Characterizing and Estimating Block DCT Image Compression Quantization Parameters

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Charactrizing and Estimating Block DCT Imag Comprssion Quantization Paramtrs Ramin Samadani Imaging Systms Laboratory HP Laboratoris Palo Alto HPL-2005-190 Octobr 20, 2005* comprssion, artifact

1 Charactrizing and Estimating Block DCT Imag Comprssion Quantization Paramtrs Ramin Samadani Imaging Systms Laboratory HP Laboratoris Palo Alto HPL Octobr 20, 2005* comprssion, artifact rduction, imag procssing, squntial stimation This papr dscribs an algorithm for stimating quantization matrics from rastr imags prviously comprssd with JPEG. First, th spac of commonly usd quantization matrics is statistically charactrizd using ovr imag fils. Th insights from this charactrization ar usd to dsign th algorithm. Th two stag algorithm first applis a nw squntial stimation procss to dtrmin som individual Q matrix ntris, and thn applis shap-gain vctor quantization to rcovr complt quantization matrics. Low avrag absolut rror valus ar found for 124 imags not usd during training. In addition, th stimatd Q matrics work wll whn applid to comprssion artifact rduction. * Intrnal Accssion Dat Only Publishd in th Asilomar Confrnc on Signals, Systms, and Computrs, 30 Octobr 2 Novmbr 2005, Pacific Grov, CA, USA Approvd for Extrnal Publication Copyright 2005 IEEE.

2 CHARACTERIZING AND ESTIMATING BLOCK DCT IMAGE COMPRESSION QUANTIZATION PARAMETERS Ramin Samadani Imaging Systms Laboratory ABSTRACT This papr dscribs an algorithm for stimating quantization matrics from rastr imags prviously comprssd with JPEG. First, th spac of commonly usd quantization matrics is statistically charactrizd using ovr imag fils. Th insights from this charactrization ar usd to dsign th algorithm. Th two stag algorithm first applis a nw squntial stimation procss to dtrmin som individual Q matrix ntris, and thn applis shap-gain vctor quantization to rcovr complt quantization matrics. Low avrag absolut rror valus ar found for 124 imags not usd during training. In addition, th stimatd Q matrics work wll whn applid to comprssion artifact rduction. Input imag Gt nxt 8x8 block Convrt RGB to YCC Forward 2D DCT no scond stp For ach cofficint not yt stimatd Updat spars structur with currnt valu Excut dcision rul on spars structur. Larg nough count to stimat? ys Estimat and incrmnt numbr of cofficint stimats compltd. Enough cofficints stimatd for xtrapolation? ys Normaliz stimatd cofficints and q tabl shap codbooks Find closst codbook and rturn with appropriat normalization no, procss nxt cofficint 1. INTRODUCTION Onc an imag is dcomprssd, xplicit information about its comprssion paramtrs is lost. Th comprssion paramtrs ar usful, howvr, to: 1) st paramtrs for comprssion artifact rduction; 2) avoid loss during rcomprssion; and 3) potntially dtrmin th camra modl usd during imag captur. For xampl, th croppd original imag in th top of Figur 2 contains comprssion artifacts. Th middl imag shows rsults of artifact rduction [1], using th known, tru comprssion paramtrs. Th bottom imag shows almost indistinguishabl rsults using comprssion paramtrs stimatd as dscribd blow. Th rsulting artifact rduction is vry similar for all th 124 imags tstd. Q matrics ar quivalnt, whn transformd by columnscanning, to q vctors, with componnts q i, i {0 63}. Known mthods for stimating q i start by r-applying 8x8 forward DCT transforms to non-ovrlapping imag blocks. Th rsulting cofficints d i (k), for imag blocks k, ar, idally, multipls of th tru q i. Prior work stimatd individual q i valus by applying maximum liklihood stimation [2], or othr objctivs [3], to histograms of d i (k). Asa first stp, this papr stimats q i valus using a nw squntial mthod without complt histogram calculation, shown in th top lft and th top right dashd block of Figur 1. With any of ths mthods, th q i cannot b stimatd for Fig. 1. Flowchart for th mthod for th Y componnt. For th color componnts, a similar algorithm may b applid with simpl modifications that tst diffrnt possibl subsamplings. som i, bcaus th d i (k) quantiz to zro throughout th imag. An xtrapolation is ndd to rcovr all th q i.to my knowldg, no prviously proposd mthods rcovr ntir vctors q whn a numbr of diffrnt typs of quantization matrics ar candidats for th comprssion. Hr, a brif ovrviw of th algorithm is givn, with dtails dsribd in latr sctions. Figur 1 shows th flowchart for th mthod for th Y componnt. Th first stps up to and including th forward 2D DCT occur onc for ach 8x8 block. Th procssing in th dashd box on th top right of th figur occurs for ach of th 64 DCT cofficints: updating th counts and valus of a spars data structur, dciding basd on th magnituds of th counts, whthr to form an stimat for th q i valu or to continu procssing. Aftr this loop, a variabl that counts th numbr of cofficints stimatd so far is chckd against a constant. If thr ar not nough cofficint stimats, th procssing continus at a nw imag block. If thr ar nough cofficint stimats, th scond and final stp conducts a vctor quantization dcoding. Th stimatd q i valus from th first stp ar

shap matchd against prviously gnratd VQ codbooks of q vctors (ths prvious codbooks wr gnratd using th LBG vctor quantizr dsign algorithm on svral thousand JPEG q vctors).

3 shap matchd against prviously gnratd VQ codbooks of q vctors (ths prvious codbooks wr gnratd using th LBG vctor quantizr dsign algorithm on svral thousand JPEG q vctors). Th closst match is normalizd and rturnd as th final q vctor stimat. Sction 2 covrs th charactrization of quantization matrics, which providd insights during th dsign of th algorithm. Thn, th first stag of th algorithm, th nw squntial mthod for stimating individual stimats q i,is dscribd in Sction 3. Sction 4 dscribs th scond stag that uss shap-gain VQ to stimating th ntir quantization vctor q. Finally, Sction 5 dscribs xprimnts validating th approach. 2. CHARACTERIZING QUANTIZATION PARAMETERS y r g n l a u id s r t iv la R Numbr of SVD cofficints Fig. 3. Th rlativ rsidual nrgy as a function of numbr of singular vctors usd to approximat th spac of q vctors. Fig. 2. Imag of a fir nar a rsidntial nighborhood. Top shows portion of original imag with comprssion artifacts. Middl shows rducd artifacts with known comprssion paramtrs. Bottom, indistinguishabl to middl, shows rducd artifacts with stimatd comprssion paramtrs. Charactrization of th quantization matrics that ar usd, in practic, is ndd. Ovr JPEG fils, mostly consisting of digital camra imags, but also of JPEG fils typically found on computr disks, wr studid. From ths fils, 1142 uniqu q vctors (fils originating from th sam sourc oftn hav th sam q vctor), for th luminanc channl, wr xtractd and usd to form a 64 by 1142 matrix M. Thn, th singular valu dcomposition, USV T = M, was computd. Figur 3 shows th rlativ rsidual nrgy, ˆM M M, whn stimating M using an incrasing numbr of singular vctors (arrangd in dcrasing ordr of corrsponding singular valu magnitud). Th figur shows that th first singular valu accounts for about 90% of th nrgy in rprsnting this spac. Intrstingly, inspction found th first singular vctor to b rmarkably similar to th JPEG dfault quantization matrix [4]. Not, howvr,

is natural to apply shap-gain vctor quantization [5] (which sparats a normalization gain α from a shap) to th stimation. This will b dscribd in Sction 4.

4 that thr is also nrgy in som of th rmaining singular vctors. With th knowldg that th first fw singular vctors contain most of th nrgy in rprsnting Q matrics, and th additional knowldg that manufacturrs oftn control comprssion ratio by scaling th Q matrics, it is natural to apply shap-gain vctor quantization [5] (which sparats a normalization gain α from a shap) to th stimation. This will b dscribd in Sction 4. First, howvr, th nxt sction dscribs th first stag of th algorithm, th squntial stimation of individual q vctor ntris. valus clustr at multipls of th tru q i valu, and that not all ths horizontal lins ar ndd to stimat q i, so that a spars data structur that can track paks may b usd instad. In addition, xamining th bottom plot of th figur shows that thr is nough information to stimat th q i valu vn by analyzing only a fraction of th blocks from th imag, suggsting th us of a squntial algorithm. Valu buffr V0 = 0 V1 = 1 V2 = 2 V3 V4 V5 Count buffr 3. INITIAL SEQUENTIAL ESTIMATION N0 N1 N2 N3 N4 N5 To stimat an unknown q, initial stimats of individual q i ar first found using th mthod shown in th top lft and th top right dashd block of Figur 1. Th initial stimat vctor, ˆq 1, has valid stimats for only a subst of componnts. G NW C X V P KG H KE G Q E 6 % & &%6EQGHHKEKGPVHQTDGIKPPKPIRQTVKQPQHKOCIG &%6DNQEMUGSWGPEGPWODGT Fig. 4. Rconstructd DCT cofficints as a function of squntial block numbr. Th top right shows a spars data structur may b sufficint for stimating th q valus. Th bottom shows that only a fraction of an imag may b ndd to stimat th q valus. Rviw of Figur 4 provids motivation for us of a spars data structur that dos not stor an ntir DCT cofficint histogram, and for th ability to stimat squntially. Th top lft of th figur shows th valus of a rconstructd DCT cofficint (on th y axis) as a function of squntial block numbr. Th top right of th figur xpands th vrtical scal, showing that th rconstructd cofficint Fig. 5. This spars data structur, contains nin storag locations (V0, V1 and V2 ar implicit and nd not b stord) vrsus th thousand that would b ndd whn storing ntir histograms (for ach of 64 cofficints). Figur 5 shows th spars data structur usd, on for ach of th 64 DCT cofficints, mant to track th maxima of th probability distribution function (pdf) for th givn cofficint. Th structur has six clls, ach cll with a valu and a count. Th first thr valus ar hardwird to valus 0, 1, and 2 and ar procssd spcially by th dcision rul (if v2 >v0it is assumd that q =1or q =2and additional ruls distinguish ths two cass). Ths spcial valus handl th spcially difficult cas of fin quantization whn th tru q i valu is vry small. Th othr valus, v3, v4 and v5, float: thy may tak any valus that occur in th data stram. A rplacmnt rul updats th data structur at ach block. If th currnt data valu occurs in th tabl, its count is incrmntd. Othrwis, th currnt data valu and th count valu 1 rplacs th cll with th smallst count. In practic, this procdur finds and kps th maximum non-zro valu of th pdf, which bcaus of th statistics of DCT cofficints of imags, most of th tim corrsponds to an stimat of q i itslf. Th dcision rul in th cas of larg q i picks th maximum valu. With this spars structur approach, th q i valu is stimatd without storing or procssing th 1000 lmnt histograms for ach cofficint, making th mthod fast in softwar and fficint in hardwar storag. 4. SHAPE VQ FOR THE FINAL ESTIMATION Givn th first stag stimats of som of th q vctor ntris, th scond stag rcovrs th complt q vctor. First, basd on th insight from Sction 2 that a fw of th singular vctors hav nrgy, during training, normalizd q vctors wr input to th LBG algorithm [5], to gnrat fiv rprsntativ, normalizd codword vctors, c j, j {0 5}.

5 Aftr th compltion of th first stag of th algorithm, th initial subst of stimats ˆq 1 ar usd to form final stimat, ˆq f. Th shap-gain dcoding shown at th bottom dashd block of Figur 1, applid to ˆq 1, is now dscribd. A diagonal wight matrix, W, with w ii = { 1 if qi initial stimat xists 0 if q i initial stimat dos not xist accounts for th fact that not all th q i hav initial stimats. Givn initial stimat, ˆq 1, for a givn a codword c j, th modifid shap-gain distortion is dfind by, (1) D(ˆq 1,αc j )={(ˆq 1 αc j ) T W (ˆq 1 αc j )}, (2) Th valu of α at th minimum cost may b shown to b, αj = ct j W ˆq 1 c T j Wc. (3) j With ths valu for α, th final stimat ˆq f = αk c k is givn by th codword k with minimum D(ˆq 1,αk c k) in Equation RESULTS Th Q matrix stimation was applid to 124 imags that wr not usd for training, and for which th tru Q matrics wr known. Statistical rsults ar shown in Figur 6. Th top of th figur shows avrags of th absolut rrors (absolut diffrnc btwn stimat and tru q i ) for ach of th 64 Q matrix cofficints stimatd. Th bottom of th figur shows th stimatd standard dviation of th man absolut rror, computd using th bootstrap mthod. For illustration, xampls of th stimation applid to two prviously comprssd imags ar now prsntd. For th first imag, th known tru Q matrix, shown on th top of Figur 7, is similar to th dfault JPEG tabl. Th rsults of th first stp of th stimation ar shown in th middl of th figur, with zros rprsnting valus without stimats. Th bottom of th figur shows th final stimat, ˆQf, that is vry clos to th original. Th scond xampl, shown in Figur 8, shows th rsult for a known tru Q matrix that is constant. In this xampl, th tru Q matrix is xactly rcovrd. As prviously mntiond, comprssion artifact rduction was conductd using both th known Q matrics and th stimatd ons, with vry similar rsults in all cass. Also, th sam approach was usd to charactriz th quantization matrics of th color componnts as wll, with th addd complication bing th unknown subsampling factor of th color componnts. ˆµ = ˆσ = Fig. 6. Th top shows th avrag of th absolut rror for 124 imags, and th bottom shows th standard dviation (computd using bootstrap) of th avrag absolut rror. 6. REFERENCES [1] R. Samadani, A. Sundararajan, and A. Said, Dringing and dblocking DCT comprssion artifacts with fficint shiftd transforms, in Procdings of th Intrnational Confrnc on Imag Procssing (ICIP), [2] Zhigang Fan and Ricardo L. d Quiroz, Idntification of bitmap comprssion history: Jpg dtction and quantizr stimation, IEEE Transactions on Imag Procssing, vol. 12, no. 2, pp , Fbruary [3] Jssica Fridrich, Miroslav Goljan, and Rui Du, Stganalysis basd on jpg compatibility, in SPIE Multimdia Systms and Applications IV, August 2001, pp [4] V. Bhaskaran and K. Konstantinids, Imag and Vido Comprssion Standards: Algorithms and Architcturs, Springr, 2nd dition, [5] A. Grsho and R.M. Gray, Vctor Quantization and Signal Comprssion, Kluwr Acadmic Publishrs, 1992.

6 Q = ˆQ 1 = ˆQ f = Q = ˆQ 1 = ˆQ f = Fig. 7. First xampl: Original Q, initial stimat ˆQ 1, and final stimat ˆQ f. Fig. 8. Scond xampl: Original Q, initial stimat ˆQ 1, and final stimat ˆQ f.

CS 361 Meeting 12 10/3/18

CS 361 Meeting 12 10/3/18 CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm