Lifting factorization-based discrete wavelet transform architecture design

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 11, NO. 5, MAY 2001 651

Lifting Factorization-Based Discrete Wavelet Transform Architecture DesignWenqing Jiang and Antonio Ortega

Abstract—In this paper, two new system architectures,overlap-statesequential andsplit-and-mergeparallel, are proposed basedon a novelboundary postprocessingtechnique for the computationof the discrete wavelet transform (DWT). The basic idea is to intro-duce multilevel partial computations for samples near data bound-aries based on afinite state machinemodel of the DWT derivedfrom the lifting scheme. The key observation is that these par-tially computed (lifted) results can also be stored back to their orig-inal locations and the transform can be continued anytime later aslong as these partial computed results are preserved. It is shownthat such an extension of the in-place calculation feature of theoriginal lifting algorithm greatly helps to reduce the extra bufferand communication overheads, in sequential and parallel systemimplementations, respectively. Performance analysis and experi-mental results show that, for the Daubechies (9,7) wavelet filters,using the proposedboundary postprocessingtechnique, the min-imal required buffer size in the line-based sequential DWT algo-rithm [1] is 40% less than the best available approach. In the par-allel DWT algorithm we show 30% faster performance than ex-isting approaches.

Index Terms—Boundary postprocessing, discrete wavelettransform, overlap-state, parallel algorithm, sequential algorithm,split-and-merge.

I. INTRODUCTION

E FFICIENT system architecture design for the discretewavelet transform (DWT) has received a lot of attention

recently [2]–[4], [1], [5]–[8] due to the success of DWT-basedtechniques in areas as diverse as signal processing, digitalcommunications, numerical analysis, computer vision andcomputer graphics [9]. Two important parameters have beenused to measure the efficiency of practical DWT systemdesigns: 1) thememorynecessary for the DWT computation(mostly in sequential algorithms) and 2) thecommunicationoverhead required by parallel DWT algorithms. As a matterof fact, memory efficiencyis one major design factor forwavelet-based image compression applications in printers,digital cameras and space-borne instruments where large sizememory leads to high cost and demands more chip design area[1], [10], [11]. Similarly,communication efficiencyis critical tothe success of parallel DWT systems built upon the network ofworkstations (NOWs) or local area multicomputers (LAMs),

Manuscript received November 9, 1999; revised September 26, 2000. Thisresearch was supported in part by the Integrated Media Systems Center, aNational Science Foundation Engineering Research Center, under CooperativeAgreement EEC-9529152. This paper was recommended by Associate EditorZ. Xiong.

W. Jiang was with the Integrated Media System Center, Department of Elec-trical Engineering-Systems, University of Southern California, Los Angeles,CA 90089 USA. He is now with C-Cube MicroSystems Inc., Milpitas, CA95035 USA (e-mail: [email protected]).

A. Ortega is with the Integrated Media System Center, Department of Elec-trical Engineering-Systems, University of Southern California, Los Angeles,CA 90089 USA (e-mail: [email protected]).

Publisher Item Identifier S 1051-8215(01)03818-6.

since in these systems cheap but slower communication linksare used (as compared with dedicated parallel systems) [12],[8], [13], [14].

The major difficulty in achieving an efficient DWT architec-ture design (both in terms of memory and communication) isthat, with the exception of trivial Haar filters, the DWT is not ablock transform. When data has to be processed one block (orone image scanline) at a time in sequential systems [1], [6] orpartitioned over multiple processors in parallel systems [4], [8],correct DWT computation near data boundaries requires extrabuffer and/or extra communication compared to that needed fora block transform such as the discrete cosine transform (DCT).In standard FFT-based filtering approaches, such a boundaryissue can be easily handled with appropriate data overlapping(e.g., theoverlap-saveor overlap-addapproaches [15]). How-ever, because the DWT consists ofrecursivefiltering operationson multileveldownsampleddata sequences, direct applicationof the overlapping techniques can be very costly in terms ofmemory and/or inter-processor communication.

Consider, for example, a-level wavelet decomposition of a-point input sequence to be performed using two processors

(assuming is even for simplicity). In this case, either the twoprocessors are given sufficient overlapped data to carry on thewhole computation without communicating with each other, oralternatively, they have to communicate samples after each levelof the decomposition has been computed. The first approach,overlapping, requires that input data near the block boundariesbe given to both processors. Since each processor has to com-pute its own transform for multiple decomposition levels, thisoverlap can be quite large. As given in the analysis of the spa-tially segmented wavelet transform (SSWT) by Kossentini [16],the buffer size for a -level decomposition of a -point se-quence is ( is the filter length). As onecan see, the overlap increases exponentially with the increaseof decomposition level , which can become significant if longwavelet filters are used and the number of levels of decompo-sition is large. Notice that the in-place lifting algorithm [17] isalready assumed to be used in our work and the focus of thispaper is on the reduction of memory at boundaries below thelevel, , required in a standard lifting approach.To the best of our knowledge, reduction of boundary memoryhas not been addressed within a lifting framework until recently.

As one alternative, theoverlappingtechnique has been alsoapplied at each decomposition level rather than once for all as inSSWT. Examples of this approach include the recursive pyramidalgorithm (RPA) by Vishwanath [3], and the reduced line-basedcompression system by Chrysafiset al. [1], [10]. These ap-proaches reduce significantly the buffer size tofor a -level decomposition of a -point sequence. Unfortu-nately, this is still a quite large overlap for some applications,for example, the line-based wavelet image compression system

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652 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 11, NO. 5, MAY 2001

Fig. 1. Example dataflow chart of a three-level wavelet decomposition. Solidlines: completely transformed data. Dashed lines: boundary samples from theneighboring block. Operations 1,3,5: communicate boundary data samples toneighboring blocks. Operations 2,4,6: transform for current level.

described in [1], [10]. In that system, image lines are stored inmemory only while they are used to generate output coefficientsand are released from memory when no longer needed. Thisleads to and the amount of memory isimage lines (due to line downsampling) at each stage. Considera color image of size , such that each color com-ponent sample is stored as a 4 bytes floating point number forDWT computation. In this case, one image scanline requires 48kB. Using the Daubechies (9,7) wavelet filterbank ( ), fora three-level decomposition, the total memory would be 588 kB.In this paper, we propose a novel technique which can help toreduce the memory to only 296 kB.

In the second approach,nonoverlapping,to parallel DWTimplementation, input data is not overlapped so the memoryrequirement is relaxed. But boundary samples need to beexchanged at each decomposition level. Such an approach isused, for example, in the design of mesh and hypercube parallelDWT architectures by Fridmanet al. [4]. Their analysis showsthat, for a -level wavelet decomposition, data exchangesare needed between neighboring processors [4] (see Fig. 1 fora three-level example). In order to reduce the communicationoverhead, Yanget al. [12] proposed to use boundary extensionsin their DWT system configured from a cluster of SGI worksta-tions. This, however, computes incorrect wavelet coefficientsnear data boundaries, which causes performance degradationin low-bit rate image coding applications [18].

This provides the motivation to study the problem of block-based DWT computation and its implications on memory andcommunication overhead in practical system designs. In thispaper, we present a novel technique,boundary postprocessing,which can help to achieve significant memory and communica-tion savings. The idea is motivated by the standardoverlap-addtechnique whichfirst performs filtering operations on neigh-boring data blocks independently and completes the computa-tion laterby summing the partial boundary results together [15].We extend this idea to the case of multilevel wavelet decomposi-tions using the lifting framework formulated by Daubechies andSweldens [17]. In the proposed approach, the DWT is modeledas afinite-state machine, in which each sample is updated pro-gressively from the initial state (the original data sample) to thefinal state (the wavelet coefficient) with the help of samples inits local neighborhood. Obviously, samples near data boundaries

cannot be fully updated due to lack of data from neighboringblocks. Rather than leaving them unchanged, as was done inprevious approaches, we propose to update these samples intointermediate states and preserve these partially transformed re-sults (state information) for later processing. The FSM modelthus ensures that correct transform can still be achieved for theseboundary samples using the preserved state information. Be-cause of the partial computation and the state preservation, wewill show that the buffer size in sequential algorithms and thecommunication overhead in parallel algorithms can be reduced.

Some recent works have also explored (independently of ourwork) the use of lifting factorizations for memory savings insequential DWT implementations [19]–[21]. The novelty ofour work is that, first, we introduce partial computations forboundary samples at multiple decomposition levels for memorysavings and second, we propose that processors exchangedata after multilevel decompositions for communication sav-ings. Application of the proposedboundary postprocessingtechnique results in two new DWT system architectures, theoverlap-statesequential, and thesplit-and-mergeparallel. Wewill show how the proposed technique can be used to reducethe memory requirement and the interprocessor communicationoverhead in the architecture designs.

We mention that, throughout this paper, we focus on theMallat tree-structured [22] multilevel octave-band wavelet de-composition system with critical sampling using a two-channelwavelet filterbank. The extensions of our work to other DWTsystems, includingstandardDWTs [23], multichannel waveletfilterbank, and wavelet packet decompositions are straight-forward. The rest of the paper is organized as follows. In thenext section, thefinite state machinemodel is introduced forthe DWT and theboundary postprocessingtechnique for thetransform near block boundaries is presented. The proposedsequential and parallel architectures are given in Sections IIIand IV, respectively, along with performance analysis andexperimental results. Section V concludes our work.

II. FINITE-STATE MACHINE MODEL FORDWT

In this section, we first introduce the FSM model for DWTbased on the lifting factorization. Then we discuss a postpro-cessing technique for DWT computation near block boundaries.

A. The Finite-State Machine Model

The polyphase matrix of any FIR wavelet filterbank hasa factorization form ([17, Theorem 7]) as

(1)

where are Laurent polynomials, which are calledpredictionandupdatingoperations, respectively. Without lossof generality, we use to represent the elementary matrices.That is

or

Let us consider the time domain filtering operations corre-sponding to . By definition, we have


where are the two polyphase compo-nents of filter . Each corresponds to two time domainfilters

or

(2)where and are the low- and high-pass filters inthe analysis filterbank, respectively. In time domain, thiscorresponds to In time domain, this corresponds to

(3)

For a -point input sequence (assume is even),and . The state transition in vector

form for the upper triangular elementary matrix (similar formfor the lower triangular elementary matrix) is

...

...

...

...

Obviously, each and every elementary matrix in the fac-torization of the polyphase matrix can be modeled as sucha state transition process.

Consider the input as a column vector, and define the in-termediate states in the process of transformation

, where is the result of applying theoperation to , and where the initial input is

. There are two important observations.

1) Every time we generate , we only need to store thisset of values, i.e., we do not need to know any of the other

, for , in order to compute the output (the finalwavelet coefficients).

2) Each sample is updated using samples only from theother polyphase component and itself. Consequently,each sample can be updatedindependently, fully orpartially, and written back to its original location, anextended in-place computation feature of the liftingalgorithm. For example, the updating of (may onlybe partially updated because insufficient boundary datais available) will not affect the updating of at thisstage.

For the polyphase matrix factorization, this requires that theelementary matrix can only be in the form of lower/uppertriangular matrices with constants on the diagonal entries. Thiskey property of the lifting factorization guarantees that theFSM structure applies. Thus, the wavelet transform based onthe polyphase factorization can be modeled as afinite statemachine, as depicted in Fig. 2, where each elementary matrix

updates the FSM state to the next higher level .The significance of such a FSM model is that the transform

Fig. 2. State transition diagram of DWT as a FSM.

Fig. 3. State transitions across block boundary usinge . (a) Partial computa-tions near boundaries. (b) After updating, boundary samples stay in intermediatestates. The “new boundary” separates fully updated output coefficients frompartially computed ones.

can be stopped at any intermediate stage. As long as the stateinformation is preserved, the computation can be continued atany later time. This is the key reason that one can defer thetransform for block boundary samples and still can obtain thecorrect result.

B. Boundary Postprocessing

In Fig. 3, we show oneupdating operation for inputsequence where even-indexed samples are updatedas . Denote

( ), then

where and are respectively the contributions fromthe anticausal and causal part of filter.

Now let us consider the computations near the blockboundary at point . Take sample , for example. Itis obvious that it cannot be updated into state becausethe anticausal filtering result cannot be computed dueto lack of data samples from the other block, i.e., samples

, as shown in Fig. 3(a).


A typical approach used in most existing DWT algorithms isto buffer and wait until the future samples are available,see, e.g., [3], [1], [4]. This direct buffering approach, however,forces one also to buffer whatever samples are necessary inthe current block for the computation of , i.e., samples

in this case. Clearly, for the futureupdating of sample , the buffer should at least be largeenough to hold three samples.

Observing the fact that is readily available (all sam-ples needed are in the same block as ), we propose tofirst update into and then buffer thispartially computed result at the same location of . Noticethat can be over-written (i.e., in-place computation) be-cause no samples in the current block or the other block need theoriginal value for their updating operations as describedbefore. As soon as samplesbecome available, can be computed and can beupdated into . Clearly,in this case we only need a one sample size buffer to store thepartially computed result at sample locationfor the future up-dating. Thus compared to the case of direct buffering of ,buffering partially transformed coefficients helps to reduce thememory requirement.

The same analysis also applies for other samples at bothsides of the block boundary. The end state after the applicationof is shown in Fig. 3(b). As one can see, the physicalboundary splits into two and extends inwards in both blocks.The next stage state transition will operate against these twonew boundaries. Notice that locations of these two new bound-aries can be derived easily using the filtering tap information

and before the stage updating operation. As a matterof fact, given the lifting factorization of the polyphase matrix

, one knows exactly the end state number of each samplein the input data sequence. Therefore, the state number of eachsample needs not to be stored for future processing.

Consequently, the buffer size at stage, , is only deter-mined by the number of samples that need to be stored. Forgiven filters , where is the numberof taps of the longest of the two filters. Assuming a total ofstate transitions, then the total buffer size is .For the (9,7) filters using the factorization given in [17], onecan obtain and thus . That is, onlyfour partially transformed samples need to be preserved ratherthan seven samples as before. We emphasize that the buffer sizereduction is based on a typical assumption in in-place liftingalgorithms that each sample (including original data samplesand partially transformed samples) needs the same amount ofstorage space. Since lifting factorization of a given polyphasematrix is not unique, one would choose the factorization whichgives the minimum if the amount of memory is limited.

We thus call this approach of preserving intermediate stateinformation and completing the transform later theboundarypostprocessingtechnique. Fig. 4 shows an example three-levelwavelet decomposition using the proposed technique. Com-pared to the approach shown in Fig. 1, one can see that, onlyone data exchange is necessary between the two blocks. Theamount of data exchanged is also reduced due to the partialcomputation as described above.

Fig. 4. Example dataflow chart of a three-level wavelet decomposition usingthe proposedboundary postprocessingtechnique. Solid lines: completelytransformed data. Dashed lines: partially transformed data. Operation 1: eachblock transforms its own allocated data independently and state informationis buffered. Operation 2: state information is communicated to neighboringblocks. Operation 3: complete transform for the boundary data samples.

Fig. 5. The proposed sequential architecture for DWT.

III. OVERLAP-STATE SEQUENTIAL ARCHITECTURE

In Fig. 5, the proposedoverlap-statesequential DWT systemarchitecture is shown. The input data sequence is segmented intononoverlapping blocks of length and fed into the DWT/FSMone block at a time. The state information is overlapped betweenconsecutive blocks. The computed wavelet coefficients are con-catenated together to give the final result.

As shown in the last section, the memory requirement de-pends on the specific lifting factorization used in the implemen-tation. Table I provides memory requirements for commonlyused wavelet filterbanks, including the Daubechies (9,7) filters,the (2,10) filters, and the cubic B-Splines CDF(4,2) filters; theirfactorizations can be found in [24]. A direct extension of 1-DDWT becomes the strip-sequential 2-D DWT as shown in Fig. 6,where the input data is transformed one strip at a time withstate information overlapped only vertically (boundary exten-sions are used horizontally). The buffer sizefor the state in-formation in a -level decomposition can be computed as

(4)


TABLE ICOMPARISON OFMEMORY REQUIREMENTS IN 1-D DWT

N: SEQUENCELENGTH. L: FILTER LENGTH. J: DECOMPOSITIONLEVEL.B: NUMBER OF PARTIALLY COMPUTED SAMPLES AT EACH LEVEL

Fig. 6. The strip-sequential 2-D DWT system on data of sizeW � H . Theinput is transformed one strip at a time from top to bottom. State information isoverlapped vertically only.

where is the data width (subscript “ ” stands forbottom side of the data strip) is the number of rows partiallycomputed at level.

A special case of this strip-sequential 2-D DWT system is theline-based DWT system described in [1], [10], which assumesall completely transformed coefficients are not buffered. Basedon this assumption, for one level decomposition using a-tapfilterbank, only rows need to be buffered in SSWT orRPA (if counting the two new input rows, the total buffer sizeshould be ). The memory requirement in the proposed systemis always upper-bounded by that of RPA and can have substan-tial reduction (approximately 40%) for commonly used waveletfilterbanks, as shown in Table II. For example, using the (9,7)filters, the proposed system only needs to buffer four rows (i.e.,

) of data at each level while RPA needs to buffer 7 rowsof data. The constant is due to line downsam-pling at each decomposition level. Refer to [24] for more detailson memory requirements and other 2-D DWT systems (e.g., theblock sequential system).

IV. SPLIT-AND-MERGEPARALLEL ARCHITECTURE

In Fig. 7, the proposed parallel DWT architecture is shown.The input data is uniformly segmented into nonoverlapping

TABLE IICOMPARISON OFMEMORY REQUIREMENTS IN LINE-BASED SYSTEMS

W: ROW WIDTH. B: NUMBER OF PARTIALLY COMPUTED SAMPLES

AT EACH LEVEL. � = (2 � 1). � = (1� 2 )

Fig. 7. Proposed parallel architecture for DWT.

blocks and allocated onto available processors. Each pro-cessor computes its own allocated data up to the requiredwavelet decomposition level. This stage is calledSplit. Theoutput from this stage consists of: 1) completely transformedcoefficients and 2) the state information (partially updatedboundary samples). In the second stage,Merge, a one-waycommunication is initiated and the state information is trans-fered to the neighboring processors. The state information fromthe neighbor processor is then combined together with its owncorresponding state information to complete the whole DWTtransform. This is further illustrated by Fig. 8, where it can beseen how the operations in each processor are carried as far aspossible in the split operation, while in the merge operation theprocessor will combine the available information to update thepartially computed outputs.

A. Communication Delay

The communication delay is the time used for communicatingdata between adjacent processors. Let be the communi-cation setup time, e.g., the handshake time in an asynchronouscommunication protocol. For a level wavelet decomposition,the total communication [4] is

(5)

where is the time to transfer one data sample andis the number of boundary samples exchanged to adjacent pro-cessor at each level. In the proposed approach, however, onlyone communication setup is necessary to communicate the stateinformation between adjacent processors. Furthermore, the sizeof the state information at each decomposition levelis upper


Fig. 8. An examplesplit-and mergeparallel DWT architecture using the Daubechies (9, 7) filters for two level decompositions. Shaded boxes represent partiallyupdated samples to be exchanged between processors in themergestage. Notice that samples {25, 26, 27} are left unprocessed for clarity and the block boundarycan be at any one of these three samples.

bounded by [24]. So the upper bound of the communi-cation delay is

(6)

As one can see, using theboundary postprocessingapproach,the communication overhead for link setup is reduced. Noticethat the comparison is based on the assumption that all datasamples which have to be buffered and exchanged (be it an orig-inal data sample or a partially computed intermediate sample)require the same storage space. This is a reasonable assump-tion when the in-place lifting algorithm is used to compute thewavelet transform. The reduction of the communication timecertainly contributes to the total DWT computation time reduc-tion. However, we mention that the overall contribution dependson the relative weight of the communication overhead in thetotal DWT computation. In general, more gain can be achievedfor parallel systems with slow communication links.

To test the efficiency of the proposed parallel architecture,four different DWT algorithms are implemented using twoSUN ULTRA-1 workstations running the LAM 6.1 parallelplatform developed in Ohio Supercomputer Center [14]. Thealgorithms compared are the sequential lifting algorithm,parallel standard algorithm (subsample-filtering approach),parallel lifting scheme, and our proposed parallel scheme. Thetest wavelet filters are the (9,7) filters and the test image isthe Lena grayscale image of size . A 2-D separablewavelet transform is implemented with strip data partition [4]between processors (refer to Fig. 6). We implemented all the al-gorithms ourselves with no specific optimizations on the codes

TABLE IIIDWT RUNNING TIME AND SPEEDUP OFDIFFERENTPARALLEL ALGORITHMS

(IN SECONDS). S: SEQUENTIAL. P: PARALLEL . L: LIFTING

except for the compiler optimization. Further improvement ofalgorithms performances is certainly possible through morerigorous code optimizations. To compare the performances, therelative speedup is computed and the averagedresults over 50 running instances for 1–5 decomposition levelsare given in Table III.

It can be seen from the results that our proposed parallel algo-rithm can significantly reduce the DWT computation time evencompared with the fastest available parallel algorithm, i.e., theparallel lifting algorithm. Notice that the improvement is notlinear with the increase of the decomposition level. The reasonis that, though the communication overhead increases with thedecomposition level, the total numerical computation also in-creases. Another observation is that the improvement of the pro-posed algorithm at one level decomposition is even greater thanthat at multiple level decompositions. It suggests that, in ourexperimental system setup, the gain due to saved amount of ex-changed data is greater than that due to saved number of com-


munication setups, though, future work is needed to further in-vestigate this issue.

V. CONCLUSIONS

To conclude, we have proposed a newboundary post-processingtechnique for the DWT computation near blockboundaries. Performance analysis and experimental resultsshow that the auxiliary buffer size for boundary DWT and thecommunication overhead can be significantly reduced by usingthe proposed technique. The results presented here can be easilyextended to 2-D or higher dimensional wavelet transforms byusing separable transform approaches [24].

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Lifting factorization-based discrete wavelet transform architecture design