Wavelet Coding of Volumetric Medical Datasets

Transcription

1 IEEE Transactons on Medcal Imagng 1 Wavelet Codng of Volumetrc Medcal Datasets Peter Schelkens, Member, IEEE, Adran Munteanu, Joer Barbaren, Mhnea Galca, Xaver Gro-Neto, IEEE Member, and Jan Cornels, Member, IEEE Abstract Several technques based on the 3D Dscrete Cosne Transform (DCT) have been roosed for volumetrc data codng. These technques fal to rovde lossless codng couled wth qualty and resoluton scalablty, whch s a sgnfcant drawback for medcal alcatons. Ths aer gves an overvew of several state-of-the-art 3D wavelet coders that do meet these requrements, and rooses new comresson methods exlotng the quadtree and block-based codng concets, layered zerocodng rncles and context-based arthmetc codng. Addtonally, a new 3D DCT-based codng scheme s desgned, and used for benchmarkng. The roosed wavelet-based codng algorthms roduce embedded data streams that can be decoded u to the lossless level, and suort the desred set of functonalty constrants. Moreover, objectve and subjectve qualty evaluaton on varous medcal volumetrc datasets shows that the roosed algorthms rovde comettve lossy and lossless comresson results when comared to the state-of-theart. Index Terms Volumetrc codng, medcal mage comresson, lossless comresson, embedded codng, quadtree codng, layered zero codng, rogressve mage transmsson, JPEG2000. I. INTRODUCTION IN many medcal alcatons comresson s ndsensable to guarantee nteractvty durng the consultaton of large sets of mages (e.g. volumetrc data sets, tme sequences of mages, mage databases), for robng context deendent detaled mage structures, and/or quanttatve analyss of the measurements. As a consequence, tradng-off mage qualty and/or mlementaton comlexty aganst bt-rate ntroduces secfc constrants. On one hand t s ntolerable to dro any nformaton when handlng medcal data. Dscardng small mage detals that mght be an ndcaton of athology could alter a dagnoss, causng severe human and legal consequences [1]. For examle, mages obtaned from rojecton radograhy may reveal lesons by mage detals Manuscrt receved Setember 29, Ths work was suorted n art by the Flemsh Insttute for the Promoton of Innovaton by Scence and Technology (IWT ), the Mnstry of the Flemsh Communty Scence, Innovaton and Meda Deartment (BIL99/61), the Federal Offce for Scentfc, Techncal and Cultural Affars (IAP Phase V Moble Multmeda) and the EC Socrates Student Exchange rogram. The authors are wth the Deartment of Electroncs and Informaton Processng (ETRO), Vrje Unverstet Brussel, Plenlaan 2, B-1050 Brussels, Belgum (corresondng author: hone: +32(2) ; fax: +32(2) ; e-mal: [email protected]). P. Schelkens has a ost-doctoral fellowsh wth the Fund for Scentfc Research Flanders (FWO), Egmontstraat 5, B-1000 Brussels, Belgum. that are extremely senstve to lossy comresson snce they are small or have oorly defned borders (e.g. some mcrocalcfcatons n mammograms, the trabecular attern of bone, the edge of a neumothorax, etc.), and are only dstngushable by subtle changes n the contrast [1]. On the other hand a concet lke rogressve data transmsson [2] and thus nherently suort for lossy codng s equally mortant. Ths methodology allows for examle to rortze low-resoluton versons of the requested mages and to rogressvely refne the resoluton of the vsualzed data by transferrng addtonal data. Ths scalablty mode s often referred to as resoluton scalablty. In a qualty scalablty scheme, the mages are decoded mmedately to the full resoluton but wth a reduced vsual qualty. Addtonally, by selectng regons that are relevant for the medcal dagnoss.e. the regons-of-nterest (ROIs) arts of the mage can be evaluated n a very early transmsson stage at full qualty. Meanwhle, the background nformaton wll be further refned. Moreover, t should be clear that we target otmal ratedstorton erformance over the comlete range of bt-rates that s requested by the alcaton. For examle, JPEG2000 [3] (based on the wavelet transform) clearly outerforms ts redecessor JPEG (based on the dscrete cosne transform) [4] at low bt-rates and has as mortant roerty ts lossy-tolossless codng functonalty; that s the caablty to start from lossy comresson at a very hgh comresson rato and to rogressvely refne the data by sendng detal nformaton, eventually u to the stage where a lossless decomresson s obtaned. Systems based on other technologes than the wavelet transform have been roosed, but they only artally facltate the requested set of functonaltes. Nonetheless, those technques erform excellent for the subclass of alcatons they are desgned for. Examles are contextbased redctve codng (CALIC) [5] for lossless mage comresson, and regon-based mage codng [6] for very low bt-rate codng. Although these coders are comettve n ther alcaton doman, they lack suort for the other functonaltes. Addtonally, the ncreasng use of three-dmensonal magng modaltes, lke Magnetc Resonance Imagng (MRI), Comutersed Tomograhy (CT), Ultrasound (US), Sngle Photon Emsson Comuted Tomograhy (SPECT) and Postron Emsson Tomograhy (PET) trggers the need for effcent technques to transort and store the related volumetrc data. In the classcal aroach, the mage volume

2 IEEE Transactons on Medcal Imagng 2 s consdered as a set of slces, whch are successvely comressed and stored or transmtted. Snce contemorary transmsson technques requre the use of concets lke rate scalablty, qualty- and resoluton scalablty, multlexng mechansms need to be ntroduced to select from each slce the correct layer(s) to suort the actually requred Qualty-of- Servce (QoS) level. However, a dsadvantage of the slce-byslce mechansm s that otental 3D correlatons are neglected. In the ast, volumetrc codng usng Dscrete Cosne Transform (DCT) based technques have been roosed (e.g. [7, 8]). However, these systems hardly meet the requrements mosed by the scalablty aradgm as reflected earler. Hence, we have evaluated and develoed other methods that meet the above-mentoned requrements. Tycal examles are octave zero-tree based wavelet codng [9-12] and layered zero wavelet codng [13]. Currently, these coder tyes delver the best erformance for lossy-to-lossless codng. In ths aer, we resent new aroaches for volumetrc wavelet codng. A frst new coder uses the cube-slttng (CS) algorthm [14, 15], whch s based on the quadtree codng algorthm resented n [16]. A second new coder called the 3D Quadtree Lmted (3D QT-L) coder combnes the basc rncles of quadtree codng [16, 17] and block-based codng of the sgnfcance mas [18]. Fnally, a thrd new coder [19] ntegrates both Cube-Slttng (CS) and Layered Zero Codng (LZC) rncles [13, 20, 21]. LZC s the core element of the JPEG2000 mage comresson standard (.e. EBCOT: Embedded Block Codng by Otmzed Truncaton) and was used n the IW44 algorthm of AT&T s DjVu document comresson system. The erformance of the roosed codng schemes wll be comared aganst an mlementaton of 3D Set Parttonng n Herarchcal Trees (SPIHT) [10-12], 3D SuBband-based Set Parttoned Embedded block codng (SB- SPECK) [22] JPEG2000 [3] and an orgnal 3D JPEG-alke codng aroach. The aer s structured as follows. Secton II revses several 2D wavelet-based embedded mage codng algorthms, ncludng two well-known reresentatves of the famly of nter-band coders, namely the Embedded Zerotree Wavelet codng (EZW) and SPIHT algorthms. It also focuses on two algorthms from the famly of ntra-band coders, namely the square arttonng (SQP) and the EBCOT coders, and descrbes the new QT-L coder. The roosed 3D DCT coder and the 3D wavelet-based codng algorthms are resented n Secton III. The lossless and lossy codng results obtaned for fve volumetrc data sets recorded wth dfferent magng modaltes are reorted n Secton IV. Fnally, Secton V summarzes the conclusons. II. 2D CODING TECHNIQUES A. EZW and SPIHT Codng A oular mage codng technque suortng rogressve data transmsson s EZW, ntroduced by Sharo n [23]. Ths codng scheme uses a smle, yet general model to characterze the nter-band deendences among wavelet coeffcents located n subbands wth smlar orentatons. The model s based on the zerotree hyothess, whch assumes that f a wavelet coeffcent w at a certan scale s nsgnfcant wth resect to a gven threshold T,.e. w < T, then all the coeffcents of the same orentaton n the same satal locaton at fner scales are also nsgnfcant wth resect to the threshold T. Ths hyothess s well confrmed for ecewse smooth mages [23], whch nclude medcal mages obtaned wth dfferent magng modaltes. EZW ales successve aroxmaton quantzaton (SAQ) to rovde a multrecson reresentaton of the wavelet coeffcents and to facltate the embedded codng. Wth SAQ, the sgnfcance of the wavelet coeffcents wth resect to a monotoncally decreasng seres of thresholds s recorded nto a set of bnary mas, called sgnfcance mas. It s roven n [23] that even for otmally chosen wavelet transform, quantzer and entroy coder, the cost of encodng the sgnfcance mas reresents an mortant orton of the total encodng cost. Hence, mrovng the codng of the sgnfcance mas can result n a sgnfcant over-all codng gan. The technque used n [23] to encode the sgnfcance mas s zerotree codng, whch allows for an effcent codng of nsgnfcant coeffcents across the scales. Wth ths technque, the cost of encodng the sgnfcance mas s reduced by groung the nsgnfcant coeffcents n trees growng exonentally across the scales, and by codng them wth zerotree symbols. A more comlex and effcent algorthm for codng of the sgnfcance mas s the Set Parttonng nto Herarchcal Trees (SPIHT) coder roosed by Sad and Pearlman n [24]. Ths algorthm uses the same underlyng model as EZW to characterze the nter-band deendences among the wavelet coeffcents. The basc rncles used by the SPIHT algorthm are artal orderng by magntude of the wavelet coeffcents (resultng from SAQ), set arttonng nto herarchcal trees (.e. at every aled threshold the trees are sorted, based on ther sgnfcance), and ordered bt-lane transmsson of the refnement bts (.e. the magntude of each sgnfcant coeffcent s rogressvely refned). The essental dfference of the SPIHT codng rocess wth resect to EZW s the way trees of coeffcents are arttoned and sorted [24]: durng the sortng ass SPIHT slts as ts name suggests the data nto herarchcal sets based on the sgnfcance of the arent node, the mmedate offsrng nodes, and the remanng nodes of the tree. B. Inter-band versus Intra-band Codng EZW, SPIHT and the algorthms fallng n the same category [25, 26] are zerotree-based coders, and ther effcency comes from exlotng nter-band deendences between the wavelet coeffcents. As already mentoned, ths s done by groung the nsgnfcant coeffcents n trees sannng exonentally across the scales, and by codng them wth zero symbols ( zerotree symbols n the termnology

3 IEEE Transactons on Medcal Imagng 3 of [23]). The man dsadvantage of ths aroach s the fact that the zero regons n the sgnfcance mas are aroxmated as a hghly constraned set of tree structured regons. As a consequence, certan zero regons that are not algned wth the tree structure, may be exensve to code, and some ortons of zero regons may not be ncluded n zerotrees at all [17]. In contrast to EZW, SPIHT and ther related algorthms [25, 26], the codng algorthms resented n the followng sectons, ncludng the standard JPEG2000-EBCOT codng algorthm [27], exlot only the ntra-band deendences. Bascally, these technques emloy ether a fxed-sze, block-based decomoston of the sgnfcance mas (e.g. the Lattce Parttonng algorthm [18] and the JPEG2000-EBCOT algorthm) ether a varable-sze block-based decomoston,.e. a quadtree decomoston (e.g. the SQP coder [16]), ether a combnaton of the two (e.g. the nested quadtree codng (NQS) [28] and the EBCOT coder [20]). These algorthms abandon the cross-subband tree-structure groung of the wavelet coeffcents of ts redecessors and ths revents them from makng use of the nter-band deendences. However, ntra-band redundances are exloted to a larger extent. The zero regons n the sgnfcance mas are reresented by a unon of ndeendent rectangular zero regons of fxed or varable szes. Ths smle model accounts for the fact that the sgnfcant coeffcents are clustered n certan areas n the wavelet doman, corresondng to the edges and textural regons n the mage. Therefore, by usng ths model, one can solate nterestng non-zero detals n the sgnfcance mas by mmedately elmnatng large nsgnfcant regons from further consderaton [16, 17]. The codng gan resultng from the use of ths model comensates (to say the least) for the losses ncurred by not exlctly exlotng the nter-band deendences. An argument s the theoretcal roof gven n [18], whch shows that the number of symbols needed to code the zero regons wth a fxed-sze block-based method s lower than the number of zerotree symbols, for block szes confned to some theoretcal bounds. And as a second argument, the excellent rate-dstorton erformance and the comettve lossless comresson results [16-18, 28] of the fxed/varable sze block-based coders obtaned n both 2D hotograhc and medcal mage codng ndcate that exlotng ntra-band redundances offers a better codng gan than exlotng nterband redundances. There s even one more ndcaton that ntra-band models should be favored over the nter-band models n wavelet mage codng, as exlaned n the followng. Defne X as a random varable reresentng the value of an arbtrary wavelet coeffcent, N X as a redefned neghborhood of X (excludng X ) and P X as the arent of X defned n the sense of Sharo [23]. Defne by I ( XY, ) the mutual nformaton [29] between two random varables X, Y, and by Ψ the summarzng functon gven by: 2, Ψ= X X N X The mutual nformaton s used n [30] as a mathematcal tool to quantfy the nter-band and ntra-band deendences between the wavelet coeffcents n emrcal data. The results reorted n [30] and extensve exermental results obtaned on a varety of mages wth several tyes of wavelet flters ncludng Daubeches wavelets, symmlets, coflets, and the borthogonal famly of wavelets reorted n [31], ndcate that the estmated mutual nformaton values satsfy: Iˆ( X; PX) < Iˆ( X; Ψ ) < Iˆ( X; Ψ; P X) Ths nequalty s constantly satsfed [30, 31] for dfferent non-arametrc densty estmators ncludng the log-scale hstogram method, the adatve arttonng method [32, 33], and the wavelet shrnkage method [34, 35]. Moreover, the Iˆ X; Ψ; P X s always sgnfcantly exerments show that ( ) larger than Iˆ ( X; X) P and slghtly larger than Iˆ ( X ; Ψ ) [30, 31]. Ths observaton ndcates that the ntra-band models cature most of the deendences between the wavelet coeffcents, and that only mnor gans can be obtaned wth comoste models that cature both tyes of deendences. It s mortant to note though that quantzaton further reduces the mutual nformaton, and ths mght exlan why n comarson wth the codng results obtaned wth the nterband models, the codng gans attaned wth the ntra-band models are not as large as ths nformaton-theoretc analyss mght ndcate. Nevertheless, trggered by the set of observatons mentoned above, we focused our attenton on several wavelet coders that exlot the ntra-band deendences between the wavelet coeffcents, and we roose three algorthms yeldng comettve results n the lossy and lossless comresson of 3D medcal data. C. Square Parttonng Codng SQuare Parttonng codng (SQP) [16], Nested Quadtree Slttng (NQS) [28], and Set Parttoned Embedded Block Codng (SPECK) [36] all researchers ndeendently develoed smlar embedded coders, whch are based on successve aroxmaton quantzaton make use of the quadtree codng of the sgnfcance mas and encode each btlane n the two classcal stages: a sgnfcance or domnant ass and a refnement ass. However, these coders do not emloy the same quadtree decomoston of the sgnfcance mas as the famly of zerotree-based coders does, as we wll see from the bref descrton of the SQP coder, gven n the followng. = 2 Denote by T the mum threshold used for successve aroxmaton quantzaton (SAQ) of the wavelet coeffcents, and by T the threshold aled by usng SAQ at a certan codng ste,0. Denote by Q ( kv, ) a quadrant wth to-left coordnates k = ( k 1, k 2 and of sze v = ( v 1, v 2 ), where v 1 and v 2 are the quadrant s wdth and heght resectvely. To smlfy the subsequent dscusson, we )

4 IEEE Transactons on Medcal Imagng 4 assume that the quadrant dmensons v and v are dentcal owers of 2,.e. v = v = for some J. The J corresondng quadrant delmtng bnary elements n the sgnfcance ma s denoted by Q ( kv, ). Fnally, we consder that the wavelet mage W s a matrx of W H elements, denoted by W = Q ( 0, V), wth 0 = (0, 0) and V = ( WH, ). The sgnfcance of a quadrant Q ( kv, ) for a gven threshold T s determned by the sgnfcance oerator σ : 1, f w ( l) Q ( k, v) : w ( l) 2 σ ( Q ( kv, )) = (1) 0, f w ( l) Q ( k, v) : w ( l) < 2 wth w ( l ) beng the wavelet coeffcent at oston l n the wavelet mage W. Note that for v = ( 1, 1), the sgnfcance oerator σ wll determne the sgnfcance of a sngle wavelet coeffcent w ( l ). 1 b Durng the frst arttonng ass S, the sgnfcance of the wavelet mage W s tested for ts hghest bt-lane (thus the aled SAQ threshold s T 2 x ma σ ( W ) = 1, the sgnfcance ma of the wavelet mage Q b Q b ( 0V, ) V ( k ) = 2,, 1 4, each havng half the sze, wth 2 ndcatng the orgn of each quadrant. The descendent sgnfcant quadrant(s) s (are) then further slced untl the sgnfcant leaf nodes (.e. xels) σ ( Q( l1, )) = σ ( w() l) = 1 are solated. Thus, the sgnfcance ass regsters the ostons l of all the leaf nodes newly dentfed as sgnfcant, usng a recursve tree structure of quadrants (or a quad-tree structure). Once the ostons and the sgns of the sgnfcant leaf nodes are encoded durng the sgnfcance ass, s set to x 1 R ma ). If s slt nto four quadrants 1 and the subsequent refnement ass s actvated for the sgnfcant leaf nodes. Next, the sgnfcance 1 ass S s restarted to udate the entre quadtree structure by dentfyng the new sgnfcant leaf nodes. Durng ths stage, only the sgnfcance of the revously nonsgnfcant nodes/quadrants,.e. those for whch 1 σ + ( Q ( kv, )) = 0, s encoded, whle the sgnfcant ones, 1.e. σ + ( Q ( kv, )) = 1, are gnored snce the decoder has already receved ths nformaton. Thus, the entre sgnfcance codng rocedure can actually be seen as a tree growng rocess. The descrbed rocedure s reeated, untl the comlete wavelet mage s encoded,.e. = 0, or untl the target btrate s met. Smle adatve arthmetc encodng has been used n the SQP coder to encode the sgnfcance, refnement and sgn nformaton, but the acheved gans are margnal n comarson wth the results obtaned wthout entroy codng k [16]. D. QT Lmted Ths secton resents a new quadtree-based codng algorthm, whch orgnates from the revously develoed SQP codng algorthm descrbed n [16]. Smlar to the SQP coder, the algorthm resented n ths secton bulds quadtrees corresondng to each sgnfcance ma : the arttonng rule [16] exlaned brefly n the revous secton s aled recurrently on quadrants selectng sets of bnary elements n the sgnfcance ma and, corresondngly, sets of coeffcents of the wavelet transform matrx. Encodng the sgnfcance mas (.e. the ostons of the sgnfcant coeffcents) s equvalent wth the encodng of the corresondng quadtrees. The frst man dfference wth resect to the SQP coder s that the arttonng rocess s lmted so that quadtrees are not bult u to the xel level. Once the area of the current node Q ( kv, ) n the quadtree s lower than a redefned mnmal b quadrant area, the slttng rocess s stoed and the entroy codng of the coeffcents wthn the quadrant Q kv, s ( ) actvated. Smlar to the SQP coder, deth-frst scannng s aled for scannng of the quadtrees corresondng to each sgnfcance ma. There s yet another asect that dfferentates ths algorthm and the SQP coder. Aart of the two codng stes,.e. the sgnfcance ass and the refnement ass that are used n SQP, a new codng stage called the non-sgnfcance ass s ntroduced. Bascally, durng the sgnfcance ass S corresondng to an arbtrary codng ste,0<, the coordnates of the coeffcents found as non-sgnfcant are aended nto a lst, called the lst of non-sgnfcant coeffcents (LNC). Durng the next codng stes k,0 k < the sgnfcance of the coeffcents recorded n the LNC s coded frst. Ths choce s motvated by the followng two facts: (1) the coeffcents recorded n the lst of non-sgnfcant coeffcents are located n the neghborhood of the coeffcents that have been already found as sgnfcant at the current or revous codng stes, and (2) there s a hgh robablty for these coeffcents to become sgnfcant at the next codng stes, due to the clusterng roerty the quadtree coders are based on [16]. After encodng the sgnfcance of the coeffcents n the LNC, the encoder erforms the next two codng asses that are smlar wth those of the SQP coder, namely the sgnfcance and the refnement asses. The detaled descrton of the codng algorthm ncludng the three man codng asses s gven n Fg. 1. In the fgure, the codng oeratons erformed at a codng stage corresondng to the current threshold ( ) T are llustrated; for = only the sgnfcance ass s erformed. The SGN and NSG are the acronyms for the sgnfcant and nonsgnfcant symbols resectvely, and Lmted _ Area ndcates the bound below whch no quadtree decomoston

5 IEEE Transactons on Medcal Imagng 5 Fg. 1. Pseudo-code descrbng the three codng asses erformed by the QT-L coder. s erformed. The thrd man dfference wth resect to the SQP coder s the more elaborated context-condtonng hase and context based entroy codng of the symbols generated n the three codng asses. Snce smle memoryless models are usually not effcent enough, context-based arthmetc encodng should be used to mrove the codng erformance. Ths technque exlots the deendences between the symbols to be encoded (the sgnal) and the neghborng symbols (the context). Context condtonng reduces the entroy and mroves the codng erformance [29]. Four dfferent sets of models S,1 4 are used to

6 IEEE Transactons on Medcal Imagng 6 encode the symbols generated by ths codng algorthm, and the encoder automatcally selects the arorate set at each codng stage, as shown n Fg. 1. These sets nclude: (1) the Quadrant_Sgnfcance set ( ) used to encode the sgnfcance of the nodes n the quadtrees, (2) the Pxel_Sgnfcance ( ) and Pxel_Sgn ( ) sets used to S 2 code the sgnfcance and the sgns resectvely of the coeffcents n the non-sgnfcance and sgnfcance asses, and (4) the Pxel_Refnement set ( ), used to entroy code the refnement nformaton generated n the refnement ass. The encoder assgns a number N of context models n (or states) C, S 0 n < N mode dfferent robablty model and thus the generated symbols are entroy coded wth an adatve arthmetc coder havng the arorate context model derved n the context-condtonng hase. Our algorthm assgns each coeffcent/quadrant to one of the several ossble contexts deendng on the values of the revously quantzed coeffcents. The basc dea for the context condtonng adoted n ths coder s to quantze nto a context number m (corresondng to a gven context model C m ), the number of sgnfcant neghborhood coeffcents for a gven coeffcent oston. The quantzaton erformed for the sets S,1 < 4 s descrbed by the followng S 1 ls S 4 S models exresson: S N sgn m = ( N m odels 1 ), 2 4, (2) Ntot where s the number of the neghborng coeffcents the total number of neghbors, and x s the nteger art of x. The total number of neghbors N n (2) s set to 8 n 2D tot S 3, for each set of models S,1 4. To each context state C n corresonds a N sgn declared as sgnfcant at the revous codng stes, N s codng. The same rncle s used to derve the contexts C 1,0 m < N m S1 models tot used to entroy code the sgnfcance of the quadrants, as one notces from the followng exresson: Q( kv, ) S m N 1 Nsgn = m 1 ( ) odels v 1 v, (3) 2 Q( k, v ) where N s the number of sgnfcant coeffcents found sgn at the revous codng stes n an arbtrary quadrant Q ( kv, ). Ths context-condtonng scheme s far smler than the context selecton system adoted by the EBCOT coder n [20] and detaled n the followng secton. We comared the codng results obtaned wth our coder mlementng the above entroy codng scheme wth the results obtaned by usng the context models and context assgnment scheme of the EBCOT coder. The gans rovded by the latter are margnal [31]. Moreover, gven ts codng erformance, ths entroy codng technque s clearly a better oton than the smle adatve fxed-context arthmetc encodng of the symbols generated n the sgnfcance ass and adatve zero-order model of the symbols generated n the refnement ass adoted n the orgnal verson of the SQP coder [16]. In the next sectons, ths codng algorthm s dentfed as the QuadTree-Lmted (QT-L) coder. E. Embedded Block Codng by Otmsed Truncaton (EBCOT) The quantzaton module and entroy coder adoted by the JPEG2000 standard [27], whch went to Internatonal Standard (IS) n December 2000, s based on the EBCOT coder ( Embedded Block Codng by Otmzed Truncaton ) as roosed n a recent aer by Davd Taubman [20]. However, ts hstory goes back n 1994 to a aer ublshed by Taubman and Zakhor concernng layered zero-codng for vdeo [21]. The EBCOT codng module conssts of two man unts. In a frst hase, Ter 1 (T1), the wavelet data s arttoned n searate, equally szed blocks, called the code-blocks B and each block s searately encoded, makng use of layered zerocodng rncles. Ths results n searate embedded btstreams for each code-block. Because multle codng asses are necessary to obtan a layered reresentaton of the codeblock data, each codng ass aled on code-block can be n assocated wth a rate contrbuton R. Addtonally, the dstorton ntroduced n the reconstructed mage for truncaton n ont n and code-block B, s denoted as D. After havng encoded all code-blocks, a ost-rocessng oeraton determnes where each code-block s embedded stream should be truncated n order to acheve a re-defned bt-rate, dstorton bound or vsual qualty level. Ths btstream reschedulng module s referred to as the Ter 2. It establshes a mult-layered reresentaton of the fnal btstream, nstallng an otmal erformance at several bt-rates, resolutons and/or vsual qualty levels. In the next aragrahs, we wll shortly dscuss the dfferent modules of the JPEG2000, EBCOT-based coder. 1) Ter 1 Codng Oeratons for Embedded Block Codng The T1 coder exsts out of a fractonal bt-lane encoder, whch encodes each bt-lane n three asses: the sgnfcance roagaton ass P, the magntude refnement ass P, 1 and the normalzaton ass P3. The three codng asses are ordered n such a way that most relevant data are encoded frst, consequently generatng otental truncaton onts n the bt-stream. The data of each code-block s scanned alyng a stre-wse scannng attern: the elements are read n grous of four vertcally algned elements. When a comlete stre s rocessed (4 lnes), the subsequent stre s rocessed. Addtonally, these codng asses call several codng oeratons (rmtves),.e. the zero codng (ZC), sgn codng (SC), magntude refnement (MR) and run-length codng B 2

7 IEEE Transactons on Medcal Imagng 7 (RLC) rmtves, whch wll not be dscussed here n detal and for whch we refer to [37]. These rmtves enable the selecton of sutable context models for the subsequent arthmetc codng and/or run-length codng stages. The chosen adatve arthmetc encoder s the MQ coder [38]. An element w ( k ) s encoded wth the sgnfcance roagaton ass P 1, f t has been revously classfed as non-sgnfcant ( σ ( w ( k )) = ), and f t has at least one sgnfcant element n ts referred neghborhood Θ,.e. σ + ( Θ ) = 1 k 1. The referred neghborhood Θ refers to the eght wavelet coeffcents surroundng the element w ( k ) beng coded. The magntude refnement ass (P ) encodes refnement nformaton for those elements that have been marked sgnfcant n revous bt-lanes, + 1 σ ( w ( k )) = 1. Fnally, the normalzaton ass (P 3 ) scans for the new sgnfcant elements wthout consderng the referred neghborhood Θ. Ths ass can be understood as a garbage collector, because t rocesses all these elements that have not been vsted yet by the sgnfcance roagaton and magntude refnement asses. For each bt-lane, the sgnfcance roagaton ass, the magntude refnement ass and the normalzaton ass are called, excet for the frst bt-lane where the frst two asses are dscarded. The latter s trval snce no sgnfcance nformaton has yet been encoded. The hlosohy of the codng ass orderng s to mrove frst the already dentfed sgnfcant areas n the wavelet mage by addng extra onts wth the ad of the sgnfcance roagaton ass, before ntroducng new solated structures (Fg. 2). Thereafter, revously detected sgnfcant coeffcents are refned n the magntude refnement ass. Only then, the rocess s actvated to look for new solated sgnfcant elements wth the normalzaton ass. In rncle, defnng more codng asses for each bt-lane, causes a hgher granularty of the embedded bt-stream, and a better aroxmaton of the 2 k k k otmal rate-dstorton curve. 2) Ter 2 Layer Formaton At the end of the Ter 1 codng ass a searate bt-stream has been generated for each code-block B, wthout utlzng nformaton from the other blocks. As already mentoned, these local embedded bt-streams have the desrable roerty that they can be truncated n several otental truncaton onts. The Ter 2 (T2) comonent of EBCOT otmzes n the truncaton rocess, and tres to reach the desred bt-rate whle mnmzng the ntroduced dstorton, utlzng Lagrangan rate allocaton (LRA) rncles. The followed rocedure s known as Post-Comresson Rate-Dstorton (PCRD) otmzaton [20] and the basc rncle s extensvely dscussed n Everett s aer [39]. Whle the PCRD otmzaton delvers a mzed suort for one qualty layer, successve alcaton of PCRD wll result n the suort for several qualty layers. Each qualty layer corresonds to the rates R, q [ 0, Q]. Furthermore, the code-block contrbutons to one qualty layer are dvded accordng to the resoluton level l they are contrbutng to. The algorthm also rovdes suort for multle comonents (e.g. color). The dfferent bt-stream chunks are groued n lc, searate ackets, each acket contrbutng to one qualty K q layer q, one resoluton level l and one mage comonent c. Ths tye of data organzaton s very ractcal: t allows easy reschedulng of the data. The end-user can secfy easly the scalablty set-u that s requred by the alcaton: for examle resoluton scalablty (llustrated n Fg. 3) or SNR scalablty. Each acket contans a header and a body. The header contans nformaton about code-blocks, whose comressed stream s ncluded n the body of the acket. The header descrbes whch code-blocks contrbute to the subband and qualty level covered by the acket. In addton, the mum bt-deth, the number of new codng asses (or truncatons onts), and the number of encoded bytes are transmtted for each code-block. q n Fg. 2. Illustraton of the actvty of the codng oeraton for the secfed bt-lane. Notce that the forward sgnfcance ass s actve n regons where revously sgnfcant xels were detected.

8 IEEE Transactons on Medcal Imagng 8 The code-block ncluson nformaton s encoded usng the tag-tree concet [20]. The basc dea s to buld a tree whose leaf nodes corresond to the code-blocks. The quanttes to be encoded and assocated wth every leaf node, descrbe the qualty layer n whch the code-block s frst ncluded. The tag tree s constructed from the leaves u to the root by groung leave nodes n blocks of 2x2 quanttes. The nformaton assocated wth every ntermedate node s the mnmal mutual quantty of all descendent nodes. The rocess s reeated untl the root node s reached. Fg. 3. Illustraton of a acket orderng suortng a resoluton rogressve lc, bt-stream ( K q ) A. 3D DCT Codng III. EXTENSION TO 3D CODING The frst coder ntroduced n the 3D test bed s a JPEGalke, 3D DCT-based coder. Ths coder was desgned n order to have a good reference for DCT-based systems. The 3D JPEG-based coder s comosed of a dscrete cosne transform, followed by a scalar quantzer and fnally a combnaton of run-length codng and adatve arthmetc encodng. The basc rncle s smle: the volume s dvded n cubes of 8x8x8 xels ( N = 8 ) and each cube s searately 3D DCTtransformed, smlar to a classcal JPEG-coder (see Fg. 4.a-b): DCT ( u ) = N 1 N 1 N 1 3 x1= 0x2= 0x3= 0 ( 2x + 1) π f ( x) C( u ) cos u 1 2N = (4) 1 u = 0 wth C( u ) = N 2 u > 0 N Thereafter, the DCT-coeffcents are quantzed usng a quantzaton matrx. In order to derve ths matrx, one has to consder two otons. One oton s to construct quantzaton tables that roduce an otmzed vsual qualty based on sycho-vsual exerments. It s worthwhle mentonng that JPEG uses such quantzaton tables, but ths aroach would requre elaborate exerments to come-u wth reasonable quantzaton tables for volumetrc data. The smlest soluton, adoted n ths work, s to create a unform quantzaton matrx as reorted n [14, 15, 40]. Ths oton s motvated by the fact that unform quantzaton s otmum or quasotmum for most of the dstrbutons [41]. Actually, the unform quantzer s otmum for Lalacan and exonental nut dstrbutons; otherwse the dfferences wth resect to an otmal quantzer are margnal [41]. A second ossblty nvolvng quantzers that are otmal n rate-dstorton sense s dscussed elsewhere [42]. The quantzed DCT-coeffcents are scanned usng a 3D sace-fllng curve,.e. a 3D nstantaton of the Morton-curve [43], to allow for the groung of zero-valued coeffcents and hence to mrove the erformance of the run-length codng (Fg. 4.c). Ths curve was oted for, due to ts smlcty comared to that of 3D zgzag curves [44]. The non-zero coeffcents are encoded usng the same classfcaton system as for JPEG. The coeffcent values are groued n 16 man magntude classes (ranges), whch are subsequently encoded wth an arthmetc encoder [45]. Fnally, the remanng bts to refne the coeffcents wthn one range are added wthout further entroy codng. The adoted entroy codng system shown n Fg. 5 s artally based on the JPEG archtecture [4], although the Huffman coder s relaced by an adatve arthmetc encoder [45]. Consequently, the large look-u tables mentoned n annex K of the standard [4] are suerfluous and moreover, adatve arthmetc encodng tends to have a hgher codng effcency. The DC coeffcents are encoded wth a redctve scheme: Fg. 4. In the 3D DCT coder the volume s decomosed n blocks of 8-by-8-by-8 xels (a-b), and each block s ndeendently transformed usng a 3D DCT (b). Thereafter for each DCT block the transformed coeffcents are scanned accordng to the Morton curve, quantzed and run-length/arthmetcally encoded.

9 IEEE Transactons on Medcal Imagng 9 Fg. 5. Run-length/arthmetc encodng system of the 3D DCT-based coder. aart from the frst DC coeffcent, the entroy codng system encodes the dfference between the current DC coeffcent and the revous one. For ths dfference, the range s determned and encoded wth an arthmetc encoder that has a DC model suortng 16 ranges. Smly transmttng the remanng bts of the coeffcent refnes the range secfcaton wthout any further entroy codng. The latter s ossble snce the robablty dstrbuton of all ossble values can be seen as unform, hence entroy codng wll not be able to further reduce the bt consumton. The AC coeffcents are encoded by secfyng frst the amount of zeros recedng the encoded symbol,.e. the run. The runs of zeros are encoded usng an arthmetc encoder wth a searate model. Runs of u to 15 zeros are suorted. Note that to ndcate the stuatons n whch 16 or more zeros recede a sgnfcant AC coeffcent, an extra symbol OVF (overflow) s used. After encodng ths symbol, the remanng zeros are mmedately encoded to avod confusng stuatons nvolvng a successon of several OVF encodngs. Fnally, the range of the encountered sgnfcant symbol s encoded, usng an arthmetc encoder wth a smlar (AC) model as n the case of the DC coeffcents, followed by the necessary refnement bts. B. The 3D Wavelet Transform Before descrbng n the followng sectons the roosed 3D wavelet-based technques, t s mortant to notce that these technques suort lossless codng, all the requred scalablty modes as well as ROI codng, and ths s a sgnfcant dfference wth resect to the 3D DCT-technque resented above, whch s not able to rovde these features. For all the 3D wavelet-based coders ncluded n ths study, a common wavelet transform module was desgned that suorts lossless nteger lftng flterng, as well as fnterecson floatng-ont flterng. A heterogeneous selecton of flter tyes and a dfferent amount of decomoston levels for each satal drecton (x-, y- or z-drecton) are suorted by ths module. Ths allows for adatng the sze of the wavelet yramd (Fg. 6.a) n each satal drecton n case the satal resoluton s lmted. For examle, fewer levels wll be requred along the slce axs f the amount of slces or the resoluton along the axs s lmted. The suorted lossless nteger lftng flters nclude the (S+P), (4,2), (5,3), and (9,7) nteger wavelet transforms. Ths selecton s based on recent ublcatons [9, 46], as well as nvestgatons erformed n the context of the JPEG2000 comresson standard. A tycal roblem encountered wth 3D lossless nteger wavelet transforms s the comlexty needed to make them untary, whch s not the case for floatng-ont transforms. Ths roerty s necessary n order to acheve a good lossy codng erformance. By calculatng the L norm of the low- and hgh-ass flters, the normalzaton factors can be determned. In 2D ths s not a roblem, snce the tycal scalng factors to obtan a untary transform are aroxmately owers of two [47]. However, n 3D the roblem os u agan, and t only dsaears f one takes care that the sum of all decomostons nfluencng each ndvdual wavelet coeffcent (.e. decomostons n both slce drectons and n the axal drecton) s an even number. Hence, some roosals have been formulated [10, 12] that make use of a wavelet acket transform [48] to acheve ths goal (Fg. 6.b), whle assumng that the -based normalzaton factors for the L 2 suorted kernels scale-u wth 2 for the low-ass and 1 2 for the hgh-ass kernels. In ractce ths seems to be an accetable aroxmaton. Nevertheless, n the resented study, whenever ossble, untary transforms wll be used (and t wll be exlctly mentoned f not). C. 3D Set Parttonng n Herarchcal Trees In the test set of wavelet coders, a 3D SPIHT encoder [11] was ncluded as a reference. An early verson of ths coder [12] has already roven to beat the erformance of a contextbased octave zero-tree coder [9]. The source code was made avalable by the authors so t could be equed wth the roosed wavelet transform front-end. The SPIHT mlementaton n ths study uses balanced 3D 2

10 IEEE Transactons on Medcal Imagng 10 satal orentaton trees. Therefore, the same number of recursve wavelet decomostons s requred for all satal orentatons. If ths s not resected, several tree nodes do not refer to or be lnked wth the same satal locaton, and consequently the deendences between dfferent tree-nodes are destroyed and hence the comresson erformance s reduced. Thus, a acket-based transform s not usable to obtan a untary transform wth ths embedded codng system. Therefore, the SPIHT coder was equed wth a non-untary transform. It s however worthwhle mentonng that solutons have been roosed utlzng unbalanced sato-temoral orentaton trees n the context of vdeo codng [49]. Fg. 6. (a) A two-level 3D wavelet transform usng a Mallat decomoston scheme. (b) A two-level wavelet acket transform that roduces a untary decomoston wth dyadc scalng factors. Each subband n the Mallat confguraton has been subject to an extra transform along the slce axs. For 3 3,1 3,2 examle, d 1 was decomosed nto d 1 and d 1. The examned 3D SPIHT algorthm [11] follows the same rocedure as ts 2D homologous algorthm, wth the exceton that the states of the tree nodes each embracng eght wavelet coeffcents are encoded wth a context-based arthmetc codng system durng the sgnfcance ass. The selected context models are based on the sgnfcance of the ndvdual node members, as well as on the state of ther descendents. Consequently, for each node coeffcent four state combnatons are ossble. In total 164 dfferent context models are used. D. Cube Slttng The cube-slttng technque s derved from the 2D square arttonng coder (SQP) roosed n secton II.C. In the context of volumetrc encodng, the SQP technque was extended to a thrd dmenson: from square slttng towards cube slttng. Cube-slttng s aled on the wavelet mage n order to solate smaller enttes,.e. sub-cubes, ossbly contanng sgnfcant wavelet coeffcents. Fg. 7 llustrates the cube-slttng rocess. Durng the frst sgnfcance ass S, the sgnfcance of the wavelet mage (volume) W s tested for ts hghest btlane wth the sgnfcance oerator σ. If σ ( W ) = 1, the wavelet mage W s slced n eght q q v sub-cubes (or octants) Q b k, 2, 1 q 8, wth toleft coordnates k = (k, k, k ) and of q q q q sze q q q 1 v2 v q v v =,, The descendent sgnfcant cube (or cubes) s (are) then further slced untl the sgnfcant wavelet coeffcents σ ( w ( k) )= 1 are solated. Thus, the sgnfcance ass S regsters sub-cubes and wavelet coeffcents, newly dentfed as sgnfcant, usng a recursve tree structure of octants (cfr. Fg. 7.a-c). The result s an octtree-structured descrton of the data sgnfcance aganst a gven threshold (Fg. 7.d). As mght be notced, equal mortance weghts are gven to all the branches. When a sgnfcant coeffcent s solated, also ts sgn for whch two code symbols are reserved s mmedately encoded. When the comlete bt-lane s encoded wth the sgnfcance ass S, s set to 1 and the 1 refnement ass R s ntated for ths bt-lane, refnng all coeffcents marked as sgnfcant n the octtree. Thereafter, the sgnfcance ass s restarted to udate the octtree by dentfyng the new sgnfcant wavelet coeffcents for the current bt-lane. Durng ths stage, only the revously non-sgnfcant nodes,.e. q 1 q σ + v Q, k = 0 j 2, 0 < j J are checked for sgnfcance, and the sgnfcant ones,.e. q + 1 q v σ Q, k = 2 j 1 are gnored snce the decoder already receved ths nformaton. The descrbed rocedure s reeated, untl the comlete wavelet mage W s encoded,.e. = 0 or untl the desred bt-rate s obtaned. To encode the generated symbols effcently, a contextbased arthmetc encoder was ntegrated. The context model s smle. For the sgnfcance ass four context models are dstngushed, namely one for the symbols generated at the ntermedate cube nodes, one for the xel nodes havng nonsgnfcant neghbors for the revous threshold, one for the xel nodes havng at least one sgnfcant neghbor for the revous threshold and fnally one for encodng the sgn of the solated sgnfcant xel nodes. Only two contexts are used

11 IEEE Transactons on Medcal Imagng 11 Fg. 7 When a sgnfcant wavelet coeffcent s encountered, the cube (a) s slced n eght sub-cubes (b), and further on (c) u to the xel level. The result s an octtree structure (d) (SGN = sgnfcant node; NS = non-sgnfcant node). In the next sgnfcance ass, the non-sgnfcant nodes that contan sgnfcant wavelet coeffcents are further refned. for the refnement ass: one for the xel nodes havng nonsgnfcant neghbors for the revous threshold, one for the xel nodes havng at least one sgnfcant neghbor for the revous threshold. Other 2D technques, lke NQS [28] and Subband Block (SB) SPECK [22], have been roosed that use smlar quadtree decomoston technques. These coders dvde the wavelet sace n blocks and actvate for each block searately a quadtree codng mechansm. In case of SB-SPECK, the block szes are also deendng on the subband szes, forcng each block to resde n one subband. Each block s searately encoded, and thereafter an EBCOT-alke reschedulng takes lace to restore the scalablty functonalty. SB-SPECK was also artally extended to 3D.e. 3D SB-SPECK codng [22] by equng the coder wth a 3D wavelet transform front-end. The transform s actvated on dscrete chunks of slces (GOFs: Grous of Frames), to mantan the accessblty of the data (tycal GOF szes are 8, 16 or 32 lanes). The oton s not mlemented n the coders we desgned. SB- SPECK does not use arthmetc encodng. However, the 3D SB-SPECK coder delvers comettve results, and we wll refer to t whenever ossble. E. 3D QT-L The QT-L coder roosed n Secton II.D has also been extended towards 3D codng. The octtrees corresondng to each bt-lane are constructed followng a smlar strategy as for the cube-slttng coder. However, the arttonng rocess s lmted n such a way that once the volume of a node 3 q v Vn =, 0 < j J becomes smaller than a redefned j = 1 2 threshold V, the slttng rocess s stoed, and the entroy th codng of the coeffcents wthn such a sgnfcant leaf node q q σ Q v, k = 2 j 1 s actvated. Smlar to the 2D verson, the octtrees are scanned usng deth-frst scannng. In addton, for any gven node, the eght descendant nodes are scanned usng a 3D nstantaton of the Morton-curve [43]. For each bt-lane, the codng rocess conssts of the nonsgnfcance, sgnfcance and refnement asses of Fg. 1 adated for 3D codng; also, for the hghest bt-lane, the codng rocess conssts of the sgnfcance ass only. The context-condtonng scheme and the context-based entroy codng are smlar wth ther 2D counterarts descrbed n secton II.D. Notce that the total number of neghbors n (2) s set to 26 n 3D codng. F. 3D CS-EBCOT B N tot The CS-EBCOT codng [19] combnes the rncles utlzed n the cube-slttng coder wth a 3D nstantaton of the EBCOT coder [20]. In the next aragrahs the nterfacng of the cube-slttng coder wth a verson of EBCOT adated to 3D s dscussed. To start wth, the wavelet coeffcents are arttoned EBCOT-wse n searate, equally szed cubes, called codeblocks. Tycally, the ntal sze of the code-blocks s 64x64x64 elements. Other szes (even dfferent ones for each dmenson) can be selected, deendng on the mage characterstcs and the alcaton requrements. The codng module CS-EBCOT agan conssts of two man unts, the Ter 1 and Ter 2 arts. The Ter 1 of the roosed 3D codng archtecture s a hybrd module combnng two codng technques: cubeslttng and fractonal bt-lane codng usng context-based arthmetc encodng. The Ter 2 art s dentcal to the one used n the 2D codng system.

12 IEEE Transactons on Medcal Imagng 12 1) Cube Slttng The cube-slttng ass S s derved from the cube-slttng technque resented n secton III.D. In the roosed codng system, the cube-slttng s aled on the ndvdual codeblocks n order to solate smaller enttes,.e. sub-cubes, ossbly contanng sgnfcant wavelet coeffcents. The smallest sub-cube sze that s suorted s 4x4x4. We wll refer to these smallest sub-cubes as the leaf nodes. B Durng the frst cube-slttng (CS) ass S, the sgnfcance of code-block s tested for ts hghest btlane wth the sgnfcance oerator σ. If σ ( B ) = 1, the code-block B s slced untl the sgnfcant leaf nodes v q q b, k Q 2 G are solated, where G secfes the mum amount of cube slttng levels. When all sgnfcant leaf nodes are solated, the fractonal bt-lane codng art s actvated for the current bt-lane and only for the sgnfcant leaf-nodes. When the comlete bt-lane s encoded utlzng the fractonal bt-lane codng, s set to 1 and the 1 subsequent CS ass, S s actvated. The descrbed rocedure s reeated, untl the comlete block s encoded,.e. = 0. Due to the lmted amount of code-symbols and ther dstrbuton, arthmetc codng s not aled. 2) Fractonal Bt-lane Codng The fractonal bt-lane coder encodes only those leaf nodes that have been dentfed as sgnfcant durng the Cube- Slttng ass. Three asses are defned er bt-lane lke n the 2D case: the sgnfcance roagaton ass magntude refnement ass P3 P2 P1, the, and the normalzaton ass (Fg. 8). Addtonally, these codng asses call several codng oeratons (rmtves),.e. the zero codng (ZC), sgn codng (SC), magntude refnement (MR) and run-length codng (RLC) rmtves. These rmtves enable the selecton of sutable 3D context models for the subsequent arthmetc codng or run-length codng stages. The chosen adatve arthmetc encoder s based on an mlementaton by Sad & Pearlman of the algorthm roosed by I.H. Wtten et al. [45], whch s dentcal to those utlzed n the revously mentoned encoders. The data resdng n each leaf-node s scanned alyng a slce-by-slce scannng attern. Wthn one slce the attern s dentcal to the JPEG2000 scannng: the voxels are read n grous of four vertcally algned voxels. When a comlete slce s stre-wse rocessed, the subsequent slce s rocessed (Fg. 9). The fractonal codng asses behave n an dentcal way as for the orgnal EBCOT mlementaton. However, the referred neghborhood refers now to the twenty-sx Θ k voxels around the voxel k beng coded (.e. the mmedate neghbors). For each bt-lane, successvely the sgnfcance roagaton ass, the magntude refnement ass, the cubeslttng ass and the normalzaton ass are called, excet for the frst bt-lane where the frst two asses are dscarded. 3) Codng Prmtves As for EBCOT, four codng rmtves are defned to suort the encodng rocess n the dfferent codng asses: the zero codng (ZC) rmtve, the run-length codng (RLC) Fg. 8. Reresentaton of one bt-lane. For each bt-lane, successvely the sgnfcance ass, the magntude refnement ass, the cube-slttng ass and the normalsaton ass are called, excet n the case of the frst bt-lane, where the frst two asses are dscarded.

13 IEEE Transactons on Medcal Imagng 13 rmtve, the sgn codng rmtve (SC) and the magntude refnement (MR) rmtve. For arthmetc encodng, the context-model selecton s based on the state of the neghborng voxels of the voxel k beng encoded,.e. the referred neghborhood Θ, and the subband tye n whch k the voxel k s located. The referred neghborhood s dvded n 7 orthogonal subsets accordng to ther oston to the voxel k [19, 42]. Each codng rmtve has got ts own look-u table to dentfy the robablty model that has to be utlzed by the arthmetc coder for a gven context stuaton [19, 42]. Addtonally, we have to remark that the comlexty of ths art of the codng engne ncreases heavly comared to the orgnal 2D mlementatons, due to the enlarged referred neghborhood (from 8 to 26 neghbors) and consequently the augmented ntrcacy of the look-u tables [19]. Fg. 9. The slce-by-slce scannng attern shown for an 8x8x8 leaf node. 4) Ter 2 Layer Formaton The followed rocedure,.e. Post-Comresson Rate- Dstorton (PCRD) otmzaton [20], s dentcal to the orgnal one. However, we have to menton one asect that s of key mortance. The PCRD routne allows comensatng for the fact that a non-untary transform has been used. By correctng the calculated dstortons for each ass n wth a scalng factor ς b Θ k, the codng system wll erform as f a untary transform was used (or aroxmated when usng nteger owers of 2 ). Hence, the dstorton wll be now descrbed by: D ˆn 2 n = ζ ( s ˆ [ k] s [ k ]), (5) B b k B where s [ k ] denotes the magntude of element k n ˆ n codeblock and s [ k ] gves the quantzed reresentaton of that element assocated wth truncaton ont n. Ths correcton enables suort for a untary transform wthout obstructng the ossblty of lossless codng, a roblem that does occur wth classcal zerotree-based coders. The orgnal 2D algorthms suort multle comonents (e.g. color), but ths feature s not retaned n the roosed 3D mlementaton. Hence, only gray-scale mages (volumes) are suorted. Nevertheless, the dfferent bt-stream chunks are now groued nto searate ackets, each acket contrbutng to one qualty layer and one resoluton level. The code-block ncluson nformaton s agan encoded usng the tag-tree concet. The only change that has been made was extendng the tag-tree concet to the thrd dmenson,.e. movng from a quadtree structure to an octtree structure. IV. EXPERIMENTAL ASSESSMENT A. Lossless Codng The lossy and lossless comresson erformances of the roosed coders were evaluated on a set of volumetrc data obtaned wth dfferent magng modaltes, ncludng: ostron-emsson tomograhy PET (128x128x39x15bts), comutersed tomograhy CT1 (512x512x100x12bts), CT2 (512x512x44x12bts), ultrasound US (256x256x256x8b), and MRI data MRI (256x256x200x12b). Lossless codng results are reorted for most of the technques dscussed u to now: CS, 3D QT-L, 3D SPIHT, 3D SB-SPECK (only for the CT2 and MRI data based on results reorted n [22]), CS-EBCOT and JPEG2000. Obvously, we dd not nclude the 3D DCT coder n the lossless comresson test due to the lossy character of ts DCT front-end. Addtonally, the codng results obtaned wth the JPEG2000 coder equed wth a 3D wavelet transform (JPEG2K-3D) are reorted. The latter s one of the functonaltes rovded by the latest Verfcaton Model software (from V7.0 on), whch was added to suort mult-sectral mage codng. For all the tests erformed n lossless codng (as well as for lossy codng later), tycally a 5-level wavelet transform (wth a lossless 5x3 lftng kernel) was aled n all satal dmensons, excet for the low-resoluton PET mage (4 levels). The same number of decomostons n all dmensons was used to allow a far comarson wth the 3D SPIHT algorthm. As mentoned earler, a non-balanced decomoston would lead to the destructon of the deendences wthn the satal orentaton trees of the 3D SPIHT coder, due to the balanced character of the latter. It s evdent that the other coders are not lmted by such drawbacks, but to ensure a far comarson, we aled the same restrcton for them too. Fg. 10 shows the ncrease n terms of ercentage of the btrate acheved n lossless comresson, wth the reference technque taken as the algorthm yeldng the best codng results for each test volume. We notce that for the US and PET volumes, the 3D QT-L coder delvers the best codng erformance, whle for the other three volumetrc data, the CS-EBCOT erforms better. If one refers to the average ncrease n ercentage takng the CS-EBCOT coder as the reference, then one can notce that the 3D QT-L yelds smlar erformance, snce the average dfference between the two s only 0.1%. The CS coder follows t, wth a dfference of 1.45%. The 3D SPIHT and the JPEG2K-3D coders rovde smlar results, wth an average

14 IEEE Transactons on Medcal Imagng 14 dfference of 3.56% and 3.65% resectvely. Fnally, the average dfference ncreases u to 7.07% and 12.78% for the 3D SB-SPECK and JPEG2K coders. One notces also from Fg. 10 that the relatve erformance of the several technques s heavly deendant on the data set nvolved. For examle, 3D SPIHT delvers excellent results for the US, CT1 and MRI sets, whle for the other ones the erformance s relatvely oor. JPEG2000 yelds the worst codng results for all, excet for the CT2 mage, whch has a low axal resoluton. One notces that actvatng the 3D wavelet transform faclty of JPEG2000 boosts the lossless codng erformance of the JPEG2000 coder (excet agan for CT2). The results of the 3D SB-SPECK have been reorted n [22] only for the CT2 and MRI data sets, and the results are stuated n between JPEG2K and JPEG2K-3D for the MRI volume. In summary, these results lead to the followng mortant observatons for lossless codng: CS-EBCOT and 3D QT-L delver the best lossless codng results on all mages; The 3D wavelet transform as such sgnfcantly boosts the codng erformance; As satal resoluton and consequently nter-slce deendency dmnshes, the beneft of usng a 3D decorrelatng transform and mlctly a 3D codng system decreases. B. Lossy codng Lossy codng exerments were carred out on the fve volumetrc data sets for the aforementoned coders, and n addton, the 3D DCT-based codng engne s ncluded. The eak sgnal-to-nose rato (PSNR) s measured at seven dfferent bt-rates: 2, 1, 0.5, 0.25, 0.125, and bts-er-xel (b). Due to sace lmtatons ths aer does not contan all the obtaned results, hence we refer to [50] for a comlete reort. Smlar to the lossless codng exerments, the erformance of the wavelet coders s evaluated usng a wavelet transform wth a lossless 5x3 flter kernel. To smulate a untary transform, the CS-EBCOT, JPEG2000, JPEG2K-3D and SB- SPECK ntrnscally comensate the non-untarness by alyng an addtonal scalng factor n the rate-dstorton otmzaton rocess see equaton (5). Unfortunately, ths aroach s not ossble for the 3D SPIHT and CS algorthms. Hence, the reorted results for these coders are sub-otmal, snce they are overemhaszng the hgh frequency comonents durng encodng. To overcome ths roblem, a sub-otmal soluton used to aroxmate (roughly) a untary transform was adoted for the 3D QT-L coder. Integer scalng factors of the form 2, Z have been used to k k + aroxmate the theoretcal scalng factors of the form m 2 ( 2 ), Z, 1, m { 1, 0, 1} that yeld a untary (but not nteger anymore) 3D wavelet transform. These nteger scalng factors have been used n the same manner as suggested by Sad and Pearlman n [47]. Fg. 11 llustrates the eak sgnal-to-nose ratos (PSNR s) n decbels calculated for the US, PET, CT1, CT2 and MRI data sets as a functon of bt-rate. Ths set of exerments delvers surrsng results. If we exclude the results for the CT2 and PET data set n a frst evaluaton, we observe that deste ts sub-otmal aroxmaton of untarness, the 3D QT-L coder outerforms all the other wavelet coders n the whole range of bt-rates. For examle, the 3D QT-L coder Fg. 10. The ncrease of the lossless bt-rate n terms of ercentage, wth the reference technque taken as the algorthm yeldng the best codng results for each test volume. All technques use a lossless 5x3-lftng kernel n the wavelet transform.

15 IEEE Transactons on Medcal Imagng 15 Fg. 11. US, PET, CT1, CT2 and MRI comressed wth JPEG2000, JPEG2K-3D, 3D DCT, CS-EBCOT, CS, 3D QT-L and 3D SPIHT. All wavelet-based methods use a 5-level transform (excet PET: 4-levels) wth a lossless 5x3 lftng kernel. Excet for CS and 3D SPIHT the transforms were untary, or aroxmately untary (3DQT-L). yelds on the US data set at 1.00 b a PSNR of 38.75dB, whch s 0.5 db better than the JPEG2K-3D, and wth 1.43 db better than CS-EBCOT. At hgher rates (2b) the dfferences between them ncreases u to 0.88 db and 1.45 db for the JPEG2K-3D and CS-EBCOT coders resectvely. The 3D QT-L and the JPEG2K-3D algorthms erform equally well at low rates (0.125b) on the US data set, and outerform the CS-EBCOT coder wth 0.37 db and 0.25 db resectvely. A smlar rankng accordng to ther erformance can be done by takng nto account the result obtaned on the MRI data set (Fg. 11). At 0.125b the results are n order db, db and db for the 3D QT-L, JPEG2K-3D, and CS- EBCOT resectvely. Note that at lower rates CS-EBCOT gves slghtly better results ( b db) than JPEG2K-3D ( b db) but stll less than those rovded by 3D QT-L wth db at the same rate. Smlarly, the 3D QT-L outerforms on the CT1 data set the next rated wavelet coder JPEG2K-3D, but the dfferences between them are smaller: from 0.27 db at 1b, to 0.57 db at b. The results obtaned on the PET volumetrc data ndcate that at rates below 0.25 b the 3D QT-L coder outerforms all the other coders; for examle at b, 3D QT-L yelds a PSNR of db n comarson to db and

16 IEEE Transactons on Medcal Imagng db rovded by the JPEG2K-3D and CS-EBCOT resectvely. However, at hgher rates the JPEG2K-3D outerforms the 3D QT-L coder on ths data, wth a dfference of 0.52 db and 0.33 db at 1b and 2 b resectvely. Note also from Fg. 11 the unexectedly good comresson results rovded by the 3D DCT coder. At hgh rates (e.g. 2b) the PSNR fgures rovded by ths coder are hgher than those obtaned wth the 3D QT-L coder on the US, CT1, and MRI data sets; however, ths stuaton changes on the PET data, for whch both JPEG2K-3D and 3D QT-L coders are better. Note also that at low bt-rates the erformance of 3D DCT tends to decrease fast, as t s also observed for ts 2D counterarts. If one refers to the standard JPEG2000, one notces that ths coder tycally delvers oor codng results on the PET, US volumes and MRI data; on CT1 t yelds good results at rates hgher than 0.25b, but the results are modest at lower rates. However for CT2, JPEG2000 s the best coder at hgh btrates, and s only beaten by JPEG2K-3D and SB-SPECK at low-bt-rates. At that moment JPEG2000 equals the erformance of CS-EBCOT. Once agan ths llustrates the effect of the nter-slce correlaton (and consequently resoluton). To further demonstrate ths, a small exerment was carred out. Four subsamled nstantatons (along the slce axs) of the MRI data set were created (decmaton factors 2, 4, 8 and 16). Thereafter, on each of these data sets the codng erformance of JPEG2000 (wth & wthout 3D wavelet transform) and CS-EBCOT was evaluated. Fg. 12 llustrates the results for the orgnal volume and the one subsamled wth a factor of 16 along the slce axs. As exected, the codng erformance of the two volumetrc codng engnes droed down sgnfcantly, whle the one of JPEG2000 was less affected. For hgh and ntermedate btrates, JPEG2000 even erforms better for the hghly subsamled MRI data set. To handle arorately a lmted axal resoluton, we can adat the wavelet kernels consstently as suggested ndrectly n [51]. In case of a reduced axal resoluton (corresondng to more sngulartes n the slce axs drecton), the suort sze of the wavelet functon must be reduced to avod large wavelet coeffcents. In the ooste case.e. an ncreased axal resoluton wavelet flters wth longer suort szes should be used n the slce axs drecton. If one refers to the other two algorthms, namely the CS and the 3D SPIHT, one notces that on the US, CT1, and MRI data sets they are constantly erformng worse than the 3D QT-L, JPEG2K-3D and CS-EBCOT algorthms at all rates. These oor results are caused by the fact that these coders were equed wth a non-untary wavelet transform. In order to assess the mortance of ths asect, we erformed a second exerment n whch a lossy 9x7 lftng-based wavelet transform ( L -normalzed) has been used for the CS, CS- 2 EBCOT, 3D SPIHT and JPEG2000. JPEG2000 wth a 3D transform was excluded from the test, snce VM8.0 was not devsed wth lossy 9x7 suort n the slce axs drecton. The PSNR versus bt-rate results obtaned for the CT1, CT2, and MRI data sets reorted n [50] show that for CT1 and MRI the 3D SPIHT mlementaton was sueror to the other technques, closely followed by CS (for CT1) and then by CS- EBCOT. Only at hgh bt-rates the 3D DCT can comete wth these technques. The erformance of JPEG2000 was oorer, esecally for MRI, havng a hgh nter-slce correlaton. For CT2, JPEG2000 delvered the best codng erformance, as t was the case for the lossless 5x3 lftng kernel. Remark however, that the technques evaluated wth the lossy 9x7 lftng-based wavelet transform do not beat JPEG2000 at low bt-rates. Fg. 12. MRI (set 1/1) and a subsamled (1/16) verson of MRI comressed wth JPEG2000, JPEG2K-3D and CS-EBCOT, usng a 5-level transform wth a lossless 5x3 lftng kernel.

17 IEEE Transactons on Medcal Imagng 17 Fg. 13. MRI comressed at (1) b and (2) b usng (a) JPEG2000, (b) JPEG2K-3D, (c) 3D DCT and (d) CS-EBCOT. All wavelet-based methods use a 5-level transform wth a lossless 5x3 lftng kernel. Slce 45 s dected. Fg. 13 llustrates the vsual erformance of JPEG2000, JPEG2K-3D, CS-EBCOT and 3D DCT for one slce taken from the MRI data set. At b the qualty of the mages comressed wth JPEG2000 clearly devates from the other technques, due to the blurrng. For the 3D DCT-based technque, blockng artfacts dstort the vsual qualty (although less dsturbng) and slght smoothng effects occur for JPEG2K-3D and CS-EBCOT. At very low bt-rates (e.g b), t s ractcally mossble to dstngush a vsual qualty dfference between CS-EBCOT and JPEG2K3D. Both methods have a sueror qualty comared to the other technques. Fg. 14 shows the vsual results of the encodng rocess for the CT2 data set. The CT2 mages llustrate that PSNR s not a suffcent crteron to evaluate mage qualty. At b JPEG2000 rovdes a hgh PSNR, whle the erceved mage qualty s much oorer than that of the other wavelet-based methods (relatvely large rngng artefacts). Addtonally, t can be observed that 3D SPIHT erforms worse than CS and CS-EBCOT. Wth CS-EBCOT the nter-ventrcular setum can stll be observed at b (Fg. 14.e), whle also the other anatomcal structures are well mantaned. V. DISCUSSION AND CONCLUSIONS Ths aer gves an overvew of several state-of-the-art 3D wavelet coders and rooses three new codng algorthms ncludng the CS coder, the 3D QT-L coder and the CSEBCOT coder. Ther codng erformance was comared wth several state-of-the-art codng technques, ncludng JPEG2000, JPEG2K-3D, 3D SPIHT and 3D SB-SPECK. Based on a test bench of 5 volumetrc data sets t was shown that CS-EBCOT and 3D QT-L delver n all cases the best lossless codng erformance. JPEG2000 tycally delvers the worst results of all. In lossy codng, 3D QT-L tends to delver the best overall lossy codng erformance for a lossless 5x3-lftng kernel. At low rates CS-EBCOT cometes wth JPEG2K-3D. Wth a lossy 9x7-lftng kernel, 3D SPIHT tycally yelds the best erformance followed closely by CS and CS-EBCOT. Vsual qualty assessments demonstrate that the 3D technques delver comarable vsual rates over the comlete bt-range, wth the remark that CS and CS-EBCOT reserve better low-frequency satal structures. Overall, one may conclude that the roosed coders demonstrate excellent lossless comresson erformance, whle n lossy comresson they rovded comettve lossy codng results when comared to the hybrd technques (3D SB-SPECK and JPEG2K-3D). Moreover, t becomes aarent that the 3D technques are senstve to a reduced satal resoluton along the slce axs. A reducton of ths resoluton works to the advantage of the classcal 2D technques (JPEG2000). Snce t falls out of the scoe of ths aer, we dd not evaluate the mlementaton comlexty and the comutatonal load of the 3D comresson schemes. However, we can observe that the comutatonal comlexty of the 3D technques s sgnfcantly hgher than that of ther 2D

18 IEEE Transactons on Medcal Imagng 18 counterarts when aled on the same data. Tycal bottlenecks are the requred memory bandwdths and memory szes. A major roblem s the concdent access of axally related data, causng huge jums n the memory and consequently cache falure, wth as a result long executon tmes. Fortunately, data transfer and storage otmsed versons of ther 2D counterarts have been roosed n the ast (e.g. [42, 52-55]) of whch the ssued rncles can be transferred to 3D. For examle, the realzaton a 3D nstantaton of the local wavelet transform [56] wll sgnfcantly mrove the memory access and consumton behavor of the resented algorthms. ACKNOWLEDGEMENT The authors would lke to thank Wllam A. Pearlman and Frederck W. Wheeler of Rensselaer Polytechnc Insttute for ther suort n relaton to the 3D SPIHT and 3D SB-SPECK coders. REFERENCES [1] T. Hamd, "DICOM requrements for JPEG2000," ISO/IEC JTC1/SC29/WG1, Reort N944, [2] K. H. Tzou, "Progressve mage transmsson: a revew and comarson of technques," Otcal Engneerng, vol. 26, , July [3] C. Chrstooulos, "JPEG2000 Verfcaton model 8.5," ISO/IEC JTC1/SC29/WG1, Reort N1878, Se [4] W. B. Pennebaker and J. L. Mtchell, JPEG stll mage data comresson standard. New York: Van Nostrand Renhold, [5] X. Wu, "Hgh-order context modelng and embedded condtonal entroy codng of wavelet coeffcents for mage comresson," n Proc. 31st Aslomar Conference on Sgnals, Systems & Comuters, 1997, vol. 2, [6] M. Kunt, A. Ikonomooulos, and M. Kocher, "Second-generaton mage codng technques," Proc. IEEE, vol. 73, , Ar [7] G. P. Abousleman, M. W. Marcelln, and B. R. Hunt, "Comresson of hyersectral magery usng the 3-D DCT and hybrd DPCM/DCT," IEEE Trans. Geosc. Remote Sensng, vol. 33, , Jan [8] A. Vlacu, S. Lungu, N. Crsan, and S. Persa, "New comresson technques for storage and transmsson of 2D and 3D medcal mages," n Proc. SPIE Advanced Image and Vdeo Communcatons and Storage Technologes, Feb. 1995, vol. 2451, [9] A. Blgn, G. Zweg, and M. W. Marceln, "Three-dmensonal comresson wth nteger wavelet transform," Aled Otcs, vol. 39, , Ar [10] Y. Km and W. A. Pearlman, "Lossless volumetrc medcal mage comresson," n Proc. SPIE Conference on Alcatons of Dgtal Fg. 14. CT2 comressed at b usng (a) 3D DCT, (b) JPEG2000, (c) CS, (d) 3D SPIHT, and (e) CS-EBCOT. All wavelet-based methods use a 5- level transform wth a lossy 9x7 lftng kernel suortng a untary transform. Slce 13 s dected.

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20 IEEE Transactons on Medcal Imagng 20 Peter Schelkens (M 99) was born n Wllebroek, Belgum n He receved hs Electronc Engneerng degree n VLSI-desgn from the Industrële Hogeschool Antweren-Mechelen (IHAM), Camus Mechelen n Thereafter, he obtaned the Electrcal Engneerng degree (M.Sc.) n aled hyscs n 1994, the Bomedcal Engneerng degree (medcal hyscs) n 1995, and the Ph.D. degree n Aled Scences n 2001 from the Vrje Unverstet Brussel (VUB). Snce October 1994, he s a member of the Deartment of Electroncs and Informaton Processng (ETRO) at VUB. Snce October 2002, he holds a ostdoctoral fellowsh wth the Fund for Scentfc Research Flanders (FWO), Belgum. Peter Schelkens s also afflated as vstng researcher to the DESICS deartment of the Interunversty Mcroelectroncs Insttute (IMEC) n Leuven, Belgum. Snce 2000, he s coordnatng a research team n the feld of mage and vdeo comresson, and related multmeda technologes. Ths team s artcatng to the ISO/IEC JTC1/SC29/WG1 (JPEG2000) and WG11 (MPEG-4) standardzaton actvtes. Peter Schelkens s the Belgan head of delegaton for the ISO/IEC JPEG standardzaton commttee, and coedtor of art 10 of JPEG2000: Extensons for Three-Dmensonal and Floatng Pont Data. Dr. Peter Schelkens s the author/co-author of more than 60 scentfc ublcatons and atent alcatons. Adran Munteanu was born n Constanta, Romana n He receved the M.Sc. degree n Electroncs and Telecommuncatons from the Poltehnca Unversty of Bucharest n 1994, and the M.Sc. degree n Bomedcal Engneerng from the Techncal Unversty of Patras, Greece, n Snce October 1996, he s a member of the Deartment of Electroncs and Informaton Processng, Vrje Unverstet Brussel, Belgum, where he s currently ursung the PhD degree. Hs research nterests nclude scalable stll mage and vdeo codng, multresoluton mage analyss, mage and vdeo transmsson over networks, vdeo segmentaton and ndexng, statstcal modelng of mages. He s the author and co-author of more than 60 scentfc ublcatons and atent alcatons, and has contrbuted to two books n hs areas of nterest. Xaver Gro-Neto (S'98) was born n Badalona, Catalona n He receved hs Telecommuncatons Engneerng degree n communcatons from the Unverstat Poltecnca de Catalunya (UPC) n 2000 after hs thess work at the Vrje Unverstet Brussels (VUB). After hs studes, he worked n the Sony Develoment Center Euroe n Brussels as a Software Engneer. Snce October 2001, he s a Ph.D. student at the Image Grou of the Sgnal Theory and Communcatons Deartment at UPC n Barcelona. Snce January 2002, he holds a re-doctoral fellowsh from the Catalan Government. Hs resent work ams to the automatc semantc ndexng of multmeda documents. He s currently nvolved n the IST Faethon roject suorted by the Euroean Unon. Jan P. H. Cornels (S 79, M 80) was born n Wlrjk, Belgum, n He obtaned hs M.Sc. degree n Electro-mechancs n 1973 and hs Ph.D. degree n Aled Scences n 1980 from the Vrje Unverstet Brussel (VUB). He s rofessor n electroncs, medcal magng and dgtal mage rocessng at the Vrje Unverstet Brussel VUB; currently also Vce-rector Research of the VUB. He s drector of the deartment of Electroncs and Informaton Processng ETRO, at the Faculty of Aled Scences, and coordnates the research grou on mage rocessng and machne vson - IRIS. Current research actvtes of the IRIS grou are n the areas of medcal magng, mage and vdeo codng, machne vson, remote sensng, mne- and mnefeld detecton, and the desgn of algorthms and comuter archtectures for mage rocessng. He s the author and co-author of more than 300 scentfc ublcatons and atent alcatons. Joer Barbaren was born n Leuven, Belgum n He receved the Electrcal Engneerng degree n nformatcs from the Vrje Unverstet Brussel (VUB) n Snce October 2000, he s a member of the Deartment of Electroncs and Informaton Processng, Vrje Unverstet Brussel, Belgum, where he s currently ursung a PhD degree. Hs research nterests nclude scalable vdeo codng, stll mage codng, vdeo segmentaton and vdeo ndexng. He s the author and co-author of several scentfc ublcatons. code otmzaton. Mhnea Galca was born n Bucharest, Romana n He receved the M.Sc. degree n Electrcal Engneerng and Comuter Scence from the "Poltehnca" Unversty of Bucharest n 2001, after hs thess work at the Vrje Unverstet Brussels (VUB). Hs man rofessonal actvtes and research nterests are n software engneerng, wavelet-based stll mage codng, statstcal modelng of mages, comuter generated magng, comler desgn and