Prediction-based Incremental Refinement For Binomially-factorized Discrete Wavelet Transforms

Transcription

1 IEEE Trasactios o Sigal Processig - T-SP , to appear. Predictio-based Icremetal Refiemet For Biomially-factorized Discrete Wavelet Trasforms Yiais Adreopoulos, Dai Jiag ad Adreas Demostheous ABSTRACT It was proposed recetly that quatized represetatios of the iput source (e.g. images, video) ca be used for the computatio of the two-dimesioal discrete wavelet trasform (2D DWT) icremetally. The coarsely-quatized iput source is used for the iitial computatio of the forward or iverse DWT ad the result is successively refied with each ew refiemet of the source descriptio via a embedded quatizer. This computatio is based o the direct two-dimesioal factorizatio of the DWT usig the geeralized spatial combiative liftig algorithm. I this correspodece we ivestigate the use of predictio for the computatio of the results, i.e. exploitig the correlatio of eighborig iput samples (or trasform coefficiets) i order to reduce the dyamic rage of the required computatios, ad thereby reduce the circuit activity required for the arithmetic operatios of the forward or iverse trasform. We focus o biomial factorizatios of DWTs that iclude (amogst others) the popular 9/7 filter-pair. Based o a FPGA arithmetic co-processor testbed, we preset eergy-cosumptio results for the arithmetic operatios of icremetal refiemet ad predictio-based icremetal refiemet i compariso to the covetioal (o-refiable) computatio. Our tests with combiatios of itra ad error frames of video sequeces show that the former ca be 70% more eergy efficiet tha the latter for computig to half precisio ad remais 5% more efficiet for full-precisio computatio. Idex Terms Approximate Sigal Processig, Discrete Wavelet Trasform, Eergy Cosumptio, Icremetal Refiemet of Computatio, Liftig Scheme EDICS:DSP-WAVL, MDS-ALGO I. INTRODUCTION The two-dimesioal discrete wavelet trasform (2D DWT) has bee established as oe of the mai tools for image compressio [], image deoisig ad other popular image processig operatios [2]. I the vast majority of applicatios, the trasform coefficiets are produced to the maximum degree of precisio ad the they are quatized ad processed as appropriate []. However, it has bee recogized that this wastes system resources for the cases where severe quatizatio would reder the majority of the coefficiets ot beig used at all, or used at Copyright (c) 200 IEEE. Persoal use of this material is permitted. However, permissio to use this material for ay other purposes must be obtaied from the IEEE by sedig a request to pubs-permissios@ieee.org. The authors are with the Uiversity College Lodo, Dept. of Electroic ad Electrical Egieerig, Torrigto Place, WCE 7JE, Lodo, U; Tel: ; fax: ; iadreop@ee.ucl.ac.uk (Y. Adreopoulos), djiag@ee.ucl.ac.uk (Dai Jiag), a.demostheous@ee.ucl.ac.uk (A. Demostheous). This work was supported by the EPSRC, grat EP/F02005/.

2 IEEE Trasactios o Sigal Processig - T-SP , to appear. 2 very low precisio [3]. For example, this is commoly the case for low-bitrate image ad video codig applicatios [3] ad resource-costraied image ad video processig operatios [4]. For this reaso, previous work proposed schemes for approximate computatio of trasforms ad sigal processig operatios [3]. A property that has bee recogized to be of great importace is icremetal refiemet of computatio [4]-[6], where the trasform represetatio of a sigal (image, video) is produced icremetally with the use of embedded (bitplae-based) quatizatio. I our recet work [4], this desig has bee theoretically aalyzed both for the forward ad the iverse two-dimesioal multilevel DWT usig the geeralizatio of the spatial combiative liftig algorithm (SCLA) of Meg ad Wag [7]. The overall framework is depicted i Figure. There, the multilevel DWT decompositio of the iput source (video frame) occurs idepedetly for each quatizatio threshold (bitplae), startig from the most-sigificat bitplae (MSB) ad goig dow to the least-sigificat bitplae (LSB). The results are accumulated after each multilevel SCLA computatio to form a icremetally-refied output. Similarly, for the DWT recostructio, the MSBs of the trasform-coefficiets are iserted first ad the multilevel iverse DWT recostructs the image icremetally. Each additioal processig step requires additioal eergy cosumptio. If the processig resources are termiated, oe receives the decomposed or recostructed image with the best-possible quality (cotrolled by the umber of bitplaes processed). Figure. Operatioal refiemet of computatio for the multilevel DWT decompositio ad recostructio. The SCLA calculatio per bitplae performs two decompositio (or recostructio) levels ad the results are accumulated at the ed before movig to the ext bitplae. Although the framework of Figure receives idividual bits (per pixel or per wavelet coefficiet), the dyamic rage of computatios performed is icreasig accordig to: (i) the liftig coefficiets of each liftig step; (ii) the

3 IEEE Trasactios o Sigal Processig - T-SP , to appear. 3 iput-source statistics; (iii) the umber of decompositio levels. I order to adapt the circuit activity accordig to varyig iput statistics, it is crucial to have arithmetic uits that perform variable dyamic-rage computatio. Xathopoulos [8] proposed a suitable framework for this purpose: for all arithmetic uits, very low-cost MSB-rejectio circuits are utilized, which idetify the exact umber of active bits withi each elemet (adder or multiplier). I this paper, we use a zero-detectio circuit to avoid performig parts of multiplicatios with zero iputs ad demostrate its effectiveess i cojuctio with icremetal computatio o a FPGA arithmetic co-processor testbed. The cotributio of this paper is twofold: firstly, we propose icremetal computatio of the DWT with the use of predictio withi each refiemet layer (bitplae) of the iput (Sectio II ad III); i additio, via the utilized FPGA co-processor (itroduced Sectio IV), we demostrate the eergy-distortio scalability offered by icremetal computatio ad the proposed predictio-based icremetal computatio i compariso to the covetioal (o-refiable) computatio (Sectio V). Our results are relevat to DWT architectures localizig memory accesses to o-chip memory [9] [0], or to cases whe the etire image ca be stored o-chip, sice eergy cosumptio stems predomiatly from arithmetic operatios ad ot memory accesses i such cases [9] [0]. II. OVERVIEW OF SCLA-BASED DWT UNDER INCREMENTAL REFINEMENT OF COMPUTATION The 2D DWT of a R C iput matrix X cosistig of image itesity values is expressed i the spatial domai by T Ε Ε, where E is the aalysis polyphase matrix cosistig of alteratig rows of low- ad S X high-pass filters shifted by two (i order to apply the DWT dowsamplig), ad S the 2D matrix of output wavelet coefficiets. The Z-domai expressio of the aalysis matrix ca be factored ito liftig steps []: 0 ap( k) ( z ) E ( z) k( z) k( z) U P k k au( k) ( z ) 0, () where P k( z ) ad U k ( z) are the k th predictio ad update liftig matrices. The lower- or upper-triagular 2 2 matrices of () cotai symmetric Lauret biomials [] [2], with a the fixed-poit represetatio of the k th update filter coefficiet (likewise for the predict step). Although the biomial liftig factorizatios represeted by () do ot cover all possible factorizatios, they do cover some of the most-popular oes foud i practical applicatios [] [] [7] [2]. As a example, for the 9/7 filter-pair we have 2, i.e. two predictio ad two update filters, each of which has two idetical o-zero ad o-uity taps, as show i (). Other popular filter-pairs also obey this rule, e.g. cubic B-splie filters [], the 5/3 filter-pair [] ad the 7/5 filter-pair [4]. I u( k) We are ot cocered with the scalig performed after the liftig aalysis ad before the liftig sythesis [] because all scalig factors ca be icorporated ito the subsequet ecodig or processig stage [2]. I additio, for otatioal simplicity, we assume that the image dimesios are iteger multiples of 2 L max, with L max the umber of wavelet decompositio levels.

4 IEEE Trasactios o Sigal Processig - T-SP , to appear. 4 additio, all symmetric or atisymmetric factorizatios ca be expressed as biomial liftig factorizatios [2]. A. Overview of Icremetal SCLA Computatio A reformulatio of the 2D liftig scheme has bee proposed i the spatial domai by Meg ad Wag [7]: ( ( ( ( ) ) ) ) T T T T P U S U P U P X P U, (2) This equatio computes the 2D DWT i a series of steps, startig from the ier part, i.e. p T M P X P, ad workig outwards toward the fial result S. The computatio is performed i squares of 2 2 iput samples (2D polyphase compoets), as it will be explaied i the followig. If we use double-deadzoe embedded quatizatio of the iput X with basic partitio cell [], each quatized coefficiet x quat [ rc, ] of iput X ( 0 with x [ rc, ] the th bit of quatized coefficiet r< R, 0 N quat x N [ rc, ] c< C ) is expressed as: x [ rc, ] ( ) 2 x [ rc, ] (3) 0 x quat [ rc, ] (where 0 [, ] x rc is the least-sigificat bit), ad x [ rc, ] is the sig bit. This is the popular case of successive approximatio quatizatio (SAQ) [], where, startig from the sig bit, each additioal bitplae correspods to icreased precisio i the approximatio of xrc [, ]. I the expressios that follow, we use the siged bits of the iput, i.e. x siged bit of quatized coefficiet x quat [ rc, ]. x N [ rc, ] x rc deotes the th [ rc, ] 2 [, ] For icremetal refiemet of the computatio of a 2D biomial factorizatio give by (2) uder the SAQ approximatio of each iput sample xrc [, ] show i (3), the computatio of the first step, each bitplae, 0 < N, is [4] ( 0 i< R 2, 0 j< C 2 ): N p T M P X P, for p [2 i,2 j] x [2 i,2 j] (4) p [2 i,2j ] x [2 i,2j ] a x [2 i,2 j] x [2 i,2j 2)] (5) p() p [2 i,2 j] x [2 i,2 j] a x [2 i,2 j] x [2 i,2 j2] p [2 i,2 j] p [2 i2,2 j] (6) p() p [2i,2 j] x [2i,2 j] a x [2 i,2 j] x [2i 2,2 j] (7) p() where p 2 i {0,}, 2 j {0,} is the output quadrat of coefficiets, as computed from the siged quatized values (siged bitplaes) x [ rc, ] of the iput. Equatio (4) is a simple copy operatio betwee iput ad output, while (5)-(7) add to the iput bit of each case a factor that depeds o the th siged bit values of the samples i the spatial eighbourhood of x 2 i {0,}, 2 j {0,} ad the liftig-filter coefficiet a p(). u p T The secod matrix product, M U M U, is produced by reusig the results of (4)-(7): u 2i,2j p 2i,2j (8) u 2i, 2j p 2i, 2 j a u() p [2i, 2j ] [2, 2 ] (9) ( p i j )

5 IEEE Trasactios o Sigal Processig - T-SP , to appear. 5 u 2 i,2j p 2 i,2 j a u() p [2 i,2j ] [2,2 ] [2,2 ] [2,2 ] (0) ( p i j u i j u i j) u 2 i,2j p 2 i,2j au() p [2i,2j ] p [2i,2j ] () where u 2 i {0,}, 2 j {0,} is the output quadrat of coefficiets, as computed for the output coefficiets p [ r, c ] of (4)-(7). Agai, (8) is a simple copy operatio, while, for (9)-(), the fixed-poit liftig coefficiet a is used to update the iput. The remaiig steps k 2,, to complete the sigle-level decompositio usig the 2D liftig formulatio of (2) are performed as i (4)-(7) ad (8)-() with the replacemet of the iput with the output of each previous step ad the replacemet of the coefficiets a p() ad a u() by p( k) a ad u() u( k) a, respectively. All steps ca be performed i-place as i the covetioal liftig decompositio, with the reuse of the memory for the p M ad u M arrays. Oce all steps are completed, we perform the reorderig ad additio to the previous results for the high-frequecy coefficiets by: HL ij, HL ij, u[2 i,2 j], LH ij, LH ij, u[2 i,2 j], HH ij, HH ij, u[2 i,2 j], where HL, LH ad HH are the three high-frequecy subbads of level oe (see Figure ) ad a b assigs b to a. Notice that the low-frequecy coefficiets ( u [2 i,2 j ]) are ot ivolved i this process, as they are the iput for the subsequet decompositio level, as described i the followig. B. Multilevel Extesio The multilevel extesio of the bitwise computatio of the DWT for L max levels was performed i our previous work [4] via a frequecy-first icremetal refiemet of computatio: for each iput bitplae, the liftig-scheme computatio is cotiued for all subsequet levels after the first level ad the results are accumulated at the ed, before movig to the ext iput bitplae. This is achieved by reformulatig the first predictio step of (4)-(7) for all levels l, 2 l Lmax by ( 0 i< R 2 l, 0 j< C 2 l ): p 2 i,2j u 4 i,4j (3) p 2 i,2j u 4 i,4j 2 ap() u[4 i,4 j] u[4 i,4j 4] (4) p 2 i,2 j u 4 i2,4 j2 ap() u[4 i2,4 j] u[4 i2,4 j4] p [2 i,2 j] p [2 i2,2 j] (5) p 2i,2j u 4i 2,4 j ap() u[4 i,4 j] u[4i 4,4 j] (6) where we use the low-frequecy outputs u 4 i {0, 2}, 4 j {0, 2} of the last update step of the previous level. The subsequet update step ad all additioal pairs of liftig steps, as well as the icremetatio of all high-frequecy subbads, HL l, LH l, HHl, are performed as show previously, with: 0 2l (2) i< R, 0 j< C 2 l. The low-frequecy subbad coefficiets produced at the last decompositio level are icremeted by: LL, LL ij, u [2 i,2 j ] (7) L ij max L max

6 IEEE Trasactios o Sigal Processig - T-SP , to appear. 6 This process is the carried out for the ext bitplaes,,0. Source code for the etire process is available [3]. C. Iverse SCLA ad Iverse DWT Cocerig the iverse SCLA, the process is exactly ati-symmetric, as i the covetioal liftig computatio: all liftig steps are performed i reverse order by solvig the forward bitwise liftig equatios for the coefficiets beig predicted or updated durig the forward trasform. More explicitly, (8)-() are iverted by solvig for p [2 i {0,},2 j {0,}] ad replacig the terms p [ ii, ] iside the paretheses of (9) ad () by u [ ii, ]. These equatios would the perform the iverse update step ad derive p [2 i {0,},2 j {0,}] from iputs u [2 i {0,},2 j {0,}]. Iversio of (4)-(7) occurs as for (8)-(), with x [ ii, ] iside the paretheses of (5) ad (7) replaced by p [ ii, ]. The iversio equatios are executed i reverse order; first the iverted (), followed by the iverted (0), up to the iverted (4). For the th pair of liftig steps, the iputs u [2 i {0,}, 2 j {0,}] cosist of the th siged bitplae of the iput wavelet coefficiets LL [ ij, ],HL [ ij, ],LH [ ij, ],HH [ ij,, ] l Lmax, Lmax,, ; they are liked via (2) ad (7) (if Lmax l l l l Lmax ). Hece, i the iverse trasform, the siged bits of the iput low ad high-frequecy wavelet coefficiets of all decompositio levels are used, startig from the coarsest level, ivertig all liftig steps, ad icreasig the resolutio up to the pixel domai. All levels are iverted withi each icremet layer, ad the fial recostructed icremets for all pixels are added to the output pixels (see the right side of Figure ) i order to progressively reduce the distortio of the recostructed image. This process is the carried out for the ext bitplaes,,0. Source code for the iverse trasform is available olie [3]. III. PREDICTIVE INCREMENTAL REFINEMENT OF COMPUTATION FOR THE DWT Sice eighborig iput pixels or trasform coefficiets are expected to be correlated 2, we ca attempt to reduce the required computatios for the icremetal computatio algorithm preseted previously by itroducig a predictio scheme betwee the pixels or coefficiets of the iput. For each of the 2 2 iput 2D polyphase compoets of each liftig step, we form the predictio i the directio of the 2D liftig-based filterig preseted i the previous sectio. A. Proposed Formulatio The first compoet of the first step [show i (4)] remais uchaged as this is simply a assigmet operatio. The secod compoet applies the liftig filter alog the j-axis. By writig (5) with the replacemet of j by j ad subtractig the resultig expressio from (5) we get the icremet of computatio for p [2 i,2j ] if we have previously computed p [2 i,2j ]. This is give by: 2 The DWT should decorrelate the iput sigal. However, i practice, eighborig subbad coefficiets remai correlated. Image compressio ad deoisig applicatios use this by formig cotext models based o each coefficiet s eighborhood.

7 IEEE Trasactios o Sigal Processig - T-SP , to appear. 7 ( x x ) ( x x ) p() p [2 i,2j ] p [2 i,2j ] [2 i,2j ] [2 i,2j ] a [2 i,2j 2] [2 i,2j 2] (8) Notice that (8) assumes that the calculatio is applied rows-first, i.e. for each positio j,, C 2 withi each row i 0,,, R 2. Furthermore, if x [2 i,2j 2] x [2 i,2j 2], we avoid the multiply-accumulate step with factor a p() i (8), which is performed by the origial icremetal liftig of (5). Hece, with a multiplier desig that is dyamically adjustable to the iput sparsity, we expect to have decreased circuit activity for the multiplicatio operatios. The third polyphase compoet [show i (6)] applies diagoal filterig alog both the i ad the j-axis by reusig the results of the secod compoet computed previously. For this reaso, we break the computatio i two parts, with the first part performig predictio across the j-axis ad the secod performig predictio across the i -axis ad icremetig the result of the previous computatio. The first part is: ( x x ) ( x [2, 2 2] x [2, 2 2] ),part,part ap() i j i j p [2i, 2j ] p [2i, 2j ] [2i, 2j ] [2i, 2j ] (9) ad it is applied for each positio j,, C 2 withi each row i 0,,, R 2. The secod part is: p(),part,part p [2i, 2j ] p [2i, 2j ] a p [2i 2, 2j ] p [2i 2, 2j ] (20) ad it is applied for each i,, R 2 withi each colum j 0,,, C 2. Similarly as before, if x [2i, 2j 2] x [2i, 2j 2] i (9), or p,part[2i 2, 2j ] p,part[2i 2, 2j ] i (20), we reduce the circuit activity required for the costly multiply-accumulate operatios performed by the icremetal liftig of (6). However, sice (9) ad (20) ivolve two multiplicatios, this may ot be cost-effective i compariso to (6). Hece, we ca selectively disable these operatios ad perform (6) istead if this is deemed to be more efficiet. Fially, the last compoet of the quadrat of computatio of the first liftig step [show i (7)] is writte i the same maer as the secod compoet, albeit applyig the predictio i the i -axis: ( x x ) ( x x ) p() p [2i,2 j] p [2i,2 j] [2i,2 j] [2i,2 j] a [2i 2,2 j] [2i 2,2 j] (2) ad it is applied for each i,, R 2 withi each colum j 0,,, C 2. The secod liftig step [update step of (8)-()] is modified i the same maer by applyig the predictio as show i (8)-(2). The expressios ca be derived followig the same rules ad are omitted for brevity of descriptio. Subsequet pairs of liftig steps follow the same rules. Oe fial aspect relates to the border treatmet of the predictio-based icremetal refiemet of the DWT computatio. This is performed by applyig the origial equatios for icremetal refiemet for the iitial row ( i 0 ) or the iitial colum ( j 0 ) of each case. The modificatio for the calculatio performed for the subsequet levels [(4)-(6)] is performed as follows [we omit (3) sice it is just a assigmet operatio]. For (4), for each j< C 2 l of each row 0 i< R 2 l :

8 IEEE Trasactios o Sigal Processig - T-SP , to appear. 8 ( ) ( ) 2,2 2,2 4,4 2 4,4 2 p() [4,4 4] p i j p i j u i j u i j a u i j u [4 i,4 j 4]. (22) For (5), for each j< C 2 l of each row 0 i< R 2 l, we first calculate: ( ) ( [4 2, 4 4] u[4i 2, 4j 4] ) p 2i,2j p 2i,2j u 4i 2,4j 2 u 4i 2,4j 2,part,part ap() u i j ad the, for each i< R 2 l of each colum 0 j< C 2 l we calculate p 2i, 2j as before, if (23) ad (20) are deemed to be iefficiet, we ca apply (5) istead. (23) with (20). Similarly Fially, for (6), for each j< C 2 l of each row 0 i< R 2 l we have: ( ) ( ) 2,2 2,2 4 2,4 4 2,4 p() [4 4,4 ] p i j p i j u i j u i j a u i j u [4 i 4,4 j ]. (24) The calculatios at the borders are performed by applyig the origial equatios for icremetal refiemet [(3)-(6)] for the iitial row ( i 0 ) or the iitial colum ( j 0 ) of each case. Idicative source code for the proposed approach (forward ad iverse) is available olie [3]. B. Discussio The proposed predictio-based icremetal refiemet of computatio is applied per iput bitplae. As such, it ca be selectively applied to some iput bitplaes, e.g. the four most-sigificat oes, ad the origial equatios (4)-() ca be applied for the remaiig bitplaes. I additio, it ca be applied selectively o some parts of the liftig steps, e.g. we ca apply (6) istead of the proposed (9) ad (20), or eve o selected iput frames where the correlatio betwee eighborig iput samples is stroger (this is actually used i the experimetal results of Sectio V). This creates a hybrid approach that uses predictio oly o some parts of the computatio. This ca be beeficial because, for example, correlatio betwee eighbourig iputs is expected to decrease whe movig to lower bitplaes, sice this correspods to the low-amplitude parts of the iput image, which are expected to cotai fie-grai oise. Similarly, decreased correlatio betwee eighborig samples is observed whe dealig with error frames istead of video frames, sice error frames are the result of temporal predictio. Eve though the descriptio of the paper focused o sigle-bitplae iputs, a umber of bitplaes ca be iserted together ad the computatio ca utilize all of them together as oe icremet layer. For example, for a 8-bit iput image, the three MSBs ca be processed first, followed by the three itermediate bitplaes ad the two LSBs. This creates oly three icremet layers of computatio. This ca reduce the expected overhead whe icreasig the umber of icremet layers. I our recet work o software desigs for icremetal image processig [6], we foud that three or four icremet layers provide for sufficiet quality-complexity scalability without sigificat overhead i the overall executio time. From the implemetatio perspective, a crucial elemet eeded i order to take advatage of the reductio of the multiplicatio bitwidth is a multiplier uit with adjustable circuit activity accordig to the iput bit patters. This is the topic of the followig sectio.

9 IEEE Trasactios o Sigal Processig - T-SP , to appear. 9 IV. FGPA-BASED ARITHMETIC CO-PROCESSOR TESTBED FOR ENERGY MEASUREMENTS I order to substatiate the potetial beefits of the proposed computatio i a real eviromet, we setup a experimetal testbed cosistig of a mai platform (persoal computer) realizig the trasform decompositio or recostructio algorithmic flow i software. All arithmetic operatios (multiplicatios, additios) are trasferred to a Xilix Virtex II XC2V500 FPGA ad are computed based o customized desigs of multipliers ad adders placed o the FPGA. We the measure ad report the eergy cosumptio of these operatios for the differet algorithms (covetioal, icremetal refiemet ad predictio-based icremetal refiemet). We used the ISE 8.2 FPGA developmet eviromet with the XPower 8.2 tool [5] for this purpose ad all our results relate to the dyamic eergy dissipatio reported by the tool. All arithmetic operatios are performed i siged fixed-poit represetatio ad require (maximally) a 2-bit iteger part plus a 2-bit fractioal part, which esures recostructio peak-sigal-to-oise (PSNR) values above 55dB for up to 6 levels of decompositio [6]. All our arithmetic uits restrict their output to 24 bits, which is comparable or less tha related desigs [6] [7]. The multiplier desig separates the iteger ad fractioal parts ito three 4-bit parts (IA3~IA ad FA~FA3) ad the performs a cascade realizatio of the multiplicatio, whose itermediate operatios are depicted i Figure 2(a). Sice the result will always be cotaied withi the 2 plus 2 bit output, oly the itermediate operatios cotributig to the retaied part of the output of Figure 2(a) are performed. Furthermore, each idividual 4-bit multiplicatio of Figure 2(a) is eabled based o a zero-detectio circuit. This circuit checks all iputs A[0:3] ad B[0:3] of the 4-bit multiplier ad raises the CE (circuit eablig) sigal oly whe the multiplicatio will have o-zero output, as show by Figure 2(b); otherwise the default zero result is obtaied without activatig the multiplier. (a) (b) Figure 2: (a) Operatios of the 24-bit multiplier. (b) Zero-detectio circuit for the activatio of each 4-bit multiplier.

10 IEEE Trasactios o Sigal Processig - T-SP , to appear. 0 The use of the zero-detectio circuit esures that the overall multiplier desig is adjusted to: (i) each liftig coefficiet s active bitwidth without customizig the multiplier desig to a particular liftig scheme; (ii) the iput s varyig statistics ad varyig dyamic rage; (iii) the output s rage (24 bits). This desig was foud to cosume less eergy tha the stadard 24-bit cascade multiplier with the same throughput ad latecy. Cocerig the adder s desig, eve though we desiged a bitwidth-adaptive adder uit, our measuremets idicated that the eergy cosumptio of this desig is comparable to that of a stadard 24-bit adder, ad overall more tha a order of magitude smaller tha the average eergy cosumptio of the adaptive multiplier desig. Hece, we resorted to usig a stadard 24-bit adder for all additios performed. Both the multiplier ad adder desigs operate i oe clock cycle. Represetative results for the dyamic eergy cosumptio measured for each combiatio of iputs whe clocked at 75MHz, as well as the layout of the multiplier desig are provided olie [3]. V. EXPERIMENTAL RESULTS We examie the eergy cosumptio of the proposed approach for idividual cases of forward ad iverse DWT applied to YUV itra frames as well as to YUV error frames produced by motio-compesated predictio. Two schemes are used for compariso purposes: covetioal (o-refiable) computatio ad icremetal refiemet of DWT computatio as proposed i our previous work [4]. All 300 frames of the CIF sequeces Stefa, Coastguard ad Forema were used for the experimets of this sectio. Image distortio is measured usig the PSNR across lumiace (Y) ad chromiace (U ad V) chaels of each video frame: 4PSNR(Y) PSNR(U) PSNR(V) mea_psnr i order to produce oe metric icludig all chaels. The PSNR of each chael C {Y,U,V} is measured as: PSNR(C) 0 log0, where MSE(C) is the mea MSE(C) squared error. For the forward DWT, PSNR is measured by ivertig the produced decompositio via a software implemetatio of the iverse DWT with double-precisio floatig-poit represetatio [3]. Eergy cosumptio is measured via the testbed preseted i Sectio IV, which icludes oe adder ad oe multiplier clocked at 35MHz for the covetioal approach ad at 75MHz for the icremetal approaches. The clock frequecies were chose so as to allow for processig of 25 frames/sec whe usig all iput bitplaes: sice the icremetal approaches are performig more arithmetic operatios (multiple icremets but with smaller bitwidth), we icreased the clock frequecy for these cases i order to esure they complete the frame processig at the same time with the covetioal approach. The 9/7 [] ad 7/5 [4] filter-pairs were chose for our comparisos sice they preset state-of-the-art compressio performace with moderate complexity. The liftig coefficiets of the 9/7 ad 7/5 filter-pairs ca be foud i [] ad [4], respectively. The covetioal approach is usig all iput bitplaes simultaeously, while the icremetal approaches are breakig the computatio ito two

11 IEEE Trasactios o Sigal Processig - T-SP , to appear. layers: for the forward DWT, the four MSBs are iserted first, followed by the 4 LSBs; for the iverse DWT, the 8 MSBs are iserted first, followed by the 4 LSBs. This creates two layers of computatio labeled as half-precisio ad full-precisio results. All utilized error frames i our experimets were produced by a spatial-domai motio-compesated predictio scheme [8] usig successive (frame-by-frame) predictio ad withi a group-of-pictures (GOP) of 6 frames. Four wavelet decompositio levels were performed. No wavelet-coefficiet codig was performed i order to avoid artificial thresholdig of wavelet coefficiets before the iverse DWT. The results are reported i Table. As see from the top half of the table, whe used for itra frames, the proposed approach provides beefits i compariso to the origial algorithm for icremetal refiemet of computatio [4] by decreasig its eergy cosumptio ad makig it comparable to covetioal computatio. I additio, the proposed scheme brigs sigificat beefits i terms of eergy reductio for half-precisio computatio. However, whe used for error frames, the proposed approach does ot provide ay improvemet i compariso to icremetal refiemet [4]; istead, eergy cosumptio is icreased, sometimes by a sigificat amout. This is attributed to the failure of the predictio scheme due to reduced correlatio betwee eighborig coefficiets of error frames. These observatios hold for both filter-pairs ad both forward ad iverse DWT. There are some mior discrepacies i PSNR betwee the differet approaches caused by the fixed-poit precisio chose for the example cases preseted, i the order of 0.03dB. All visual artifacts observed i the recostructed images ad error frames at each termiatig quatizatio precisio (bitplae) are typical of low-bitrate wavelet-based quatizatio ad codig approaches studied i the literature [9]. Frame Full Precisio Half Precisio Type Covetioal Icremetal [4] Proposed mea_psnr Icremetal [4] Proposed mea_psnr 9/7 forward DWT Itra % 4.6% 56.7dB -34.4% -43.5% 29.5dB Error % 9.2% 57.7dB -74.5% -63.% 37.2dB 9/7 iverse DWT Itra % 0.4% 57.6dB -39.4% -45.7% 33.3dB Error % 0.3% 57.7dB -82.4% -52.3% 36.7dB 7/5 forward DWT Itra % 6.6% 56.7dB -34.8% -43.6% 29.5dB Error % -04.0% 57.7dB -77.5% -68.3% 37.2dB 7/5 iverse DWT Itra % 2.7% 57.6dB -36.2% -48.% 33.3dB Error % -30.% 57.7dB -84.4% -50.7% 36.7dB GOP with: Itra (I) & 5 Error (P) frames Covetioal Proposed (I)Icremetal (P) mea_psnr Proposed (I)Icremetal (P) mea_psnr 9/7 forward DWT % 57.4dB -69.4% 33.7dB 9/7 iverse DWT % 57.7dB -72.8% 35.4dB Table. Eergy cosumptio (i micro-joule per frame) ad PSNR results per itra frame ad error frame. The results of icremetal refiemet [4] ad the proposed predictio-based icremetal refiemet approaches are preseted as the percetile differece to the eergy cosumptio reported for the covetioal approach.

12 IEEE Trasactios o Sigal Processig - T-SP , to appear. 2 The bottom half of Table shows the eergy cosumptio i a video codig sceario where itra frame ad 5 error frames are trasformed, whe the proposed approach is used for the itra frames ad icremetal refiemet [4] is used for the error frames. This hybrid approach provides for lower eergy cosumptio for full-precisio computatio i compariso to the covetioal (o-refiable) computatio. Moreover, i such video applicatios, eergy cosumptio is reduced sigificatly (by more tha 70% o average) by termiatig at half-precisio (first icremet layer), as show by the bottom half of Table. VI. CONCLUSIONS We propose a predictio-based method for icremetal computatio of the forward ad iverse discrete wavelet trasform (DWT) uder a bitplae-based formulatio of the 2D liftig decompositio. The proposed approach applies predictio of eighborig pixels or wavelet coefficiets i the directio of the liftig-based filterig. This is performed for each of the 2D polyphase compoets of the direct 2D computatio of the multilevel DWT decompositio. Based o FPGA-based comparisos with covetioal computatio, we verified that the proposed approach ad icremetal refiemet has comparative eergy cosumptio i full-precisio computatio, with the proposed approach providig a improvemet for itra frames. At the same time, these approaches ca termiate at itermediate eergy-distortio poits (e.g. half-precisio) with sigificat decrease i eergy cosumptio. REFERENCES [] JPEG2000: Image Compressio Fudametals, Stadards ad Practice, D. Taubma, M. Marcelli, luwer, [2] S. Mallat, A wavelet tour of sigal processig. Academic Press, Sa Diego CA, 998. V.. Goyal, ad M. Vetterli, Maipulatig rates, complexity ad error-resiliece with discrete trasforms, Proc. IEEE Asilomar Cof. o Sigals, Syst. ad Comput., vol., pp , Nov [3] Z. Wag, Pruig the fast discrete cosie trasform, IEEE Tras. o Comm., vol. 39, o. 5, May 99. [4] Y. Adreopoulos ad M. va der Schaar, "Icremetal refiemet of computatio for the discrete wavelet trasform," IEEE Tras. o Sigal Processig, vol. 56, o., pp , Ja [5] J. Wiograd ad S. H. Nawab, Icremetal refiemet of DFT ad STFT approximatios, IEEE Sigal Proc. Letters, vol. 2, o. 2, pp , Feb [6] D. Aastasia ad Y. Adreopoulos, Software desigs of image processig tasks with icremetal refiemet of computatio, Proc. IEEE Workshop o Sigal Process. Systems (SIPS), Tampere, Filad, Oct. 2009, pp [7] H. Meg ad Z. Wag, Fast spatial combiative liftig algorithm of wavelet trasform usig the 9/7 filter for image block compressio, IEE Electroics Letters, vol. 36, o. 2, pp , Oct [8] T. Xathopoulos, Low power data-depedet trasform video ad still image codig, Ph.D. Thesis, Massachusetts Istitute of Techology, Feb. 999, [Olie]. Available: [9] P.-C. Tseg, Y.-C. Chag, Y.-W. Huag, H.-C. Fag, C.-T. Huag, ad L.-G. Che, Advaces i hardware architectures for image ad video codig a survey, Proc. of the IEEE, vol. 93, o., 84-97, Ja [0] M. Agelopoulou,. Masselos, P. Y.. Cheug, ad Y. Adreopoulos, "Implemetatio ad compariso of the 5/3 liftig 2-D discrete wavelet trasform computatio schedules o FPGAs," VLSI Sigal Processig, vol. 5, o., pp. 3-2, April [] I. Daubechies ad W. Sweldes, Factorig wavelet trasforms ito liftig steps, J. Fourier Aal. Appl., vol.4, pp , 998. [2] C.-T. Huag, P.-C. Tseg ad L.-G. Che, Efficiet VLSI architectures of liftig-based discrete wavelet trasform by systematic desig method, Proc. IEEE It. Symp. o Circ. ad Syst., ISCAS, vol. 5, pp , [3] [Olie]. Available: [4] C. Brislaw, Iterim report o (Part 2) core experimet CodEff07, 7-tap/5-tap Filter Bak optio, ISO/IEC JTC/SC29/WG, N76, JPEG, July [5] [Olie]. Available: [6] V. Spiliotopoulos, et al, Quatizatio effect o VLSI implemetatios for the 9/7 DWT Filters, Proc. IEEE Iteratioal Cof. o Acoustics, Speech ad Sigal Processig, ICASSP'0, Salt Lake City, UT, vol. 2, pp , May 200. [7] S. L. Bishop, S. Rai, B. Guturk, J. L. Traha, ad R. Vaidyaatha, Recofigurable implemetatio of wavelet iteger liftig trasforms for image compressio, Proc. IEEE Iterat. Cof. o Recofig. Comput. ad FPGA's, pp. -9, [8] Y. Adreopoulos, et al, Scalable wavelet video-codig with i-bad predictio - Implemetatio ad experimetal results, Proc. IEEE Iteratioal Cof. o Image Processig, ICIP'02, vol. 3, pp , Sept [9] A. B. Watso, G. Y. Yag, J. A. Solomo ad J. Villaseor, Visibility of wavelet quatizatio oise, IEEE Tras. o Image Process., vol. 6, o. 8, pp , Aug. 997.