250 弼 192 120 There hav been many result* on the 2-dimeTiBiunal

31、Arnold iranBformMtion and its applies- iiont(I9*n 03. In ref* _ 10 the 2-djmensional Arnold tranaformaLion Ik generttliKed lo the 3-di- Tnenflional cac and ?ome algorithms to calculate the period of I he trannfbrmadon art given. 3 Fibonacci* Q matrix tnuiNlortnation Fibonacci sequence a very importa

32、nt topic in mathematics bcaufi* it always has some unexpected ontlerfu properties and a. Lui of applications. But only recently da people relaie iL with computer graphics . In nefe. k6 11the Arnold tran&EormaLion uimac a nJ its scrambling action were considered + But they are related with Fibonacci

33、sequence r We will consider the H- bunaci matrix and define the u-called Fibunaeci -Lr&nfbnnatLori and ih application in the scrambling action of digital image in thifi sertian. Wp aho explain the relationship between (h? 卜 ibonacci (?-Lranicjrmaiin Etnil the Amtdd trrkinnaiian Fibonacci sequence Se

34、t F(j = I , F| = I ,= 2 * , generally 札 * ? = F. *、+ F . Then the | FB (也 called Fh bunacri sequence. Matrix Q =丨 lion, we can prove an important property of Q ： (P in ralleii Fikinari -matrix. It is ohvifliis tkrnt dci(/ - - 1 . Ry indue- ； 尸J 山肌Q=化* i *幾 3,3 Generalized P ibonacci y-rnalrix Set =

35、(1) S 耳 Vn = 1 0 P 二 0 0 1 0 0 0 0 1 J 0 0 OJ 1 a 1 0丿 * 0 J u i 0 0 0 0 1 Q = 0 0 0 I 0 0 0 0 kl 0 0 0 1 0： 0 1 0 0； 0、 0 0 . I 0丿 Then Q$ is calkii the generalized Fibonacci (-matrixt p - 01. It is phsj to verify that d匕tQ戶= 1 when p is an even inLeger and det = - 1 when p is tin odd one. Definiti

36、on 4* For & given positive integer ：V2, the following transiformatiQn called the Fihnnacn tran$formfihnn (8) (mo 21 the fbllowmg transformation 諒 called the Fibonacci Q-transformation: * X2 9 wher Qp is lhe generaliied Fibonacci (?-matrix, jq *耳抽，升 0,1 $2號/V“ 1 丨 * Snuna* If the period of the transf

37、onnaejon I i M 1 ih iHr t then lhe following trannformaiitm has perindicity and iLa period is aJao mvi CM： ；)(；)(嘰心疔心 (/) = b JI JZE * The proof of tbis Jemma is easy sc we omit it here * 哈爾濱工業(yè)大學本科畢業(yè)設(shè)計（論文）（外文文獻） By the above lemma, we can eaaily have ThMrem 2. For a fixed positive integer N 2, if th

38、e period of the 2-dimensjona Arnold tranAlbrmaiion ls him* then th& Fibonacci transloTmatiori 2m r BecBuse delQp= 1,祥亡 have Corollary 2. The Fibonacci Q-transformation has periodicity (ubles 4t5). Tle 4 TW period, of iKe Fibcruicdi QtrHMfonnaiioM to Mxnc j(p 2) N2 3 4 5 6 7 8 9 10 )1 12 叫 7 B 14 站 5

39、6 57 2B 24 2)7 60 56 N 13 30 60 IQO 120 25 128 256 480 513 血 W 畫 L73A 2170 1736 775 S96 3472 1792 Td3 The periodn at the FibaMcca Q tmii&rnwtint k? Km N(p -3) 2 3 4 5 6 7 B 9 10 11 12 吋 b 30 312 240 M2 6G 240 1540 1330 240 25 50 60 100 阿 125 256 4S0 512 15O 1340 )130 15W 3U0 TOTO wo 192fi 3120 3 W0

40、4 The Image scrambling based on the phase itpaces Now we r 世qul to the period of thf* m -dimensional Arnold transformation J7tHut it is possible ihut diFfeirm APS iraixsfnrTnationfl have diffurtnt periods, 4.2 FPS tran&formHiion Defiikitlon 7 * Hie following trajiformarion calleci the FPS transfcrma

41、iion (ihr Fibonacci Q-ttanskinnahon based on the ph space): Pf -mod T) T where Qn _, is the Ki- bonacei Q-matrix. Similar li Lhc APS irunsfonnaiion t ih亡 FPS has also periodicity by Theorem 1. -ii - 哈爾濱工業(yè)大學本科畢業(yè)設(shè)計（論文）（外文文獻） 5 Examples of scrambling trAnsfbrmAtions ot images 5*1 An example of scrambli

42、ng transformation by 3-dimensional Arnold transforrrialion (7) TIik originaJ image is 256 x 380 in 訪produced on PCS86 by C * * * Tlie left one is iht a- Figinal imagf! The other two ar* resjiectively the images alter one time and Iwq limes of transformations 5 .2 An example of scrambUn Lransfortnati

43、on by lhe APS IraiisfonnAtiofl (9) The oriinai imac is 256 x 380 in 対ze, produced on PCS86 by C * . The lefl one is the a* ripna imagp. The othsr two re respectively the images after one time and two times of transfor- mulionM we perform APS and FPS traWurmAticn to digital imagcu, thtn the Level of each image i?leinnt point depends on Lhe ftnlumn it belong to. We can also let the point depend on the row it belonR to or even th(* whole image by nhanging