IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 1, JANUARY 2007 71

Discrete Lattice Wavelet TransformJuuso T. Olkkonen and Hannu Olkkonen

Abstract—The discrete wavelet transform (DWT) has gained awide acceptance in denoising and compression coding of imagesand signals. In this work we introduce a discrete lattice wavelettransform (DLWT). In the analysis part, the lattice structure con-tains two parallel transmission channels, which exchange informa-tion via two crossed lattice filters. For the synthesis part we showthat the similar lattice structure yields a perfect reconstruction(PR) property. The PR condition can be used to design half-bandfilters, which effectively eliminate aliasing in decimated tree struc-tured wavelet transform. The DLWT can be implemented directlyto any of the existing DWT algorithms.

Index Terms—Finite-impulse response (FIR) filters, lattice fil-ters, quadrature mirror filters (QMFs), wavelet transform.

I. INTRODUCTION

DISCRETE wavelet transform (DWT) has attained anenormous usage in signal processing society since the

discovery of the compactly supported Daubechies wavelets [1].Applications of the wavelet theory are exponentially increasingin image compression [2]–[4], noise cancellation [5], multi-scale signal analysis [6]–[8], adaptive system identification[9], and signal approximation and interpolation [10]. Manytheoretical papers treat the solution of the scaling (low-pass)and wavelet (high-pass) filters and , which areeither quadrature mirror filters (QMFs) or conjugate quadraturefilters (CQFs). The scaling and wavelet filters are orthogonaloriginating from the same prototype filter. In QMF bank

and in CQF bank[1], [11]. It is well known that only the QMF Haar wavelettransform has the perfect reconstruction (PR) property. Theorthogonal CQF filter banks obey the PR condition, but thescaling and wavelet filters have no symmetrical coefficients andhence the nonlinear effects cause image distortions and spatialinaccuracies in multiscale analyses.

Sweldens [12] discovered the lifting scheme based on thebiorthogonal wavelets. The lifting wavelet transform consists ofthe ladder network, where the low-pass filtered and high-passfiltered channels exchange information sequentially. Laddernetwork can be divided into the up and down lifting steps andthe reconstruction of the signal can be done by carrying outthe lifting steps in reverse order. The lifting scheme yieldsbiorthogonal scaling and wavelet filters, which are no moreoriginated from the same prototype filter. The support (length

Manuscript received November 17, 2005: revised March 24, 2006 and June16, 2006. This work was supported by the National Technology Agency of Fin-land (TEKES). This paper was recommended by Associate Editor Y.-P. Lin.

J. T. Olkkonen is with VTT Technical Research Centre of Finland, 90571Oulu, Finland (e-mail: juuso.olkkonen@vtt.fi).

H. Olkkonen is with the Department of Applied Physics, University ofKuopio, 70211 Kuopio, Finland (e-mail: hannu.olkkonen@uku.fi).

Digital Object Identifier 10.1109/TCSII.2006.883097

Fig. 1. Analysis and synthesis parts of the two-channel DWT.

of the filter) and the frequency response of the scaling andwavelet filters differ. This may lead to difficulties when thetotal energy of the wavelet coefficients in different scales is ofessential importance. On the other hand, when the scaling andwavelet filters are symmetric and linear phase, the lifting stepscan be carried out with only register shifts and summations inhardware implementations [13].

The two-channel DWT structure is described in Fig. 1. Ac-cording to Daubechies regulatory condition [1], [14] the fre-quency characteristics of the low-pass scaling and thehigh-pass wavelet filters are related to the binomial term

, which has the th-order root at . The bino-mial term has relatively poor low- and high-pass filter charac-teristics, which introduces aliasing in the tree structured waveletchain. In this work, we describe a discrete lattice wavelet trans-form (DLWT) which improves the low- and high- pass filtercharacteristics of the scaling and wavelet channels. We intro-duce a new design method for the transmission and lattice filtersbased on the lattice structure.

II. LATTICE STRUCTURE

In the analysis part, the two-channel DLWT consists of thescaling and wavelet filters and the lattice network,where the parallel channels having the transmission filtersand exchange information via two crossed lattice filters

and (Fig. 2). In the synthesis part the lattice con-sists of the two transmission filters and and crossedfilters and , and finally the reconstruction filters

and .The demand for the DLWT is the PR property, i.e., the output

of the DLWT chain should be only a delayed version of the inputsignal (delay , ). We may assume that the scaling,wavelet and the reconstruction filters obey the PR condition [14]

(1)

The PR condition for the lattice network (Fig. 3) is

(2)

1057-7130/$25.00 © 2007 IEEE

72 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 1, JANUARY 2007

Fig. 2. General structure for the two-channel DLWT.

Fig. 3. Lattice network.

where and are the -transforms of the input signalsand and are the -transforms the output signals. Inthe general case (Fig. 3) we have

(3)The antidiagonal elements of the network matrix are zero if westate

(4)

From the diagonal elements (3) we then have the PR condition

(5)

III. DESIGN OF LATTICE FILTERS

For the DLWT structure, there appears several approaches inthe design of the lattice filters obeying the PR condition (5). Themost general case is obtained if we define a polynomial(for odd)

(6)

where is a polynomial in . The PR condition (5) is validwhen we relate

(7)

which yields the final PR condition

(8)

We may parameterize the as

(9)

which has the frequency response

(10)

can be designed by the Parks–McClellan-type algorithm.The frequency response is exactly symmetrical with re-spect to . Then by a spectral factorization method isfactored into and . With increasing we have a va-riety of polynomials obeying (8). In the following we show thatthere exists a short polynomial solution, which yields the low-and high-pass transmission and lattice filters.

IV. HALF-BAND TRANSMISSION AND LATTICE FILTERS

As a special case we have studied a polynomial

(11)

The roots of the polynomial ( odd) are evenly distributedon the unit circle in the -plane. Hence, can be separatedinto two filters

(12)

It is reasonable to select such that the complex roots occuras complex conjugate pairs and the coefficients of the separatedfilters and are real. has all roots on the lefthalf-circle and it has low-pass filter characteristics and the cu-foff frequency at approximately . has all roots

OLKKONEN AND OLKKONEN: DLWT 73

Fig. 4. Magnitude responses of the low-passT (z) and the high-pass T (z) transmission filters for k = 21. The corresponding lattice filters areL (z) = T (�z)and L (z) = T (�z).

on the right half-circle and it has the high-pass filter charac-teristics. The lattice filters and are the QMFs inrespect with and (7). Hence,has low-pass and high-pass filter character-istics. The following Matlab™ program separate.m computesthe and filter coefficients. Real coefficients are ob-tained for ,5,9,13,17,21 etc. Fig. 4 shows an example for

.

1) Design Example: Let us consider a wavelet transformobeying the PR condition (1). We select the scaling and waveletfilters as and and the reconstructionfilters as and . If the input signal is

, the decimated scaling and waveletcoefficients are (Fig. 2)

(13)

Since and are low-pass half-band filters, scalingcoefficients contain the low-frequency components of theeven and odd parts of the signal. On the other hand,

and are high-pass half-band filters andthe low-frequency components of the even and odd parts of thesignal are predominating in the wavelet coefficients . Fig. 5demonstrates the DLTW analysis of the 1024 point video signal.Fig. 5(a) and (b) shows the scaling and wavelet coefficients com-puted via (13) using the transmission and lattice filters producedby the Matlab program separate.m for . The next scaleis illustrated in Fig. 5(c) and (d), where the scaling coefficients[Fig. 5(a)] are decomposed into the scaling and wavelet coeffi-cients.

In the case of the general QMF bankand . For the input

signal we have (Fig. 2)

(14)

The decimated scaling and wavelet coefficients are then

(15)

In the case of Haar wavelet transform and.

74 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 1, JANUARY 2007

Fig. 5. Tree structured DLWT analysis of a 1024-point video signal. The scaling (a) and the wavelet coefficients (b) of the original signal. The second scale scaling(c) and wavelet coefficients (d) are computed from the first scale scaling coefficients (a).

Fig. 6. DLWT-based construction of the dual-channel multiplexer-demultiplexer unit.

V. DISCUSSION

In this work, we have introduced a new two-channel DLWTalgorithm, where the parallel channels exchange informationsimultaneously via the lattice filters. The PR condition of theDLTW yields effective half-band transmission and lattice filters,which are QMFs. From the design example we may observe thatthe even and odd parts of the original signal are filtered by theQMFs to yield the scaling and wavelet coefficients (13). Hence,the total energy of the scaling and wavelet coefficients is pre-served. This is a valuable advantage in multi-scale analysis ofsignals, where the energy of the wavelet coefficients is statisti-cally compared in different scales. The total energy of the coef-ficients is also preserved in the case of the general QMF filterbank (15). However, the only real-valued QMF filter bank so-lution is only known for the Haar wavelettransform. For the DLWT of the complex signals we know sev-eral complex QMF filter bank solutions [8] obeying the PR re-lation (1) and we may apply the result (15).

The half-band filtered parallel information channels haveminimally overlapping frequency bands, which diminishchannel cross-talk in wavelet transmission systems. The DLWT

offers a novel tool for design of multiplexer-demultiplexer(mux-demux) information channels such as data transmissionin optical fiber networks and telemetric multi-channel equip-ments, where parallel data channels should be multiplexed intoone signal, transmitted and then reconstructed. Fig. 6 shows anexample of the dual-channel mux-demux arrangement usingthe DLWT synthesis and analysis parts.

In the tree structured wavelet transform, the half-band filteredscaling coefficients introduce no aliasing when they are fed tothe next scale. This is an important feature when the frequencycomponents of the wavelet coefficients are considered. From theDLWT analysis of the video signal (Fig. 5) we may observe thatscaling and wavelet signals have distinctly different waveformcharacteristics. On the other hand, the second scale scaling coef-ficients [Fig. 5(c)] appear as a smoothed version of the first scalecoefficients [Fig. 5(a)]. The wavelet coefficients in both scales[Fig. 5(b) and (d)] have plenty of small values, which allow ef-fective denoising and compression algorithms.

In this work, we have solved the PR condition of the latticenetwork [(5), (8)] by introducing a simple polynomial solution(11). However, we have a family of polynomials obeying PRcondition (8), which can be designed by the Parks–McClellan-

OLKKONEN AND OLKKONEN: DLWT 75

type method. The constants in (9) can be solved fromthe boundary values of the polynomial. This can be madein frequency domain using (10). We may select the zeros onthe left half plane and equal number of zeros at the right halfplane. If we want an exact solution, the number of zeros mustequal the number of constants in (9). The number of differentsolutions is countless, since we may vary the position of thezeros and the lengths of the resulting and . We mayconclude that the new lattice structure, which can be realized byfrequency selective FIR filters, extends the DWT concept. Thismay give rise to the more effective methods in wavelet-basedsignal analysis and transmission systems.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their helpfulcomments.

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