BLIND COMPRESSED IMAGE WATERMARKING FOR NOISY COMMUNICATION CHANNELS

Transcription

1 BLIND COMPRESSED IMAGE WATERMARKING FOR NOISY COMMUNICATION CHANNELS Jonathan Me Carnege Mellon Unversty Department of Electrcal & Computer Engneerng Pttsburgh, PA 523 Scott Pudlewsk Lncoln Laboratory Massachusetts Insttute of Technology Lexngton, MA 242 ABSTRACT Vsually-dstorted mages can contan valuable nformaton. Indeed, n tactcal MANET networks where throughput s extremely valuable and dffcult to come by, guaranteeng the delvery of every packet of an encoded mage s mpractcal. However, desgnng a watermark that s reslent to the types of vsual dstorton mparted on an mage or vdeo due to channel losses s a dffcult task. In ths work, we ntroduce a new watermarkng scheme for JPEG-compressed mages that ncorporates deas from compressed sensng (CS) to acheve robustness aganst certan types of errors nduced by nosy communcaton channels. Ths work uses CS technques to embed a sparse watermark nto L randomly selected quantzed JPEG mage coeffcents. Sparse reconstructon technques are then used to reconstruct the watermark from the coeffcents that were receved, ncludng those that were ncorrectly decoded. Through the development of ths watermarkng scheme, we would lke to demonstrate and explore the effect of the error reslence propertes of CS encoded sgnals on the mage watermarkng problem.we show through smulaton that even wth sgnfcant vsual dstorton n the receved mage, the CS encoded watermark can be detected wth very hgh probablty. Index Terms Watermarkng, Error Codng, Compressed Sensng, JPEG Compresson, Erasure Channel. INTRODUCTION Robust dgtal mage watermarkng [], or the act of hdng a hdden sgnal wthn an encoded mage that s reslent to both nose and tamperng, has applcatons n both commercal and mltary systems. In ths work, we wll focus on mltary applcatons where an mage s generated at some remote Dstrbuton A:Publc Release. Ths work s sponsored by the Department of Defense under Ar Force Contract FA872-5-C-2. Opnons, nterpretatons, conclusons, and recommendatons are those of the author and are not necessarly endorsed by the Unted States Government. magng sensor (.e., a persstent survellance system or an unmanned aeral vehcle (UAV)) and transmtted n part over very lossy tactcal MANET lnks. Whle tradtonal dgtal sgnatures can be used to verfy that an mage was receved from the stated sender, ths may not go far enough to ensure that the mage was created by the sender. In addton, t may be useful to be able to verfy the source of an mage long after that mages was transmtted, for nstance n legal proceedngs or to determne the source of an nformaton leak. Based on these observatons, the watermark should be embedded wthn the mage data so that after any transformaton or dstorton that produces a useful mage, the watermark s preserved. In ths work, we explore the use of compressed sensng (CS) [2, 3, 4] n robust watermarkng of JPEG encoded mages [5]. Whle there are a number of alternatves that have shown promse n these scenaros (specfcally spread spectrum technques [6, 7]), we beleve that the combnaton of the error reslency of CS encoded sgnals along wth the mnmal complexty requred for CS encodng [8] justfy ts consderaton as a watermarkng technology. Compressed sensng s a compresson scheme n whch sparse sgnal reconstructon technques are used to reconstruct a sparse message sgnal x from a small number of nose-lke random lnear combnatons y = Φx, where Φ R M N, M < N s a nose-lke samplng matrx. Whle CS as a compresson technque has a number of obvous applcatons n multmeda sgnal compresson [9], n ths work we are gong to explore the mpact of two other propertes of CS encoded sgnals. Specfcally, we wll look at the error reslence of CS encoded sgnals, along wth the ablty to nterchange receved CS samples, whch removes the dependence on any one specfc sample. In ths work, we wll show that these two propertes can be used to desgn a watermarkng system that s surprsngly reslent to packet erasures. The goal of ths work s to determne whether the error reslence propertes of a CS encoded watermark justfy the desgn of yet another data hdng scheme. We beleve that, based on the reslence to extremely hgh error rates wthn the JEPG encoded mage, future work s ndeed justfed.

2 The remander of ths paper s organzed as follows. In Secton 2, we explan the structure of the watermark and the desgn of watermarkng parameters, whle Secton 3 explans the detecton scheme. Secton 4 descrbes the expermental setup and dscuss the smulaton results. Fnally, we dscuss future work and draw the man conclusons n Secton WATERMARK DESIGN 2.. Encodng the Watermark To create the watermark, we generate a message vector x {,,} N such that x = K, N K. Let A R L N be a full rank matrx wth L > N. Such a matrx can be created wth full rank wth hgh probablty [3] usng a seeded RNG to generate each entry. The watermark vector y s then generated as y = Ax. By desgn of theamatrx,y s a real-valued vector, whle the mage coeffcents z are quantzed and coded dgtally. Thus, n order to practcally mplement the watermarkng scheme, we need to quantze y. Usng a pre-computed codebook wth quantzaton steps, we can fnd the quantzed watermark vectory (q) = y+η, such that Q(y (q) ) < for {,,...,L }, where Q( ) s the quantzaton functon and η represents the nose ntroduced by the quantzaton step. There are varous optons to create the watermarked coeffcents w( ). Our mplementaton uses nteger modulo maskng: w (z) = z +Q(y (q) ) () for {,,...,L }. Ths s analogous to k least sgnfcant bt maskng when = 2 k for ntegerk Embeddng the Watermark In ths work we embed the watermark vector nto a JPEG encoded mage. However, the basc watermarkng framework s not specfc to the compresson technque. The watermark can be embedded nto any set of mage transform coeffcents (or even the orgnal mage). We chose to use the JPEG standard prmarly for ts popularty. In JPEG encodng, the DCT coeffcents wthn each of the J macroblocks are scaled through element-wse nteger dvson wth qualty matrx B(q), where < q s the qualty level of the mage. To embed the watermark, we modfy these scaled block-dct coeffcents of macroblock ntenstes. In partcular, we modfy the subset S of the J scaled DC coeffcents z of ths block-dct (the DC coeffcent of each macroblock s the lowest frequency component of the DCT of that macroblock). Let S {,,...,J } such that S = L, and S() denotes the th element of S. Set S of L mage coeffcent ndces can be randomly selected usng the seeded RNG. The vector z represents the quantzed DC coeffcents wth the watermark wll be embedded. The coeffcents z S() = w (z S() ) for {,,...,L } are watermarked, whle the coeffcents z j = z j for j / S reman unmodfed. By watermarkng a subset of all coeffcents, our scheme allows for some addtonal securty aganst falsfyng. However, note that f L = J, then ths s equvalent to modfyng all coeffcents. Once the watermarked quantzed coeffcents z have been computed, they are compressed usng varable-length Huffman encodng and perodcally marked wth reset markers (RST) (to mark the ponts at whch the Huffman decodng process can realgn n the case of corrupted or mssng bts). 3. WATERMARK DETECTION In ths work we assume the worst-case scenaro n that our watermark decoder has no cross-layer nformaton to determne whch packets were dscarded at the lower layers. Whle some of the corrupted macroblocks wll have a zero ampltude and can therefore easly be removed, we can clearly see n Fg. that many packet losses wll result n a vald but ncorrect decoded macroblock. Whle t would be feasble to desgn a party based detecton scheme to fnd the errors n CS encoded sgnal (as n []), we show that even n ths worst-case scenaro our scheme s able to mantan hgh detecton rates at hgh packet error rates wth moderate sparsty and expanson levels. After havng removed the detected corrupt macroblocks, we have M watermarked macroblocks remanng. Due to the use of RST, the decoder can determne whch M of the remanng blocks orgnally contaned the embedded watermark coeffcents. As long as enough blocks are receved such that M Klog(N), the recever can correctly reconstruct the watermark. 3.. Decodng the Watermark Let U be the set of ndces of the M watermarked block- DCT coeffcents not classfed as corrupted after detecton, and U() be the th element of U. The receved sgnal ŷ s then defned element-wse as ) ẑu() ŷ = Q (ẑ U(). (2) Then let  = A(U,:) be the correspondng matrx wth row ndcesu froma. In other words, the mssng samples (those that were detected as ncorrect) are removed from the set of receved samples along wth the correspondng rows n the samplng matrx. Snce CS only requres that enough samples are receved for decodng (.e., the samples are nterchangable), the watermark wll be able to be detected as long as enough correct samples are receved and the magntude of the errors s bounded and small [3].

3 We then fnd ˆx by solvng the optmzaton problem mnmze x x subject to: y Ax 2 2 < ǫ, (3) whch can be shown to be equvalent to fndng the sparsest vector x that matches the receved sgnal vector ŷ wthn some error tolerance ǫ. Fnally, snce the recovered ˆx R wll be contnuousvalued whle the orgnal message only takes values x {,,}, we wll need to quantze ˆx. Lettng max(x,k) be the set of x wth the largest K absolute values, we set ˆ x (q) = otherwse. We measure the MSE between the quantzed recovered message and orgnal message = sgn(x ) forx max(x,k) and ˆx (q) MSE = x x(q) 2 2. (4) K From ths defnton, t s clear that MSE 4. We note that typcal values range from MSE 2 snce MSE = 4 x (q) = x, whle MSE = 2 lkely mples that the recovery chose all ncorrect ndces as support for our estmated message. Thus we clam to detect the orgnal messagexwhenmse < Selectng Watermark Parameters Snce ths scheme requres the watermark to be restrcted to be ak-sparse sgnal, we need to verfy that that restrcton does not make the resultng watermark too easy for an adversary to guess usng brute-force. We wsh to desgn our watermark such that a thrd party who knows the watermarkng scheme but does not have full nformaton about the shared secrets (A or x) wll not be able to generate a correct but unauthorzed watermark. In the case where the thrd party only has knowledge of the message, then the matrx A must be guessed. However, snce guessng each element of the samplng matrxan nfeasble, we only consder the more tractable case n whch the thrd party has knowledge of the matrx A but not of the messagex. The total ( number ) of messages of lengthn and sparstyk N st = 2 K. Assumng that every message has an equal K chance of beng generated by the thrd party, the probablty of guessng the exact message s the recprocal of ths quantty. However, snce we do not requre perfect recovery for detecton of the message, we examne the probablty of guessng a detectable message. The number of detectable messages s Whle our prelmnary studes have shown that some nformaton can be ganed f an attacker can correctly guess 8% of the samplng matrx elements wthn ǫ, for even our smallest case, the samplng matrx A s a real valued matrx. Ths would requre an attacker to correctly guess at least 3,8 real valued elements, whch s clearly not practcal. N K p Table. Probabltes of randomly generatng detectable message by guessng for dfferent values of K and N gven by R = K = K/2 K/2 j= 2 K j ( K )( )( ) K N K j K j ( (5) n where by conventon = f r > n. Then the probabl- r) ty of generatng a detectable message to spoof an unauthentcated watermark s R/T. Probabltes for randomly generatng a detectable message s shown n Table. Ths also shows that a number of dstnct messages {x,x,...,x P } can be desgned for a sngle matrx A to generate watermarks for fferent authentc sources wthout beng detected as a dfferent authentc message. 4.. Experments 4. SIMULATIONS AND RESULTS We use the classc grayscale Lenna test mage [] for our smulatons, and gradent projecton for sparse reconstructon (GPSR) [2] to solve the convex optmzaton problem. We chose a packet sze of 28 bytes and nserted a RST marker every 64 macroblocks whch corresponds to one at the end of each horzontal row. We assume the JPEG header s transmtted ntact. Fgure shows a typcal watermarked mage and a receved mage at PER =. For ths work, we assume a Gaussan packet erasure channel. We see the structure of the errors from the lost packets n (b). Note that even n the presence of relatvely hgh PER, there s some vsually sgnfcant nformaton present n the transmtted mage that may be worth verfyng. Fgure 2 shows the watermark recovery step performed on a typcal nstance of a receved mage. The recoveredx s contnuous-valued but sparse wth the correct support, so the quantzaton to x q stll matches the orgnal message exactly and s detectable. In our smulatons, we examned the effects of several parameters of the detecton rate of the watermark across varyng PER. These parameters ncluded sparsty level N/K, expanson level L/N, and number of quantzaton steps. The

4 Detecton of Watermark vs Packet Error Rate for varyng Expanson Levels L/N = 4. L/N = 4.8 L/N = 5.6 L/N = 6.4 L/N = 2.8 L/N = 25.6 (a) (b) Fg.. (a) shows the watermarked mage wth no transmsson errors; (b) shows the receved mage at PER =. Value.5.5 Orgnal vs Recovered Message Orgnal Sparse Soluton Recovered 3 2 PER Fg. 4. Detecton rate across packet error rates at varyng expanson levels L/N Detecton of Watermark vs Packet Error Rate for varyng Quantzaton Levels Q = 2 n = 3 = 4 = n Fg. 2. Recovery step detecton rate for each set of parameters was calculated as the rato between the number of tmes the true watermark was detected and the total number of trals. Each settng was run wth Monte Carlo trals. Fgure 4 shows that the detecton rates across all packet error rates can be ncreased by ncreasng the expanson level L/N. Fgure 3 smlarly shows that detecton rates across packet error rates can be ncreased by ncreasng the sparsty N/K of the messagex. Comparng ths result to the analyss from Table, we see that there s a tradeoff between securty and error protectng performance n desgnng the sparsty of the message. Detecton of Watermark vs Packet Error Rate for varyng Sparsty Levels N/K = 32 N/K = 6 N/K = 8 N/K = 4 N/K = PER Fg. 3. Detecton rate across packet error rates at varyng sparsty levelsn/k 3 2 PER Fg. 5. Detecton rate across packet error rates at varyng quantzaton levels Fgure 5 demonstrates ncreasng the quantzaton levels used to embed the watermark can also ncrease performance. However, there seems to be a pont at whch ths ncrease no longer results n sgnfcant ncrease n detecton rate. Ths could be due to the fact that ncreasng the quantzaton levels ncreases the magntude of the gross errors. 5. CONCLUSION In ths work we ntroduced the applcaton of CS to robust mage watermarkng. Ths was done by ntroducng a watermarkng scheme for JPEG encoded mages and showng that t was effectve n nosy communcaton channels. Our scheme takes advantage of sparse recovery deas borrowed from compressed sensng to acheve performance n very nosy condtons. We demonstrated ths performance under certan parameters and analyzed the securty of the authentcaton provded by the scheme. Future work ncludes extendng our scheme to vdeo watermarkng. We beleve our scheme s partcularly well-suted for streamng vdeo, snce our protocol can handle dscarded packets and does not rely on knowledge of the orgnal mage. In addton, alternate embeddng technques and types of attacks could be studed.

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