Effectve wavelet-based compresson method wth adaptve quantzaton threshold and zerotree codng Artur Przelaskowsk, Maran Kazubek, Tomasz Jamrógewcz Insttute of Radoelectroncs, Warsaw Unversty of Technology, Nowowejska 15/19, 00-665 Warszawa, Poland ABSTRACT Effcent mage compresson technque especally for medcal applcatons s presented. Dyadc wavelet decomposton by use of Antonn and Vllasenor bank flters s followed by adaptve space-frequency quantzaton and zerotree-based entropy codng of wavelet coeffcents. Threshold selecton and unform quantzaton s made on a base of spatal varance estmate bult on the lowest frequency subband data set. Threshold value for each coeffcent s evaluated as lnear functon of 9-order bnary context. After quantzaton zerotree constructon, prunng and arthmetc codng s appled for effcent lossless data codng. Presented compresson method s less complex than the most effectve EZW-based technques but allows to acheve comparable compresson effcency. Specfcally our method has smlar to SPIHT effcency n MR mage compresson, slghtly better for CT mage and sgnfcantly better n US mage compresson. Thus the compresson effcency of presented method s compettve wth the best publshed algorthms n the lterature across dverse classes of medcal mages. Keywords: wavelet transform, mage compresson, medcal mage archvng, adaptve quantzaton 1. INTRODUCTION Lossy mage compresson technques allow sgnfcantly dmnsh the length of orgnal mage representaton at the cost of certan orgnal data changes. At range of lower bt rates these changes are mostly observed as dstorton but sometmes mproved mage qualty s vsble. Compresson of the concrete mage wth ts all mportant features preservng and the nose and all redundancy of orgnal representaton removng s do requred. The choce of proper compresson method depends on many factors, especally on statstcal mage characterstcs (global and local) and applcaton. Medcal applcatons seem to be challenged because of restrcted demands on mage qualty (n the meanng of dagnostc accuracy) preservng. Perfect reconstructon of very small structures whch are often very mportant for dagnoss even at low bt rates s possble by ncreasng adaptablty of the algorthm. Fttng data processng method to changeable data behavour wthn an mage and takng nto account a pror data knowledge allow to acheve suffcent compresson effcency. Recent achevements clearly show that nowadays wavelet-based technques can realse these deas n the best way. Wavelet transform features are useful for better representaton of the actual nonstatonary sgnals and allow to use a pror and a posteror data knowledge for dagnostcally mportant mage elements preservng. Wavelets are very effcent for mage compresson as entre transformaton bass functon set. Ths transformaton gves smlar level of data decorrelaton n comparson to very popular dscrete cosne transform and has addtonal very mportant features. It often provdes a more natural bass set than the snusods of the Fourer analyss, enables wden set of soluton to construct effectve adaptve scalar or vector quantzaton n tme-frequency doman and correlated entropy codng technques, does not create blockng artefacts and s well suted for hardware mplementaton. Wavelet-based compresson s naturally multresoluton and scalable n dfferent applcatons so that a sngle decomposton provdes reconstructon at a varety of szes and resolutons (lmted by compressed representaton) and progressve codng and transmsson n multuser envronments. Wavelet decomposton can be mplemented n terms of flters and realsed as subband codng approach. The fundamental ssue n constructon of effcent subband codng technques s to select, desgn or modfy the analyss and synthess flters. 1 Wavelets are good tool to create wde class of new flters whch occur very effectve n compresson schemes. The choce of sutable wavelet famly, wth such crtera as regularty, lnearty, symmetry, orthogonalty or mpulse and step response of correspondng flter bank, can sgnfcantly mprove compresson effcency. For compactly supported wavelets correspondng flter length s proportonal to the degree of smoothness and regularty of the wavelet. But
when the wavelets are orthogonal (the greatest data decorrelaton) they also have non-lnear phase n the assocated FIR flters. The symmetry, compact support and lnear phase of flters may be acheved by borthogonal wavelet bases applcaton. Then quadrature mrror and perfect reconstructon subband flters are used to compute the wavelet transform. Borthogonal wavelet-based flters occurred very effcent n compresson algorthms. A constructon of wavelet transformaton by fttng local defned bass transformaton functon (or fnte length flters) nto mage data characterstcs s possble but very dffcult. Because of nonstatonary of mage data, mscellaneous mage futures whch could be mportant for good reconstructon, sgnfcant varous mage qualty (sgnal to nose level, spatal resoluton etc.) from dfferent magng systems t s very dffcult to elaborate the constructon method of the optmal-for-compresson flters. Many ssues relatng to the choce of the most effcent flter bank for mage compresson reman stll unresolved. 2 The demands of preservng the dagnostc accuracy n reconstructed medcal mages are exactng. Important hgh frequency coeffcents whch appear at the place of small structure edges n CT and MR mages should be saved. Accurate global organ shapes reconstructon n US mages and strong nose reducton n MN mages s also requred. It s rather dffcult to magne that one flter bank can do t n the best way. Rather choosng the best wavelet famles for each modalty s expected. Our am s to ncrease the mage compresson effcency, especally for medcal applcatons, by applyng sutable wavelet transformaton, adaptve quantzaton scheme and correspondng processed decomposton tree entropy codng. We want to acheve hgher acceptable compresson ratos for medcal mages by better preservng the dagnostc accuracy of mages. Many bt allocaton technques appled n quantzaton scheme are based on data dstrbuton assumptons, quantser dstorton functon etc. All statstcal assumptons bult on global data characterstcs do not cover exactly local data behavour and mportant detal of orgnal mage, e.g., dfferent texture small area may be lost. Thus we decded to buld quantzaton scheme on the base of local data characterstcs such a drect data context n two dmensons mentoned earler. We do data varance estmaton on the base of real data set as spatal estmate for correspondng coeffcent postons n successve subbands. The detals of quantzaton process and correlated codng technque as a part of effectve smple wavelet-based compresson method whch allows to acheve hgh reconstructed mage qualty at low bt rates are presented. 2. THE COMPRESSION TECHNIQUE Scheme of our algorthm s very smple: dyadc, 3 levels decomposton of orgnal mage (256 256 mages were used) done by selected flters. For symmetrcal flters symmetry boundary extenson at the mage borders was used and for asymmetrcal flters - a perodc (or crcular) boundary extenson. LL top level mddle level bottom level Fgure 1. Dyadc wavelet mage decomposton scheme. - horzontal relatons, - parent - chldren relatons. LL - the lowest frequency subband.
Our approach to flters s utltaran one, makng use of the lterature to select the proper flters rather than to desgn them. We conducted an experment usng dfferent knds of wavelet transformaton n presented algorthm. Long lst of wavelet famles and correspondng flters were tested: Daubeches, Adelson, Brslawn, Odegard, Vllasenor, Splne, Antonn, Coflet, Symmlet, Beylkn, Vad etc. 3 Generally Antonn 4 flters occurred to be the most effcent. Vllasenor, Odegard and Brslawn flters allow to acheve smlar compresson effcency. Fnally: Antonn 7/9 tap flters are used for MR and US mage compresson and Vllasenor 18/10 tap flters for CT mage compresson. 2.1 Adaptve space-frequency quantzaton Presented space-frequency quantzaton technque s realsed as entre data pre-selecton, threshold selecton and scalar unform quantzaton wth step sze condtoned by chosen compresson rato. For adaptve estmaton of threshold and quantzaton step values two extra data structure are buld. Entre data pre-selecton allows to evaluate zero-quantzed data set and predct the spatal context of each coeffcent. Next smple quantzaton of the lowest frequency subband (LL) allows to estmate quantzed coeffcent varance predcton as a space functon across sequental subbands. Next the value of quantzaton step s slghtly modfed by a model buld on varance estmate. Addtonally, a set of coeffcents s reduced by threshold selecton. The threshold value s ncreased n the areas wth the domnant zero-valued coeffcents and the level of growth depends on coeffcent spatal poston accordng varance estmaton functon. Frstly zero-quantzed data predcton s performed. The step sze w s assumed to be constant for all coeffcents at each decomposton level. For such quantzaton model the threshold value s equal to w/2. Each coeffcent whose value s less than threshold s predcted to be zero-valued after quantzaton (nsgnfcant). In opposte case coeffcent s predcted to be not equal to zero (sgnfcant). It allows to create predctve zero-quantzed coeffcents P map for threshold evaluaton n the next step. The process of P map creaton s as follows: f c < w / 2 then p = 0 else p = 1, (1) where = 1, 2,..., m n; m, n horzontal and vertcal mage sze, c - wavelet coeffcent value. The coeffcent varance estmaton s made on the base of LL data for coeffcents from next subbands n correspondng spatal postons. The quantzaton wth mentoned step sze w s performed n LL and the most often occurrng coeffcent value s estmated. Ths value s named MHC (mode of hstogram coeffcent). The areas of MHC appearance are strongly correlated wth zero-valued data areas n the successve subbands. The absolute dfference of the LL quantzed data and MHC s used as varance estmate for next subband coeffcents n correspondng spatal postons. We tested many dfferent schemes but ths model allows to acheve the best results n the fnal meanng of compresson effcency. The varance estmaton s rather coarse but ths smple adaptve model bult on real data does not need addtonal nformaton for reconstructon process and ncreases the compresson effcency. Let lc, =1,2,...,lm, be a set of LL quantzed coeffcent values, lm - sze of ths set. Furthermore let mode of hstogram coeffcent MHC value be estmated as follows: f ( MHC) = max f ( lc ) and MHC Al, (2) nlc where Al - alphabet of data source whch descrbes the values of the coeffcent set and f ( lc ) =, n lc - number of lm lc -valued coeffcents. The normalsed values of varance estmate ve s for next subband coeffcents n correspondng to spatal postons (parent - chldren relatons from the top to the bottom of zerotree - see fg. 1) are smply expressed by the followng equaton: lc Al ve s = lc MHC ve max. (3) These set of ve s data s treated as top parent estmaton and s appled to all correspondng chld nodes n wavelet herarchcal decomposton tree.
= 9-th order context model s appled for coarser data reducton n unmportant' areas (usually wth low dagnostc mportance). The unmportance means that n these areas the majorty of the data are equal to zero and sgnfcant values are separated. If sngle sgnfcant values appear n these areas t most often suggests that these hgh frequency coeffcents are caused by nose. Thus the coarser data reducton by hgher threshold allows to ncrease sgnal to nose rato by removng the nose. At the edges of dagnostcally mportant structures sgnfcant values are grouped together and the threshold value s lower at ths felds. P map s used for each coeffcent context estmaton. Noncausal predcton of the coeffcent mportance s made as lnear functon of the bnary surroundng data excludng consdered coeffcent sgnfcance. The other polynomal, exponental or hyperbolc functon were tested but lnear functon occurred the most effcent. The data context shown on fg. 2 s formed for each coeffcent. Ths context s modfed n the prevous data ponts of processng stream by the results of the selecton wth the actual threshold values at these ponts nstead of w/2 (causal modfcaton). Values of the coeffcent mportance - cm are evaluated for each c coeffcent from the followng equaton: 9 cm = coeff 1 ( 9 p j ), where, = 1, 2,..., m n. (4) j 1 Next the threshold value s evaluated for each c coeffcent: th = w / 2 ( 1 + cm w ( 1 ve )), (5) s where = 1, 2,..., m n, s - correspondng to LL parent spatal locaton n lower decomposton levels. The modfed quantzaton step model uses the LL-based varance estmate to slghtly ncrease the step sze for less varance coeffcents. Threshold data selecton and unform quantzaton s made as follows: each coeffcent value s frstly compared to ts threshold value and then quantzed usng w step for LL and modfed step value mw s for next subbands. Threshold selecton and quantzaton for each c coeffcent can be clearly descrbed by the followng equatons: where f c LL then c = c / w else f c < th then c = 0 else c = c / mw s, (6) mws = w ( 1+ coeff 2 ( 1 ves )). (7) The coeff 1 and coeff 2 values are ftted to actual data characterstc by usng a pror mage knowledge and performng entre tests on groups of smlar characterstc mages. a) b) 0 0 0 0 1 1 1 1 1 Fgure 2. a) 9-order coeffcent context for evaluatng the coeffcent mportance value n procedure of adaptve threshold value estmaton; - ponts of coeffcent context causal modfcaton, b) example of bnary P map context of sngle edge coeffcent.
2.2 Zerotrees constructon and codng Sophstcated entropy codng methods whch can sgnfcantly mprove compresson effcency should retan progressve way of data reconstructon. Progressve reconstructon s smple and natural after wavelet-based decomposton. Thus the wavelet coeffcent values are coded subband-sequentally and spectral selecton s made typcally for wavelet methods. The same scale subbands are coded as follows: frstly the lowest frequency subband, then rght sde coeffcent block, down-left and down-rght block at the end. After that next larger scale data blocks are coded n the same order. To reduce a redundancy of such data representaton zerotree structure s bult. Zerotree descrbes well the correlaton between data values n horzontal and vertcal drectons, especally between large areas wth zero-valued data. These correlated fragments of zerotree are removed and fnal data streams for entropy codng are sgnfcantly dmnsh. Also zerotree structure allows to create dfferent characterstcs data streams to ncrease the codng effcency. We used smple arthmetc coders for these data streams codng nstead of appled n many technques bt map (from MSB to LSB) codng wth necessty of applyng the effcent context model constructon. Because of refusng the successve approxmaton we lost full progresson. But the smplcty of the algorthm and sometmes even hgher codng effcency was acheved. Two slghtly dfferent arthmetc coders for producng endng data stream were used. 2.2.1 Constructon and prunng of zerotree The dyadc herarchcal mage data decomposton s presented on fg. 1. Decomposton tree structure reflects ths herarchcal data processng and strctly corresponds to created n transformaton process data streams. The four lowest frequency subbands whch belong to the coarsest scale level are located at the top of the tree. These data have not got parent values but they are the parents for the coeffcents n lower tree level of greater scale n correspondng spatal postons. These correspondence s shown on the fg. 1 as parent-chldren relatons. Each parent coeffcent has got four drect chldren and each chld s under one drect parent. Addtonally, horzontal relatons at top tree level are ntroduced to descrbe the data correlaton n better way. The decomposton tree becomes zerotree when node values of quantzed coeffcents are sgned by symbols of bnary alphabet. Each tree node s checked to be sgnfcant (not equal to zero) or nsgnfcant (equal to zero) - bnary tree s bult. For LL nodes way of sgnfcance estmaton s slghtly dfferent. The MHC value s used agan because of the LL areas of MHC appearance strong correlaton wth zero-valued data areas n the next subbands. Node s sgned to be sgnfcant f ts value s not equal to MHC value or nsgnfcant f ts value s equal to MHC. The value of MHC must be sent to a decoder for correct tree reconstructon. Next step of algorthm s a prunng of ths tree. Only the branches to nsgnfcant nodes can be pruned and the procedure s slghtly other at dfferent levels of the zerotree. Procedure of zerotree prunng starts at the bottom of wavelet zerotree. Sequental values of four chldren data and ther parent from hgher level are tested. If the parent and the chldren are nsgnfcant - the tree branch wth chld nodes s removed and the parent s sgned as pruned branch node (PBN). Because of ths the tree alphabet s wdened to three symbols. At the mddle levels the prunng of the tree s performed f the parent value s nsgnfcant and all chldren are recognsed as PBN. From conducted research we found out that addng extra symbols to the tree alphabet s not effcent for decreasng the code bt rate. The zerotree prunng at top level s dfferent. The checkng node values s made n horzontal tree drectons by explotng the spatal correlaton of the quantzed coeffcents n the subbands of the coarsest scale - see fg. 1. Sequentally the four coeffcents from the same spatal postons and dfferent subbands are compared wth one another. The tree s pruned f the LL node s nsgnfcant and three correspondng coeffcents are PBN. Thus three branches wth nodes are removed and LL node s sgned as PBN. It means that all ts chldren across zerotree are nsgnfcant. The spatal horzontal correlaton between the data at other tree levels s not strong enough to ncrease the codng effcency by ts utlsaton. 2.2.2 Makng three data streams and codng Pruned zerotree structure s handy to create data streams for endng effcent entropy codng. Instead of PBN zero or MHC values (nodes of LL) addtonal code value s nserted nto data set of coded values. Also bt maps of PBN spatal dstrbuton at dfferent tree levels can be appled. We used optonally only PBN bt map of LL data to slghtly ncrease the codng effcency. The zerotree codng s performed sequentally from the top to the bottom to support progressve reconstructon. Because of varous quantzed data characterstcs and wder alphabet of data source model after zerotree prunng three separated dfferent data streams and optonally fourth bt map stream are produced for effcent data codng. It s well known from nformaton theory that f we deal wth a data set wth sgnfcant varablty of data statstcs and
dfferent statstcs (alphabet and estmate of condtonal probabltes) data may be grouped together t s better to separate these data and encode each group ndependently to ncrease the codng effcency. Especally s true when context-based arthmetc coder s used. The data separaton s made on the base of zerotree and than the followng data are coded ndependently: - the LL data set whch has usually smaller number of nsgnfcant (MHC-valued) coeffcents, less PBN and less spatal data correlaton than next subband data (word- or charwse arthmetc coder s less effcent then btwse coder); optonally ths data stream s dvded on PBN dstrbuton bt map and word or char data set wthout PBNs, - the rest of top level (three next subbands) and mddle level subband data set wth a consderable number of zero-valued (nsgnfcant) coeffcents and PBN code values; level of data correlaton s greater, thus word- or charwse arthmetc coder s effcent enough, - the lowest level data set wth usually great number of nsgnfcant coeffcents and wthout PBN code value; data correlaton s very hgh. Urban Kostnen arthmetc coder (DDJ Compresson Contest publc doman code accessble by nternet) wth smple btwse algorthm s used for frst data stream codng. For the second and thrd data stream codng 1-st order arthmetc coder bult on the base of code presented n Nelson book 5 s appled. Urban coder occurred up to 10% more effcent than Nelson coder for frst data stream codng. Combnng a rest of top level data and the smlar statstcs mddle level data allows to ncrease the codng effcency approxmately up to 3%. The procedure of the zerotree constructon, prunng and codng s presented on fg. 3. Zerotree prunng: Constructon of - remove the branches Data selecton based on bnary zerotree wth nsgnfcant nodes statstcal dfferences - sgn a parent node as PBN LL data + bt map of PBN dstrbuton (optonally) rest of top and mddle level data bottom level data Btwse arthmetc codng Word- or charwse arthmetc codng Fnal compressed data representaton Fgure 3. Quantzed wavelet coeffcents codng scheme wth usng zerotree structure. PBN - pruned branch node. 3. TESTS, RESULTS AND DISCUSSION In our tests many dfferent medcal modalty mages were used. For chosen results presentaton we appled three 256 256 8-bt mages from varous medcal magng systems: CT (computed tomography), MR (magnetc resonance) and US(ultrasound) mages. These mages are shown on fg. 4. Mean square error - MSE and peak sgnal to nose rato - PSNR were assumed to be reconstructed mage qualty evaluaton crtera. Subjectve qualty apprecaton was conducted n very smple way - only by psychovsual mpresson of the non-professonal observer.
Applcaton of adaptve quantzaton scheme based on modfed threshold value and quantzaton step sze s more effcent than smple unform scalar quantzaton up to 10% n a sense of better compresson of all algorthm. Generally applyng zerotree structure and ts processng mproved codng effcency up to 10% n comparson to drect arthmetc codng of quantzed data set. The comparson of the compresson effcency of three methods: DCT-based algorthm, 6,7 SPIHT 8 and presented compresson technque, called MBWT (modfed basc wavelet-based technque) were performed for effcency evaluaton of MBWT. The results of MSE and PSNR-based evaluaton are presented n table 1. Two wavelet-based compresson technques are clearly more effcent than DCT-based compresson n terms of MSE/PSNR and also n our subjectve evaluaton for all cases. MBWT overcomes SPIHT method for US mages and slghtly for CT test mage at lower bt rate range. The concept of adaptve threshold and modfed quantzaton step sze s effectve for strong reducton of nose but t occurs sometmes too coarse at lower bt rate range and very small detals of the mage structures are put out of shape. US mages contan sgnfcant nose level and dagnostcally mportant small structures do not appear (mage resoluton s poor). Thus these mages can be effcently compressed by MBWT wth mage qualty preserved. It s clearly shown on fg. 5. An mprovement of compresson effcency n relato to SPIHT s almost constant at wde range of bt rates (0.3-0.6 db of PSNR). a) b) c) Fgure 4. Examples of mages used n the tests of compresson effcency evaluaton. The results presented n table 1 and on fg. 5 were acheved for those mages. The mages are as follows: a ) echocardography mage, b) CT head mage, c) MR head mage.
Table 1. Comparson of the three technques compresson effcency: DCT-based, SPIHT and MBWT. The bt rates are chosen n dagnostcally nterestng range (near the borders of acceptance). Modalty - bt rate DCT-based SPIHT MBWT MSE PSNR[dB] MSE PSNR[dB] MSE PSNR[db] MRI - 0.70 bpp 8.93 38.62 4.65 41.45 4.75 41.36 MRI - 0.50 bpp 13.8 36.72 8.00 39.10 7.96 39.12 CT - 0.50 bpp 6.41 40.06 3.17 43.12 3.18 43.11 CT - 0.30 bpp 18.5 35.46 8.30 38.94 8.06 39.07 US - 0.40 bpp 54.5 30.08 31.3 33.18 28.3 33.61 US - 0.25 bpp 91.5 28.61 51.5 31.01 46.8 31.43 The level of nose n CT and MR mages s lower and small structures are often mportant n mage analyss. That s the reason why the benefts of MBWT n ths case are smaller. Generally compresson effcency of MBWT s comparable to SPIHT for these mages. Presented method lost ts effectveness for hgher bt rates (see PSNR of 0.7 bpp MR representaton) but for lower bt rates both MR and CT mages are compressed sgnfcantly better. Maybe the reason s that the coeffcents are reduced relatvely stronger because of ts mportance reducton n MBWT threshold selecton at lower bts rate range. SPIHT MBWT 37,5 36 PSNR n db 34,5 33 31,5 30 0,2 0,3 0,4 0,5 0,6 0,7 0,8 Rate n bts/pxel Fgure 5. Comparson of SPIHT and presented n ths paper technque (MBWT) compresson effcency at range of low bt rates. US test mage was compressed. 4. CONCLUSIONS Adaptve space-frequency quantzaton scheme and zerotree-based entropy codng are not tme-consumng and allow to acheve sgnfcant compresson effcency. Generally our algorthm s smpler than EZW-based algorthms 9 and other algorthms wth extended subband classfcaton or space -frequency quantzaton models 10 but compresson effcency of presented method s compettve wth the best publshed algorthms n the lterature across dverse classes of medcal mages. The MBWT-based compresson gves slghtly better results than SPIHT for hgh qualty mages: CT and MR and sgnfcantly better effcency for US mages. Presented compresson technque occurred very useful and promsng for medcal applcatons. Approprate reconstructed mage qualty evaluaton s desrable to delmt the acceptable lossy compresson ratos for each medcal modalty. We ntend to mprove the effcency of ths method by: the desgn a constructon method of adaptve flter banks and correlated more suffcent quantzaton scheme. It seems to be possble by
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