Comparative performance of watermarking schemes using Mary modulation with binary schemes employing error correction coding

Transcription

1 Comparatve performance of watermarkng schemes usng Mary modulaton wth bnary schemes employng error correcton codng Brett Bradley, Hugh Brunk Dgmarc Corporaton, SW 72 nd Ave., Sute 250, Tualatn, OR ABSTRACT A common applcaton of dgtal watermarkng s to encode a small packet of nformaton n an mage, such as some form of dentfcaton that can be represented as a bt strng. One class of dgtal watermarkng technques employs spread spectrum lke methods where each bt s redundantly encoded throughout the mage n order to mtgate bt errors. We typcally requre that all bts be recovered wth hgh relablty to effectvely read the watermark. In many watermarkng applcatons, however, straghtforward applcaton of spread spectrum technques s not enough for relable watermark recovery. We therefore resort to addtonal technques, such as error correcton codng. As proposed by M. Kutter 1, M-ary modulaton s one such technque for decreasng the probablty of error n watermark recovery. It 1 was shown that M-ary modulaton technques could provde performance mprovement over bnary modulaton, but drect comparsons to systems usng error correcton codes were not made. In ths paper we examne the comparatve performance of watermarkng systems usng M-ary modulaton and watermarkng systems usng bnary modulaton combned wth varous forms of error correcton. We do so n a framework that addresses both computatonal complexty and performance ssues. Keywords: Dgtal watermarkng, Error Correcton Codng, M-ary Modulaton, Dgmarc 1. INTRODUCTION In stll mage dgtal watermarkng, a varety of factors can make t very dffcult to recover the watermark message. Clearly, one such factor s that the watermark message power must be much less than that of the host mage n order to mantan the orgnal mage fdelty. Another factor s that the watermarked mage s lkely to undergo a seres of degradatons: lossy compresson, analog prntng and dgtal recapture, and flterng are some examples. A thrd problem s that, n some watermarkng schemes, the watermark message shares the channel wth a synchronzaton sgnal, whch may nherently nterfere wth t. Fnally, some applcatons demand that we are able to recover the watermark message from a very small mage, or a very small sub-secton of the orgnal mage. In some applcatons of dgtal watermarkng, the objectve s to embed N bts n a dgtal mage that can later be recovered after varous mage degradatons. In ths paper we explore possbltes for the encodng of data that wll be subject to the effects of the watermark channel. The channel, whch s nextrcably connected to our choce of a varant on spatal spread spectrum watermarkng schemes, s very often characterzed by extremely low sgnal to nose ratos (SNR). In order to get even a modest number of bts as throughput, one s forced to resort to somewhat complcated methods of data embeddng. We wll begn the paper by drawng a dstncton between the watermark pxel doman, and the raw watermark bt doman. The two are related through a varant of classc drect sequence spread spectrum communcatons, and we wll use ths relatonshp to develop the parameters for possble error correcton schemes. We wll also address uncoded borthogonal M- ary modulaton and determne whether or not t s a vable alternatve to tradtonal error correcton codng technques n the context of watermarkng. A descrpton of the expermental setup wll ensue. Next, we wll provde smulaton results, and address ssues of robustness and computatonal complexty. Fnally, conclusons wll be drawn.

2 2. THE WATERMARK PIXEL DOMAIN AND ITS RELATIONSHIP TO DIRECT SEQUENCE SPREAD SPECTRUM 2.1 Embedder Ths paper wll concern tself entrely wth technques based on Drect Sequence Spread Spectrum. We wll use, for the most part, the format descrbed by Dr. Adnan Alattar n hs Smart Images paper 2. In secton 3.1 of that paper, a mathematcal descrpton of how each bt s spread throughout the mage s gven. To summarze the relevant aspects as they pertan to ths paper, a pseudo random bnary key of length J s used to represent each bt. In other words, one watermark bt maps to J watermark chps. Dependng upon the sgn of the bt, the watermark key or ts nverse s used. Each of the J components of the key s assocated wth a physcal pxel locaton n the N x M mage. The process s analogous to drect sequence spread spectrum communcatons n that a sngle bt s mapped to a chppng sequence of length J. In ths paper we wll concern ourselves wth the transton from the watermark pxel doman (chps) to the Raw Bt watermark doman where we can apply error correcton. The characterstcs of the transton process are mportant because they dctate what s possble n the raw doman. 2.2 Model for nose n the watermark pxel doman We begn our analyss by composng a model that descrbes how the watermark reader sees the watermarkng channel. To ths end, we wll gnore ssues related to the very mportant ntal steps of watermark readng, whch nclude pre-flterng, synchronzaton, etc. so that we may concentrate on the process of watermark chp extracton and bt recovery. Suppose that we are gven, by magc, an area of N x M pxels n the watermark doman..e. we have synchronzed to the watermark coordnate system. The total receved sgnal at each pxel locaton, denoted by parameters and k, can be descrbed by the followng equaton: r = w + I + g + n ( k ) ( k ) k k k, (1) where I s the host mage, w s the watermark message, g s the synchronzaton sgnal, and n s addtonal nose due to the channel. The varables and k denote that the pxel locaton n queston s assocated wth the k th chp of watermark bt. In general there are N total bts n the watermarked mage, and there are J chps that belong to each bt. The total nose power at each pxel, whch ncludes everythng but the watermark tself, s typcally enormous compared wth the watermark chp power. 2.3 Classc correlaton recever The successful retreval of a watermark bt depends upon the nose present at each of the watermark chps. In classc drect sequence spread spectrum communcatons the relatonshp between chps and bts s defned n terms of sgnal to nose rato and processng gan. The sgnal to nose rato at the chp level s SNR 0 and the correspondng probablty of error s Q ( SNR0 ) (2) f one s to attempt to determne the polarty of one of the chps. In such classc drect sequence spread spectrum systems, the nose at the chp level s typcally close to whte Gaussan, and therefore t behooves us not to make hard decsons at the chp level n order to determne the value of a partcular bt. Rather, all chps are taken nto account together by usng a lnear correlaton recever. In whte Gaussan nose, such a recever s optmum 5, 6. The parameters of the recever are descrbed as follows: f K s the chppng sequence of length J that s used to represent the th bt, then the receved sgnal, R for that bt can be descrbed by R = b K + N (3) R n equaton (3) s really just the vector form of equaton (1). The watermark vector s now represented by b K and all nose components ncludng the mage and synchronzaton sgnal have been collapsed nto the nose vector N. The lnear correlaton recever performs the nner product between the receved sgnal, R, and the chppng sequence K. Ths procedure results n the detecton statstc s Jb + ϕ, (4) =

3 where the scalar value, φ = <K,N >, represents the output nose of the correlator. The estmated value of the bt, b, s obtaned by takng the polarty of s. The estmated polarty of a bt, after lnear correlaton, s much more relable than the estmated polarty of an ndvdual chp. The net effect of the lnear correlaton process s that the sgnal to nose rato s ncreased by a factor equal to the number of chps per bt (SNR 0 -> J*SNR 0 ). The factor J, whch corresponds to the number of chps, s called the processng gan. The probablty of error s decreased from that at the chp level by substtutng the new SNR nto (2). Note: f the bt s of a partcular type of error correcton bts, we do not make hard decsons on the detector statstc. Rather, the decodng process uses the magntude of the resultng nformaton to make better decsons downstream. 2.4 A recever wth pre-flterng Due to the host of factors lsted n the ntroducton, straghtforward applcaton of the lnear correlaton recever smply wll not suffce. The probablty of error s substantal enough for a watermark bt that the payload s unrecoverable even when usng error correcton codng. In order to mtgate the problem, a flter that takes advantage of local mage pxel correlaton s appled to the neghborhood of each of the chps belongng to the bt we are nterested n recevng 1, 2. By flterng the chps pror to applyng the lnear correlaton recever, the probablty of error s reduced sgnfcantly. For the purpose of ths paper, we wll use a flter that has a smple response for all nput values wthn the range of nterest;.e. any local neghborhood of pxels. In our notaton N x s an arbtrary neghborhood of mage pxels centered on pxel x, F() denotes the flter, and R s the fltered verson of R, whch s defned n equaton (3). F ( ) { 1,0,1} (5) N x R ' = F( ) (6) N R The flter makes hard decsons about the polarty of the message at the chp level takng nto account each chp pxel s neghborhood. We could just as well call ths a bnary symmetrc channel wth error probablty, p, and nonzero erasure probablty. For the present, we wll gnore erasures rentroducng the state at a later tme for completeness. By droppng the zero state of the output of the tr-level watermark pxel doman flter, ts response to the neghborhood can be descrbed by the bnary set of outputs {-1,1}. Gven that our watermark message at the chp level s a bnary antpodal sgnal, the probablty of successfully recoverng the message chp polarty s descrbed by a Bernoull random varable. After extractng the chp, the correlaton recever s used to obtan a soft estmate of the correspondng bt. Due to the type of nose present at each watermark pxel, and the type of recever flter used, we do not beneft from the same amount of processng gan that conventonal spread spectrum systems obtan. Because we make hard decsons at the pxel or chp level pror to usng the correlaton recever, we suffer an approxmate 2dB penalty compared wth the correlaton recever n whte gaussan nose (The watermark together wth mage and other nose s decdedly NOT descrbed by a Gaussan whte channel, and therefore the 2dB loss does not apply. Rather, we have found that we beneft substantally n most stuatons when such a flter s used). After applcaton of the lnear correlaton recever, the probablty of error n the Raw Bt watermark doman s descrbed by a dstrbuton that s the sum of J Bernoull random varables. If the Bernoull probablty p s the same for all chps that comprse the bt of nterest, a Bnomal random varable wll descrbe the probablty of error. Specfcally, we wll have a watermark bt error when less than J/2 of the fltered chps agree wth ther correspondng key entres. J / 2 N! k N k k!( N k )! p (1 p) k= 0 P = (7) In truth, the Bernoull probablty, p, vares wth each chp s poston, and hence the above equaton only roughly descrbes the stuaton. If we defne the mean Bernoull probablty for the chps, p mean, then the probablty of error n (7) s pessmstcally descrbed by p mean replacng p. The reason s that the mean of the bt estmate wll be the same, but the varance wll be smaller than that of the Bnomal dstrbuton when p vares across the watermarked meda. err The characterstcs of the chp flter lend tself to easy analyss at later stages. By specfyng a range of Bernoull probabltes for makng a chp error and ncludng the dmensons of the total watermarked area n terms of pxels, we can calculate the net probablty of error for the entre system (spread spectrum through error correcton). However, we wll fnd t even more convenent to break thngs down at a level beyond that of the chp. We wll refer to ths level as the Raw Bt doman.

4 Input Image Fg.1, Key Aspects of Watermark Detecton Synchronze Pxel Doman Correlator Raw Bts Payload Decoder Decoded Payload 2.5 The Raw Bt doman We wll fnd t convenent to ntroduce an ntermedate doman that les between the watermark reader, and the fnal decoded payload bts called the Raw Bt doman. The relatonshp between the Raw Bt doman, and the other domans mentoned s shown n fgure 1. The fgure encapsulates all error correcton codng methods that we wll consder; t wll need to be adjusted for M-ary modulaton. In the Raw Bt doman, a bt s composed of J chps. As we shall see, the converson to Raw Bts allows us to express our channel as a whte Gaussan nose channel governed by nput SNR nstead of by Bernoull probabltes. The mportance of the converson s that the gans from varous codng schemes are easly determned when the Raw Bts are subject to whte Gaussan nose. Smlar observatons were reported under the context of a dfferent embeddng scheme 3. For large enough J, the modfed bnomal dstrbuton, whch descrbes our bt error statstc, approxmates a Gaussan dstrbuton wth mean and varance descrbed by the equatons below. µ = 2Jp J (8) = 4Jp(1 p) 2 σ (9) We defne the sgnal to nose rato as SNR = σ µ ( p 2) = J p(1 p) (10) The mean and varance are dfferent than those of a normal Bnomal dstrbuton because the Bernoull alphabet we use s {-1, 1} nstead of {0,1}. The SNR s proportonal to J, whch makes t very easy to make smple adjustments to the SNR when J s changed. For the case of chps descrbed by Bernoull random varables wth varyng p, the resultant bt error probablty s not Bnomal, but t too s approxmately Gaussan for large enough J. In ths case, the resultng SNR wll be slghtly better than equaton (10). Another reason t s a good dea to defne thngs at the Raw Bt level s that any change n the pxel flter would change the behavor and probablty of error at the pxel level, but n the Raw Bt doman the statstcs would reman approxmately Gaussan. The net change at the Raw Bt level would be a shft n SNR, whch would not requre any sort of re-analyss. Generally, error corrected bts wll consst of a dfferent number of chps than J. We can accommodate such cases by adjustng the sgnal to nose rato approprately. Ths can be done very easly because, as descrbed above, the processng

5 gan (SNR) s very smply related to the number of chps used n the bt. The overall system works as follows. Gven the performance of the watermark reader at the pxel level we can defne a range of Bernoull parameters, p mean, for the gamut of watermarkng channel condtons wthn nterest. We can then translate ths number to a sgnal to nose rato at the Raw Bt level, and see what the probablty of error for the canddate error correcton scheme s operatng at that level of SNR and defne our probablty of error for a gven decodng scheme. Agan, for each p mean wthn the range of nterest, the correspondng SNR s a lower bound. At ths tme we rentroduce our watermark chp flter as a bnary symmetrc channel wth nonzero erasure probablty. Erasures wll reduce the number of watermark chps n a Raw Bt by an amount proportonal to the erasure probablty. If the erasure probablty s p e then the number of chps, J, per Raw Bt wll be reduced, on average, to J(1-p e ). The net effect s a decrease n SNR n the raw doman typcally a margnal one because p e s small. Snce we have converted all components of the watermark doman nto a Raw Bt doman representaton, we wll drop our treatment of the former n what follows. 3. ERROR CORRECTION CODES AND M-ARY MODULATION 3.1. Convolutonal codes and Reed-Solomon codes Suppose we have a payload of L bts that we would lke to embed n an mage. Error correctng codes ncrease the payload sze by addng redundant nformaton. It s the redundant nformaton that allows the code to do ts job, to correct errors. In watermarkng the cost assocated wth ncreasng the payload sze s that there wll be a smaller SNR per coded bt. Error correctng codes are useful f and only f the codng gan acheved by ntroducng redundancy more than makes up for the loss n SNR. The amount of redundant nformaton, and hence the expanson n payload sze, s expressed by the code s rate. For every k bts of uncoded data, n bts are embedded n the watermarked meda. The code s rate s defned as R = k/n. Error correcton decoders operate upon the Raw Bt values (output of the correlator defned above.) Each of the Raw Bts takes on a number from J to J, where J s the length of the chppng sequence. Some decoders make hard bnary decsons on each of the Raw Bts where the sgn of the value s the bass for the decson. The decoder then works ts magc on the resultng sequence. Generally speakng, there s some loss of nformaton n makng hard decsons on the Raw Bts pror to decodng. It s possble to mplement decoders that operate on the Raw Bts themselves. Such decoders, termed soft decoders, typcally perform better than ther counterparts hard decoders. However, n most cases, the extra computatonal cost of soft decodng s prohbtve. In ths paper we consder two dstnct classes of error correcton codes, Reed-Solomon block codes and Convolutonal codes. Reed-Solomon codes are a partcular type of block code. A typcal (n,k) block coder maps blocks of k bts nto blocks of n coded bts. The coder s sad to be memory-less because the n coded bts depend only on the k source bts 5. Hard decodng s almost always used n practce wth block codes, and n such cases the block decoder wll correct up to t bt errors. Reed- Solomon codes can be thought of as a generalzaton of block codes where symbols, nstead of bts, make up block elements. In other words, the coder maps blocks of k symbols nto blocks of n coded symbols. Each of the n coded symbols s taken from an alphabet of 2 m orthogonal symbols. For most cases that we wll consder symbols correspond to groups of bts; for example, L bts wll be represented by k symbols composed of n bts each. One reason Reed Solomon codes are often used n practce s because they have good mnmum dstance propertes. The mnmum dstance, d mn = n k + 1, s drectly related to the code s capacty to correct symbol errors. The code s guaranteed to correct up to t = ½(d mn + 1) symbol errors. Another reason these codes are often used s that effcent hard-decodng algorthms are avalable 5. Convolutonal codes are used more often than block codes because they are conceptually and practcally smpler to mplement, and ther performance s often superor. Convolutonal codes have memory; passng an L bt nformaton sequence through a lnear shft regster generates encoded data. The nput bts are shfted nto the regster, k bts at a tme, and n coded bts are output to produce the n/k ncrease n redundancy. The maxmum delay (memory) of the shft regster, and hence the code, s called the constrant length. The fact that convolutonal codes are lnear codes wth memory makes them sutable for effcent soft decodng algorthms, e.g. the Vterb algorthm 6. One pecular thng about convolutonal codes, n partcular, when they are used for short payload lengths, s ther behavor n terms of the total decode at both low and hgh SNR. The success of the total decode s a much more mportant quantty to montor than the bt error rate. For watermarkng applcatons, we typcally wll accept the result of the decode only when

6 that result s error free. Any number of bt errors s too many. For the uncoded case, the probablty of correct payload retreval s drectly related to the probablty of a bt error. For example, we know that n uncoded sgnalng the probablty of no bt errors (complete message retreval) s (1-p b ) N, where N s the number of bts. However, for convolutonal codes, the probablty of a bt error does not te n neatly to the probablty of correctly decodng. At low SNR the probablty of a bt error wll be hgh, but the probablty of decodng the payload correctly wll be much hgher than one mght expect. Errors are not ndependent; they almost always occur n groups. Another curous behavor occurs at hgh SNR. To see ths, consder a Vterb decoder operatng on the k th bt of a payload of arbtrary length. The coded verson of the k th bt s represented by r encoded bts begnnng at ndex r(k 1) + 1 of the coded sequence. It s well known that a fnal decson on any gven payload bt should be made much later than the above correspondence would ndcate 6. Specfcally, f the group of r coded bts begnnng at ndex r(k-1) + 1 s the latest to enter the decoder, then decsons on the k jd th payload bt wll be near optmal, where D s the constrant length of the code and j s an nteger typcally j equal to 4 or 5 wll suffce 6. In terms of decodng a watermark message at hgh SNR, the probablty of a bt error should be very low. However, because we have a small payload, decsons regardng the fnal bts n the payload must be made prematurely, effectvely ncreasng the error probablty. Decsons are made prematurely because there s no decodng delay avalable. 3.2 Concatenated Codes In some applcatons convolutonal codes are combned wth Reed Solomon codes to form what s called a concatenated code. A Reed-Solomon encoder s used on the payload bt strng; t s referred to as the outer code. The Reed-Solomon encoded data s then tself encoded usng a convolutonal code; the nner code. The method has been used for deep space communcatons (defntely a low SNR envronment!). The dea s to explot the strengths and cater to the weaknesses of each type of code. Reed-Solomon codes offer performance superor to that of convolutonal codes, but only for reasonably good channels. For very poor channels convolutonal codes work better. Usng the convolutonal code to clean up the raw channel allows the Reed-Solomon code to work where t can perform ts best. Another synergy between the two codes s that the convolutonal decoder tends to produce bursty errors, whch s what the Reed-Solomon code s best at correctng. 3.3 M-ary methods In hs paper, M. Kutter ntroduced a sgnalng scheme for watermarkng that was a generalzaton of bnary sgnalng at the bt level 1. Ths scheme, called borthogonal M-ary sgnalng, maps groups of log2(m) bts to one of M symbols. The beneft of dong so s that under certan condtons the probablty of symbol error becomes arbtrarly small as M goes to nfnty. Specfcally, f the energy per bt s greater than -1.6dB, the Shannon lmt, the above statement holds 5. The larger the value of M the fewer symbols we are requred to hde n an mage to convey the same number of bts. The fewer symbols we have to hde n an mage, the more locatons we can use per symbol 1. In other words, the symbol energy ncreases for ncreasng M (Although the energy per bt must reman fxed). We wll fnd t useful to relate the net symbol SNR to Raw Bt SNR, developed earler. One repetton of a symbol requres M/2 chps or watermark pxel locatons. Just as there are many chps per watermark bt n the bnary case, each symbol s typcally repeated many tmes n the watermark doman to ncrease ts aggregate SNR. We wll use an example to ease the process of development. Suppose we have an L x L pxel area to be watermarked; call ths the TotalPxelArea. Further suppose that we would lke to embed N bts. In the bnary case t s easy to see that each bt wll get (TotalPxelArea /N) chp repettons. The SNR per bt s related to the Raw Bt SNR by a multplcaton factor, SNR b = asnr raw, (11) where a = (TotalPxelArea /N) /J. For example, f TotalPxelArea = 128x128, N = 64 bts, and J = 32, then a = 8, a shft of +9dB. The more general case s descrbed by the equatons below. a = TotalPxelArea M N J M 2 log2( ) In equaton (12), M/2 s the number of pxels requred per symbol repetton, N/log2(M) s the total number of symbols needed to represent the requred number of embedded bts, and J s the number of pxels (chps) per Raw Bt. (12)

7 M SNRSymb = asnrraw (13) 2 Results usng the above equatons are tabulated for some sample parameters, below. Table 1 N = 60-64bts, rawbtsnr = -2dB, chps/rawbt = 32, Symbol Sze = M/2 M Num Symbols Symbol Reps Symbol SNR(dB) Here we summarze the process of embeddng and detectng M-ary symbols. M dstnct sgnals are produced by computng an M/2 order Hadamard matrx, H M/2. To generate the addtonal M/2 borthogonal entres, we append the sgn-reversed verson of H M/2 to tself 1. The detecton process works by garnerng all avalable symbol repettons from the watermark pxel doman to form an aggregate nosy estmate of the symbol, S + N. We would use, for example, the same chp fltered descrbed by equatons (5, 6) to preflter each contrbuton to a symbol element. M/2 correlators of length M/2 are used n the detecton process, whch s descrbed n the fgure below. Fg.2 M-ary Symbol Detecton S 1 +N x r 1 Aggregate Symbol Repettons S 1 x r M/2 Select largest r ; Choose S or S Accordng to sgn of r S M/2 The SymbolSNR s the SNR of each symbol element pror to match flterng wth the bank of M/2 correlaton recevers, R = S + N and snr = s 2 /var(n ). Selectng the symbol s a two-step process: It requres largest magntude determnaton followed by sgn determnaton. Makng an error at the symbol level can be accomplshed by confusng the correct symbol wth ts antpodal verson, or mstakng one of the other possble M/2 symbols for the correct one. Errors almost always occur due to the second condton mentoned because the dstance n sgnal space between orthogonal sgnals s half that of antpodal sgnals 1,5. The probablty of error for borthogonal M-ary sgnals, P M, s gven n Proaks book as a functon of symbol sgnal to nose raton, SNR S 5. We reproduce the equaton here. P M = π 2SNR 2π S v+ ( v+ 2SNRS 2SNRS ) e 2 x / 2 dx) M / 2 1 e 2 v / 2 dv (14)

8 The equaton may be evaluated numercally for dfferent values of M and SNR S. We are more nterested n the probablty of not decodng the entre watermark, P W. Every symbol s represented ndependently n the mage and each has the same SNR S (the reason for equal SNR s s a result of an nterleavng procedure 2 ). We requre the successful demodulaton of all symbols that make up the bt payload to be successful. The equaton governng our success s N / log2 ( M ) P = 1 (1 ) (15) W P M 3.4 Combnng M-ary sgnalng wth error-correcton codng So far we have treated error correcton codng and M-ary modulaton as f they are mutually exclusve methods. Yet, snce we have entertaned the dea of concatenated codes, t seems reasonable to thnk of possbltes employng M-ary modulaton wth error correcton. We do not treat any such technques n the smulaton, but for the sake of completeness we develop the basc premse. We consder two examples of hybrd M-ary and error correcton technques. Reed Solomon codes are partcularly suted to M-ary modulaton when the full alphabet of 2 k symbols s used 5. To embed L bts wth standard M-ary modulaton would requred K, M-ary symbols, where Klog2(M) s equal to L. Usng Reed Solomon codes, the K, M-ary symbols are converted to a larger number, N, for embeddng. Snce more symbols are requred to embed the same number of bts, the SNR per symbol wll decrease. The probablty of symbol error at the output of the decoder s upper-bounded by P es = N 1 N N ( N ) PM (1 PM ) = t+ 1, (16) where P M s gven n equaton (14), but the quantty must be adjusted for the loss n SNR. SNRdB -> SNRdB - 10log10(N/K) (17) The probablty of decodng the payload s gven by (15). As we shall see, at low SNR the uncoded case s superor. The stuaton s reversed for hgher SNR. In lmted bandwdth stuatons, trells codng s often used to enhance performance. Trells codng conssts of frst convolutonally encodng the nformaton bts, and then usng lne codng to select a symbol to transmt from the avalable constellaton. For example, n dgtal communcatons, one mght use a rate 1/3 convolutonal code, and then map the trplet of bts to a symbol n an 8-PSK constellaton. On the decodng sde a Vterb soft decoder would map the symbols to receved bts. The symmetry n the constellaton makes for a large mnmum dstance error event, and hence the gan over an uncoded system s substantal. It s possble to construct a quas-trells codng scheme usng convolutonal codes and M-ary modulaton. In the M-ary orthogonal scheme, however, all symbol errors barrng the antpodal symbol are equally lkely, and therefore the mnmum dstance error event wll be closer than n a more conventonal trells-codng scheme. Nevertheless, let us proceed wth an llustraton of such a scheme. Suppose we apply 8-ary modulaton to rate 1/3 convolutonally encoded bts. Coded bts, n groups of three, are mapped to a unque symbol. The detector wll perform M-ary detecton as normal usng the bank of correlaton recevers. Instead of selectng the symbol where the correlaton s maxmum and makng hard decsons on the coded bts, the decoder wll use the strength of that correlaton value wth reference to the next hghest n order to assgn a soft relablty metrc to the chosen symbol. The resultng soft values are fed to a Vterb soft decoder. 4. ERROR CORRECTION CODING AND UNCODED M-ARY SIMULATION RESULTS In the dscusson that follows, the word protocol s synonymous wth a data-encodng scheme. In fact, t supersedes t snce M-ary modulaton s not truly an error-correcton encodng scheme. Table 2 shows a lstng of the varous protocols that have been consdered n ths study. Protocols wth convolutonal codes, Reed-Solomon codes, and also concatenated Reed- Solomon and convolutonal codes have been examned. In addton, we smulated several dfferent levels of M-ary sgnalng. For the concatenated codes, an effort was made to look at a varety of code rate allocatons between the Reed-Solomon and convolutonal code. For all types of protocols, the number of chps, or equvalently Raw Bts, per protocol bt was also vared to dentfy the best compromse between codng and repetton. In all protocols the total number of chps used s 128x128. All convolutonal codes used were memory 8 codes (wth 128 states), except a sngle protocol where a memory 7

9 code was used to see f t paradoxcally could perform better due to the small decodng trells. Most protocols were desgned to allow approxmately 64 bts for the payload; small devatons sometmes had to be made to accommodate the avalable Reed-Solomon codes, and varous M-ary levels. In order to measure the performance of each of the protocols we used smulaton. Snce only a small number of varables need to be n the model, the smulaton task s tractable. Refer to fgure 1. It shows a dagram of the reader broken nto three components: synchronzaton, watermark readng, and payload decodng. The only nput to the payload-decodng block s the sequence of raw values, one for each bt or symbol of the protocol. The factor that determnes the performance of the payload-decodng block s the statstcal relatonshp between the orgnal protocol bts and the Raw Bts that are nput to the decodng block. We have chosen a range of SNR that truly stresses the varous methods consdered. Before gong nto detal regardng smulaton results, we perform a cursory comparson wth expected theoretcal results nvolvng the bt error rate, whch s a conventonal measure of relablty n dgtal communcatons, and pont out a few problems wth relyng entrely on the measurement. In ths example the payload s 56 bts, and three dfferent methods are compared: bnary modulaton, 32-ary borthogonal modulaton, and bnary modulaton wth rate 1/3 convolutonal codes employng soft Vterb decodng. The three methods are compared over the same range of Raw Bt SNR, where a Raw Bt s comprsed of 32 chps. For bnary modulaton, the actual SNR per bt s the Raw Bt SNR dB due to the extra space avalable per bt for processng gan. The rate 1/3 convoluton codng method gets half that, an addtonal 4.5db of processng gan per coded bt; the other 4.5dB was used for codng gan. The SNR per bt of the M-ary scheme s essentally the same as that for the bnary scheme, but the SNR per symbol s log2 (M) greater. Fgure 3a, Bt Error Rate from Smulaton Fgure 3b, Payload Error Rate Comparng the bnary method wth that usng convolutonal codes, we see that the former has a lower bt error rate untl ~- 6dB, after whch t s supplanted by the codng scheme. The M-ary symbol error rate s hgher than the bnary bt error rate untl ~-4dB. The M-ary bt error rate s always lower than ts symbol error rate, snce each symbol conssts of a group of bts. It s approxmated by ½ the symbol error rate for large M 5. Vewng Fgure 3b, whch depcts the probablty of a correct decode for each of the methods, we notce that for low Raw Bt SNR the probablty of error for the bnary scheme s enormous. For the range of SNR where t appeared better than the convolutonal codng technque n terms of bt error rate, the method s essentally useless f there s no tolerance for bt errors a stpulaton that seems reasonable when the payload s small. Hernandez, et al. have ponted out ths problem 3. More nterestng, and less obvous, s that the probablty of a payload error for the convolutonal codng method s always less than that of the bnary method, even n the regon where bt error probablty would suggest otherwse. In fact the

10 probablty of a decode error s arguably reasonable, for some applcatons, when the bt error rate s stll margnally hgh. For example at 6dB the bt error rate s roughly 0.1 and the correspondng probablty of decodng the entre payload correctly s as hgh as 0.6. The reason for ths phenomenon s that, usng our example, 60% of the nosy payload data wll succeed n beng decoded as a whole. The other 40% wll be characterzed, n general, by bursts of errors. Ths all or nothng result s a consequence of consderng the data n one lumped sum. A concluson that can be drawn from ths s that bt error rate s not a very nterestng statstc when appled to convolutonal codng of small payloads. Another observaton about convolutonal codes s that the probablty of error for hgh SNR s worse than experence wth such a technque would ndcate. The reason for ths s that Vterb soft decoders need a delay of about 4-5 tmes the constrant length pror to makng bt decsons n order to operate at ther optmum rate 6. We, however, are forced to make mmedate decsons for data at the tal of the bt stream because of the very small payload sze, a fact that sgnfcantly ncreases the probablty of error. The performance of each of the protocols was smulated for SNR rangng from 7dB to 2dB n steps of 1 db. The SNR s gven n terms of a Raw Bt wth 32 chps; for smulaton purposes the SNR was adjusted as approprate for other number of chps/bt. Each data pont was obtaned from smulatng 8000 randomly chosen payloads and passng them through the AWGN channel. For reasons mentoned n the precedng paragraphs, we quote the performance of the total decode and gnore the bt error rate. Results showng the probablty of correct payload decodng are shown n Table 3. A few very general remarks can be made concernng the varous classes of protocols. Protocols employng pure Reed- Solomon block codes dd not perform well over the chosen range of SNR. Our result s consstent wth that posted by others who have tred BCH codes, a bnary block code 3. The two protocols that used pure convolutonal codes of dfferng rates performed well, better than those that used Reed-Solomon codes for all SNR n the smulaton. Several of the concatenated codes acheved very good rates of decodng the payload correctly at hgh SNR. The hgher order M-ary schemes dd reasonably well at all SNR, and they were among the best at hgher SNR n partcular. Protocols that use convolutonal codes as ther prmary ngredent are not a bad choce for watermarkng. Comparng Protocols 29 and 30, the two that used convolutonal codes of rates 1/3 and ¼ respectvely, we can say that the overall performance s about the same. The hgher rate (1/3) s slghtly better at low SNR, but at a hgher and more useable SNR, the rate ¼ code mght be margnally superor. Protocols 11, 12, and used concatenated codes and they performed at least as well as Protocols 29 and 30 at hgh SNR. The reason for ths s that at approxmately -4dB we begn to see a transton, whereas the error characterstcs of protocols 29 and 30 change from error patterns characterzed by bursts that plague the entre payload to shorter error events at the tal of a payload. The short error events, whch are a result of early trells truncaton n the Vterb Algorthm, are corrected f an outer code s used, provded we can accommodate the loss n SNR requred by the extra codng. In lght of ths dscusson, one mght argue that a concatenated code operatng upon the entre payload s overkll. If nstead of usng a tradtonal concatenated code, we apply a BCH code to the tal of a convolutonally encoded payload we would protect the most error prone part of the payload. The mpact n terms of SNR would be less than that of the concatenated code because we would be requred to embed fewer coded bts. Protocols that use pure M-ary modulaton, wth large M, are also a reasonable choce for watermarkng under the smulated condtons. Protocol 36, whch has M = 256, s arguably the best of all protocols for the smulated range of SNR. It s, of course, possble to consder values of M larger than 256 f processng tme s of a lesser concern. The appendx shows that Protocol 36 requres almost 5 tmes the number of operatons that Protocol 29 does. In the appendx, however, we dd not consder the possblty of usng a Fast Walsh Transform 7, whch can sgnfcantly curtal the number of requred operatons. The Fast Walsh Transform s possble when a Walsh-Haddamard set of bass functons s used the case we smulated here. Havng dentfed some promsng protocols, further smulaton was performed. Protocols #11, #30, and #36 were smulated over a range from 10dB to 1dB n steps of.75db. Each data pont s the result of smulaton trals. In fgure 5, we plot the estmated probablty of not correctly decodng the payload n ts entrety. Comparng Protocols #11 and #30, the two based on error-correcton schemes, we see that whch of the two protocols s best depends on the SNR. For poor effectve channels, the protocol wth a convolutonal code outperforms the protocol wth a concatenated code. For better channels the stuaton s reversed. Protocol #36, whch s an M-ary method, performs better than the two error-correcton codng technques, throughout the smulated range of SNR. We conclude that M-ary modulaton s certanly a sutable method for watermarkng.

11 Fgure 5, Detaled Comparson of Promsng Protocols 5. CONCLUSIONS The class of dgtal mage watermarkng technques based on spread spectrum methodology s often characterzed by very low SNR. Methods have been proposed that ether enhance the basc technque by applyng error correcton codes, or generalze the spatal spread spectrum prncple usng M-ary modulaton to obtan better results 3,4,1. Comparsons between dfferent error correcton schemes have been made prevously 3, but ths paper expands on the theme by testng multple code rates, explorng parameter allocaton, and ntroducng concatenated codes. Furthermore, M-ary schemes are compared aganst the varous error correcton codng methods. In the context of our comparson, we have chosen to ntroduce a framework called the Raw Bt doman that allows us to dstll down the elements of competng schemes so that we can make drect comparsons gven channel condtons (SNR), and payload sze. As a result of our smulatons, we beleve that convolutonal codes alone or concatenated wth Reed-Solomon codes can be a good choce to ncrease payload robustness. M-ary methods, for M greater than or equal to 256, are at least as good of a choce, consderatons of computatonal complexty asde. Soft Convolutonal decodng technques suffer a bt more than one would expect at hgh SNR when the payload s small, a fact that makes them nferor to hgher order M-ary technques. Some types of concatenated codes are able to mtgate ths problem at hgh enough SNR. In watermarkng the channel nose characterstcs are lkely to vary substantally, a fact that should be kept n mnd when choosng a codng scheme.

12 Table 2a, Protocol Descrptons and Parameters (Error Correcton Codng) Protocol # # Payload Bts #chps/protocol bts RS symbol RS N RS K CC rate CC memory Table 2b, Protocol Descrptons and Parameters (M-ary) Protocol # Number of Bts M-ary Level

13 Table 3, Comparatve Rates of Correct Payload Decodng Protocol # -7dB -6dB -5dB -4dB -3dB -2dB > > APPENDIX In the appendx we address ssues of computatonal complexty for each method. 1. Convolutonal Codng usng the Vterb Algorthm complexty ncreases lnearly wth payload sze. Refer to references 5,6 for mplementaton detals. K s the constrant length and L s the ntroduced codng redundancy. The followng s for rate 1/L convolutonal codes. 2 L Eucldean dstance calculatons per payload bt. 3L operatons per Eucldean dstance calculaton. 3L or 3L 1(L adds + L mults + L-1 adds). 2 x 2 k adds per payload bt. 2 k Compares per payload bt. NumOps = N x ( 2 L x3l + 2x2 k + 2 k ) Example: Protocol number 29 has N = 64, L = 3, and K = 8. The number of operatons s on the order of

14 2. Reed Solomon Codes- Vares greatly dependng upon algorthm used. In general, the complexty s less than that of convolutonal codes. 3. M-ary Sgnalng- M/2 adds plus M/2 multples per correlaton wth 1 of M/2 symbols. M/2 symbols to be correlated aganst. M/2 +1 operatons to determne the embedded symbol after correlaton (choose the symbol based on the sgn of the maxmum correlaton magntude). N/log2(M) symbols requred. NumOps Mary = N/log2(M) * (M x M/2 + M/2 + 1) Example: Protocol number 36 has M = 256, and N = 64. The number of operatons requred s on the order of 263,000, almost 5 tmes that of the Convolutonal code example. REFERENCES 1. M. Kutter, "Performance Improvements of Spread Spectrum Based Image Watermarkng Schemes Through M-ary Modulaton", Prelmnary Proceedngs of the Thrd Internatonal Informaton Hdng Workshop, pp , Dresden, A. Alattar, Smart Images, Proceedngs of SPIE Vol. 3971, pp.?, San Jose, J.R. Hernández, J. Delagle, B. Macq, Improvng Data Hdng by Usng Convolutonal Codes and Soft-Decson Decodng, Proceedngs of SPIE, Png Wah Wong, Edward J. Delp, Edtors, Vol. 3971, pp.?, San Jose, S. Baudry, P. Nguyen, H. Maître, Channel Codng n Vdeo Watermarkng: Use of Soft Decodng to Improve the Watermarkng Retreval, Proceedng ICIP 2000, Vancouver, Canada, J.G. Proaks, Dgtal Communcatons, 3 rd Edton, Chapters 5, 8, McGraw-Hll, E.A. Lee, D.G. Messerschmtt, Dgtal Communcaton, 2 nd Edton, Chapters 9, 13, 14, Kluwer Academc Publshers, MA, R.C. Gonzalez, R.E. Woods, Dgtal Image Processng, Chapter 3, Addson-Wesley, 1992.