DTIC ADA455107: Fast Reconstruction of Subband-Decomposed Progressively Transmitted Signals

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T R R
ECHNICAL ESEARCH EPORT
Fast Reconstruction of Subband-Decomposed
Progressively- Transmitted, Signals
by H. Jafarkhani, N. Farvardin
T.R. 97-18
ISR
INSTITUTE FOR SYSTEMS RESEARCH
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the National Science Foundation
Engineering Research Center Program,
the University of Maryland,
Harvard University,
and Industry
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Fast Reconstruction of Subband-Decomposed Progressively-Transmitted
5b. GRANT NUMBER
Signals
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unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
(cid:3)
Fast Reconstruction of Subband-Decomposed Progressively-Transmitted Signals
Hamid Jafarkhani
Nariman Farvardin
Department of Electrical Engineering and
Institute for Systems Research
University of Maryland
College Park, MD 20742
ABSTRACT roughreproductionofamedicalimageinatelemedicine
application, the radiologist, at the receiver, may want
In this paper we propose a fast reconstruction method
to highlight a part of the image and request a higher
for a progressive subband-decomposed signal coding sys-
(cid:12)delity (or even lossless) replica of only the highlighted
tem. It is shown that unlike the normal approach which
area.
contains a (cid:12)xed computational complexity, the compu-
Theuseofsubbanddecompositionordiscretewavelet
tational complexity of the proposed approach is propor-
transform(DWT) inimage coding systems has received
tional to the number of re(cid:12)ned coe(cid:14)cients. Therefore,
muchattentioninrecent years[1]-[3]. Notonlydothese
using the proposed approach in image coding applica-
codingtechniquesprovidegoodcompressionresults,but
tions, we can update the image after receiving each new
also they are inherentlymulti-resolution. Recently, sev-
coe(cid:14)cient and create a continuously re(cid:12)ned perception.
eral rate-scalable subband image coding systems have
This can be done without any extra computational cost
beenproposedintheliteraturewhichprovideverygood
compared to the normal case where the image is recon-
performances [4, 5, 6, 7].
structed after receiving a prede(cid:12)ned number of bits.
One of the problems in a wavelet-based progressive
1.INTRODUCTION transmissionscheme isthat thedecoder, uponreceiving
new bits, has to perform the inverse (cid:12)ltering operation
Progressive transmission of signals is a mechanism by
to reconstruct the image. The normal approach for do-
which the encoder’s output is transmitted in groups of
ingthisistoapplytheinverse(cid:12)ltersonthedecodedver-
bits (packets) and the decoder produces a higher qual-
sions of all wavelet coe(cid:14)cients to reconstruct a replica
ity replica of the signal based on receiving each new
of the image. Inthisapproach, even ifone wavelet coef-
packet. Progressive transmission is a desirable feature
(cid:12)cient is re(cid:12)ned, the reconstruction complexity will be
in many practical signal transmissionsituations such as
the same as in the case when all coe(cid:14)cients are re(cid:12)ned.
telebrowsing and database retrieval. Progressive trans-
In a progressive transmission scenario however, at
mission also provides the opportunity for interrupting
each step of progression, only some of the wavelet coef-
transmissionwhenthe qualityof the received signalhas
(cid:12)cients are re(cid:12)nedand the other coe(cid:14)cients remain un-
reached an acceptable level or when the receiver de-
changed. Therefore, only a portion of the image pixels
cidesthat the received signalis of no interest. Likewise,
will change after receiving the new bits. To reduce the
in applications where the receiver is more interested in
complexity of reconstruction, one can recompute only
speci(cid:12)c parts of the signal rather than the entire signal
those pixels of the image that need to be changed. Not
(e.g., content-based image transmission), a valid ques-
only does our proposed scheme provide a fast recon-
tion is how to transmit and reconstruct a signal with
struction of the output, but also it can update the out-
di(cid:11)erent levels of quality (distortion) in di(cid:11)erent tem-
putaseachwavelet coe(cid:14)cientarrivesanddoesnotneed
poral or spatial regions. For example, after receiving a
to wait for receiving all packets before it starts the pro-
(cid:3)
Prepared in part through collaborative participation in cess of reconstruction. This can be done without any
the Advanced Telecommunications/Information Distribution Re-
increase in complexity. This feature of our reconstruc-
searchProgram(ATIRP)ConsortiumsponsoredbytheU.S.Army
tion scheme is not limited to progressive transmission
Research Laboratory under Cooperative AgreementDAAL01-96-
2-0002 and in part through support received from the National systems and can be used for any packetized bit stream
Institute of Standards andTechnology grant 70NANBSH0051. for on-line reconstruction of the output.
1
In Section 2, we propose a new way of reconstruct- Now, let us consider a progressive transmission scheme
ing the signal which allows for updatingonly the neces- where yl(n), l =1;2;(cid:1)(cid:1)(cid:1);K, represents the di(cid:11)erent re-
saryportionsofthesignal,insteadofreconstructingthe (cid:12)nements of the output. The re(cid:12)nement of the out-
whole thing. The complexity of thisnew reconstruction put can be achieved by re(cid:12)ning the input coe(cid:14)cients
i
approach is proportional to the number of re(cid:12)ned coef- successively using bl(n) bits/sample to quantize xi(n)
i
(cid:12)cients. Section 3 generalizes the approach of Section (i = h;g) at level l. The values of bl(n) are de-
2 to a general (cid:12)lter bank structure for subband recon- (cid:12)ned by a bit allocation algorithm or are speci(cid:12)ed in-
i
struction and Section 4 contains concluding remarks. herently in the coding system. Note that bl(n) (cid:20)
i
bl+1(n) and it is possible { and crucial to note {
i i
2.ONE LEVEL OF DECOMPOSITION that we might have bl(n) = bl+1(n), for some i. We
i i i
use xi(n;b1);xi(n;b2);(cid:1)(cid:1)(cid:1);xi(n;bK) to represent the de-
In signal coding based on the DWT, the input signal is coded versions of the subband coe(cid:14)cients xi(n), at dif-
i
typically decomposed into two components: (i) a low- ferent levels of re(cid:12)nement (xi(n;b1) is the coarsest rep-
i
resolution approximation and (ii) a detail signal. This resentation of xi(n) using b1(n) bits for quantization
i i
results in decomposing the input signal into low-pass and xi(n;bK) is the (cid:12)nest representation using bK(n)
and high-pass versions, generally referred to as sub- bits for quantization).
bands. Each of the resulting subbands can be further Using the normal method of reconstruction in sub-
decomposed using the same approach. In this manner, band coding systems, for each l = 1;2;(cid:1)(cid:1)(cid:1);K, upon
the DWT decomposes a given input signal into a num- obtaining xh and xg, the decoder reconstructs the lth
ber of frequency bands [2]. At the receiver, the signal re(cid:12)nement of the output, yl, according to:
canbereconstructedusingappropriateinverseDWT(cid:12)l-
Nh n(cid:0)m h
ters. Fig. 1 shows one level of two-band decomposition yl(n)= m=(cid:0)Nhxh( 2 ;bl)h(m)+
and reconstructionusinglinear-phase,(cid:12)nite-impulsere- Ng n(cid:0)m g (5)
mP=(cid:0)Ngxg( 2 ;bl)g(m):
sponse (cid:12)lter banks [3].
In this section, to convey the basic idea behind the In other wordPs, given that yl is reconstructed at the
proposed fast reconstruction approach, we limit our- decoder, upon receiving additional bits from which
h g
selves to a one-level decomposition as in Fig. 1 where xh(n;bl+1) and xg(n;bl+1) { the (l+1)st re(cid:12)nement of
the reconstructed output signal can be written as: xh and xg { are obtained, the inverse DWT (cid:12)lters are
used anew to compute an updated replica of the signal
Nh 0
y(n)= m=(cid:0)Nhxh(n(cid:0)m)h(m)+ { the (l+1)st re(cid:12)nement of y.
Ng 0 (1) In this paper, we propose an alternative method
mP=(cid:0)Ngxg(n(cid:0)m)g(m);
in which we de(cid:12)ne (cid:1)yl(n) = yl+1(n) (cid:0) yl(n) and
where P (cid:1)xi(n;bil) = xi(n;bil+1) (cid:0) xi(n;bil), i = h;g. Clearly,
(cid:1)yl, l =1;2;(cid:1)(cid:1)(cid:1);K, can be computed as follows:
n
x0h(n) = xh(2) n even ; Xh0(z) = Xh(z2); Nh n(cid:0)m h
( 0 n odd (cid:1)yl(n)= m=(cid:0)Nh(cid:1)xh( 2 ;bl)h(m)+
0 xg(n2) n even 0 2 Nmg=P(cid:0)Ng(cid:1)xg(n(cid:0)2m;bgl)g(m): (6)
xg(n)= ; Xg(z) =Xg(z ); (2)
( 0 n odd P
Then, to reconstruct a new re(cid:12)nement of the signal,
and 2Nh +1 and 2Ng +1 are the lengths of the h and yl+1, it su(cid:14)ces to use
g (cid:12)lters, respectively.
yl+1(n) =yl(n)+(cid:1)yl(n): (7)
Tosimplifythenotation, wede(cid:12)nexh(a) = xg(a) =0
if a 2= ZZ. So,
Nowletusconsiderthecomplexityimplicationofthis
0 n new approach and its comparison with the normal ap-
xi(n) =xi( ); i=h;g: (3)
2 proach using (5). If the number of samples of the input
signal (x) is N, then the number of samples in xh and
Usingthe above notation, the output can bewrittenas: N
xg willbe 2. If, at each levelof re(cid:12)nementl, we use(5)
N N
Nh n(cid:0)m to compute yl, it costs L = 2(2Nh +1)+ 2(2Ng +1)
y(n)= m=(cid:0)Nhxh( 2 )h(m)+
Ng n(cid:0)m (4) multiplications, a quantity which is independent of the
mP=(cid:0)Ng xg( 2 )g(m): number of re(cid:12)ned samples. Therefore, after K levels of
P
2
1
re(cid:12)nement the computational complexity is KL multi- the symmetric extension of the input signal . Receiv-
plications. Using(6)and(7)toupdatetheoutputsignal ing (cid:1)xh(k) for the coe(cid:14)cients su(cid:14)ciently far from the
Nh N(cid:0)Nh
costs 2Ni+1multiplicationspereachnonzero (cid:1)xi coef- boundaries, 2 < k < 2 , a(cid:11)ects only the output
N N
(cid:12)cient, i = h;g (number of nonzero (cid:1)xi (cid:20) 2). If (cid:11)il 2 values y(2k +n); n = (cid:0)Nh;(cid:0)(Nh (cid:0)1);(cid:1)(cid:1)(cid:1);Nh. When
Nh
wavelet coe(cid:14)cients (xi, i = h;g) are re(cid:12)ned at the lth the received coe(cid:14)cient is near the origin, 0 (cid:20) k (cid:20) 2 ,
level of re(cid:12)nement, then the total number of multipli- the output samples y(n); n = 0;1;(cid:1)(cid:1)(cid:1);Nh+2k, are af-
N(cid:0)Nh N
cations using (6) and (7) to update the output signal is fected. For the remaining coe(cid:14)cients, 2 (cid:20) k < 2,
Kl=1(cid:11)hlN2(2Nh+1)+ Kl=1(cid:11)glN2(2Ng +1). Note that the output values y(2k+n); n=(cid:0)Nh;(cid:1)(cid:1)(cid:1);N (cid:0)2k, are
K a(cid:11)ected. Replacingxh withxg andNh withNg provides
Pfor large K’s, l=1(cid:11)il <P< K for existing rate-scalable
the same indices for the (cid:12)lter g and coe(cid:14)cients xg.
image coding systems [4, 5, 6, 7] and therefore the com-
P
Thesameprocedurecanbeappliedtomorethanone-
plexity of the proposed approach is much less than the
levelofdecomposedsignals. Thetwo-dimensional(2-D)
complexityofthenormalapproach. Forexample,inthe
extension of the approach is also straightforward when
embedded zerotrees wavelet (EZW) coding approach of
we use a separable 2-D transform.
[4], not only are there many insigni(cid:12)cant coe(cid:14)cients at
lowbitrateswhicharenottransmittedandareassumed
tobezero,butalsothesigni(cid:12)cantcoe(cid:14)cientsarere(cid:12)ned 3.GENERAL FILTER BANK
in dominant and subordinatepasses (at each pass some
of the \signi(cid:12)cant" coe(cid:14)cients remain unchanged). In this section, we extend the idea for complexity re-
duction developed in the previous section to general (cid:12)l-
In addition to reducing the complexity of reconstruc-
ter banks. Fig. 2 illustrates a general decomposition
tion, using (6) and (7) to reconstruct the output pro-
and reconstruction structure using a set of passband
vides the capability to update the output signal upon (cid:12)lters called (cid:12)lter banks. This structure can be used
receivingthe newre(cid:12)nement ofeach wavelet coe(cid:14)cient,
to perform linear transformations like the discrete co-
thusallowingforanon-lineupdateoftheoutput. Typi-
sine transform, lappedorthogonal transform, Laplacian
cally,foranysignalcodingsystemtheoutputbitstream
pyramid, Gabor transform, quadrature mirror (cid:12)lters,
is transmitted in the form of packets. Consider a gen-
and DWT [8]. Although there exist \fast implemen-
eral signal coding system (or one level of re(cid:12)nement of
tations" for many of these transformations, the study
a progressive system) and assume that the output is
of this general structure is worthy of consideration as
transmitted using M packets. The size of each packet
it uni(cid:12)es the application of our approach for di(cid:11)erent
and thus the value of M depends on the network char-
transformationsandprovidesageneralmethodologyfor
acteristics. At the receiver, either we should wait to
progressive transmission situations. Furthermore, our
receive all M packets or we can reconstruct M inter-
approach does not depend on the speci(cid:12)c transforma-
mediate versions of the output { one for each newly re-
tion used. In fact, the ratio of the computational com-
ceived packet. Usingthe normalreconstruction formula
plexity of the proposed approach to that of the normal
(5), resultsina delay intheformercase andanincrease
approach for reconstruction is (cid:12)xed for any transforma-
in computational complexity by a factor of M in the
tion. So, the choice of the transformation a(cid:11)ects the
latter case. Using (6) and (7) to reconstruct the out-
complexity of both approaches in the same manner.
put will provide M intermediate versions of the output
To make the argument more precise, (cid:12)rst, let us con-
without increasing the complexity or delay. Also, note
sidertheexampleillustratedinFig.3consistingofthree
thatifafterreceivingaroughreproductionofthesignal,
levels of the two-band structure in Fig. 1. Later, we
we just need to re(cid:12)ne a portion of wavelet coe(cid:14)cients
generalize our fast reconstruction approach to the (cid:12)lter
(correspondingtoaspatiallocation),thecomputational
bank shown in Fig. 2. The example in Fig. 3 is equiv-
cost using (6) and (7) is proportional to the number of
alent to Fig. 2 when k1 = 2, k2 = 4, k3 = k4 = 8 and
transmitted coe(cid:14)cients ((cid:12)lter inputs) and less than the
(cid:12)lters gi and hi, i = 1;2;3;4, are chosen appropriately.
computational cost of the normal scheme.
Given (cid:12)lters h and g inFig. 3, the correspondingrecon-
struction (cid:12)lters in Fig. 2 can be found. The main equa-
To update the output, we need to know the set of
tion governing the relationship between gi, i = 1;2;3;4
output samples whose values are in(cid:13)uenced by the re-
and g and h is the equivalence of the two structures in
(cid:12)nement of each coe(cid:14)cient. This set depends on the
1
location of the re(cid:12)ned coe(cid:14)cient. Also, the samples Tohaveasmoothboundaryafterreconstruction,thesignalis
neartheboundariesneedaspecialtreatment becauseof symmetricallyextendednear theboundaries.
3
Fig. 4 [8]. Using Fig. 4, one can establish the following Now, we concentrate on the mainissueof thissection
equations relating di(cid:11)erent (cid:12)lters in Figs. 2 and 3: and show that an approach similar to what was pro-
posedinSection 2 can beused for the (cid:12)lterbank shown
G1(z) = G(z); (8)
in Fig. 2. To simplify the notation, we de(cid:12)ne xi(a) = 0
2
G2(z) = G(z )H(z); (9) if a 2= ZZ. So,
4 2
G3(z) = G(z )H(z )H(z); (10) 0 n 0 n
xi(n) =xi( i); i=1;2;3; and x4(n) =x4( ): (15)
2 8
and
4 2 Usingthe above notation, the output can be writtenas:
G4(z) =H(z )H(z )H(z): (11)
N1
n(cid:0)m
Although the reconstruction operation in Figs. 2 and y(n)= x1( )g1(m)+
3 are the same, the number of multiplications in Fig. 2 m=(cid:0)N1 2
X
is more than the number of multiplications in Fig. 3. N2
n(cid:0)m
Speci(cid:12)cally the output signal in Fig. 2 can be written x2( )g2(m)+
as: m=(cid:0)N2 4
X
N3
N1 n(cid:0)m
0 x3( )g3(m)+
y(n)= x1(n(cid:0)m)g1(m)+ m=(cid:0)N3 8
m=(cid:0)N1 X
X N4
N2 n(cid:0)m
0 x4( )g4(m): (16)
x2(n(cid:0)m)g2(m)+ m=(cid:0)N4 8
m=(cid:0)N2 X
X
N3 In the normal approach for reconstructing the output
0
x3(n(cid:0)m)g3(m)+ of a progressive transmission scheme, the di(cid:11)erent lev-
m=(cid:0)N3 els of re(cid:12)nement of the output, y1;y2;(cid:1)(cid:1)(cid:1);yK, are com-
X
N4 0 putedafter receivingxi(n;bi1);xi(n;bi2);(cid:1)(cid:1)(cid:1);xi(n;biK) as
x4(n(cid:0)m)g4(m); (12) follows:
m=(cid:0)N4
X N1
n(cid:0)m 1
where yl(n)= x1( ;bl)g1(m)+
m=(cid:0)N1 2
n X
0 x1(2) n even 0 2 N2
x1(n) = ; X1(z) =X1(z ); n(cid:0)m 2
( 0 n odd x2( ;bl)g2(m)+
m=(cid:0)N2 4
x02(n) =( x02(n4) not=her4wkise ; X20(z) =X2(z4); XN3 x3(n(cid:0)m;b3l)g3(m)+
n m=(cid:0)N3 8
0 x3(8) n=8k 0 8 X
x3(n) = ; X3(z) =X3(z ); N4
( 0 otherwise n(cid:0)m 4
x4( ;bl)g4(m): (17)
x04(n) = x4(n8) n=8k ; X40(z) =X4(z8); (13) m=X(cid:0)N4 8
( 0 otherwise As before, we de(cid:12)ne (cid:1)yl(n) = yl+1(n) (cid:0) yl(n) and
i i i
and 2Ni + 1 is the length of gi and N1 = Ng, N2 = (cid:1)xi(n;bl) = xi(n;bl+1) (cid:0) xi(n;bl), where (cid:1)yl; l =
1;2;(cid:1)(cid:1)(cid:1);K; is given by:
2Ng +Nh, N3 = 4Ng +3Nh, and N4 = 7Nh. So, using
N1
all coe(cid:14)cients to reconstruct the output, we need L1 = n(cid:0)m 1
N[2N12+1 + 2N42+1 + 2N38+1 +12N48+11] 1multiplications1in (cid:1)yl(n)= m=(cid:0)N1(cid:1)x1( 2 ;bl)g1(m)+
Fig. 2 andL2 = N[(2Nh+1)(8+4+2)+(2Ng+1)(8+ X
1 1 N2
4+2)] multiplicationsin Fig. 3. After computing L1 in n(cid:0)m 2
(cid:1)x2( ;bl)g2(m)+
terms of Nh and Ng, we can show that m=(cid:0)N2 4
X
L1 =N[1+3Ng+3Nh] >L2 = N[74(Nh+Ng+1)]; (14) N3 (cid:1)x3(n(cid:0)m;b3l)g3(m)+
m=(cid:0)N3 8
X
establishing that the number of multiplications corre- N4
n(cid:0)m 4
sponding to the implementation of Fig. 3 is less than (cid:1)x4( ;bl)g4(m): (18)
that of Fig. 2. m=(cid:0)N4 8
X
4
Then, given that yl(n) is available at the decoder, We have proposed a scheme for fast reconstruction of a
yl+1(n) can be computed by adding (cid:1)yl(n) to yl(n). subband-decomposed progressively transmitted signal.
Equation (18) is a generalization of (6) and therefore Using the proposed approach, we can update the image
provides a fast reconstruction scheme as was discussed after receiving each new coe(cid:14)cient and create a contin-
in Section 2. uously re(cid:12)ned perception without any extra computa-
To update the output, we need to know the list of tional cost (compared to the case that we reconstruct
outputsampleswhichmustbeupdateddueto receiving the image after receiving each packet). In existing scal-
each coe(cid:14)cient. Table 1 shows the requiredupdates for able image coding systems [4]-[7], insigni(cid:12)cant coe(cid:14)-
the coe(cid:14)cients that are su(cid:14)ciently far from the bound- cients remain zero at the decoder for a few steps of re-
aries. The (cid:12)rst row contains the (cid:12)lter coe(cid:14)cients which (cid:12)nement. Also, at each step of the re(cid:12)nement, some
are used to update the corresponding outputs in the of the coe(cid:14)cients remain unchanged. Unre(cid:12)ned coe(cid:14)-
second row. To make the details of the reconstruction cients do not add to the computational complexity of
scheme more clear, let us provide a numerical example. our new approach for reconstructing the image. On the
otherhand,thecomplexityofthenormalreconstruction
3.1.EXAMPLE
scheme does not depend on the value of the coe(cid:14)cients
Although our scheme works for any scalable transform (or re(cid:12)nements).
coding system, we pick a particular coding system to In some applications, after receiving a preliminary
showthe practicalusefulnessof ourscheme. We usethe draft of the image, the receiver only needs the upgrade
EZW approach of [4] for this purpose. In [4], an 8(cid:2)8 ofaparticularportionoftheimage. Usingourproposed
wavelet decomposed imagehasbeenusedto presentthe approach, the complexity reduction in thiscase is equal
details of the EZW algorithm. We use the same exam- to the ratio of the number of pixelsin the required area
ple (Table 2) and assume that re(cid:12)ning each coe(cid:14)cient to the number of pixels in the whole image.
a(cid:11)ects all output pixels in the same row or column (a Futureworkincludesthegeneralizationofthescheme
logical assumption on the length of (cid:12)lters for an 8(cid:2)8 for (cid:12)lter banks with integer coe(cid:14)cients which replace
2
image). Thenormalreconstructionapproachtakes1024 multiplications by binary shifts.
multiplicationsto create each re(cid:12)nement of the output.
Without going through the details of the EZW, as it
is shown in [4], the (cid:12)rst dominant pass results in re- References
(cid:12)ning coe(cid:14)cients in locations 11, 12, 13, and 54 (the
numbers indicate the row and column indices, respec- [1] J.W.Woods, andS.D.O’Neil,\SubbandCodingof
tively). Reconstructing the image using our approach Images," IEEE Trans. Acoust., Speech and Signal
takes 160 multiplicationsinsteadof 1024. The(cid:12)rstsub- Proc., vol. ASSP-34, pp. 1278{1288, Oct. 1986.
ordinatepass(thesecondlevelofre(cid:12)nement)re(cid:12)nesthe
same four coe(cid:14)cients [4] and thus results in the same [2] S. G. Mallat, \A Theory for Multiresolution Sig-
amount of computational reduction. The second dom- nalDecomposition: TheWavelet Representation,"
inant pass (the third level of re(cid:12)nement) changes coef- IEEE Trans. Pattern Anal. and Mach. Intel., vol.
(cid:12)cients in locations 21 and 22. Our approach requires 11, pp. 674{693, Jul. 1989.
80 multiplications to reconstruct the third image. The
second subordinatepass (the fourth levelof re(cid:12)nement) [3] M. Antonini, M. Barlaud, P. Mathieu, and I.
re(cid:12)nes the aforementioned six wavelet coe(cid:14)cients. To Daubechies, \Image Coding Using Wavelet Trans-
create the fourthimage, ourapproachrequires240 mul- form," IEEE Trans. Image Proc., vol. IP-1, pp.
205{220, Apr. 1992.
tiplications. Therefore, to reconstruct the (cid:12)rst four im-
ages,thefastschemeneedsatotalof640multiplications
[4] J. M. Shapiro, \Embedded image coding using ze-
compared to 4096 multiplications. Also, note that since
rotrees of wavelet coe(cid:14)cients," IEEE Trans. on
we have had 16 coe(cid:14)cient re(cid:12)nements in the (cid:12)rst four
Signal Proc., vol. 41, pp. 3445-3462, December
passes,wecouldhave reconstructed16 intermediateim-
1993.
ages without any additional computational complexity.
Constructingthese16imageswouldcost16384multipli- 2
The views and conclusions contained in this document are
cations using normal wavelet reconstruction formulas. those of the authors and should not be interpreted as represent-
ing the o(cid:14)cial policies, either expressed or implied, of the Army
4.CONCLUSIONS Research Laboratory or the U.S. Goverment.
5
[5] A. Said and W. A. Pearlman, \A new fast and ef-
(cid:12)cient image codec based on set partitioning in hi-
erarchical trees," submitted to IEEE Trans. Circ.
& Syst. Video Tech..
[6] D. Taubman and A. Zakhor, \Multirate 3-D sub-
band coding of video," IEEE Trans. on Image
Proc., vol. IP-3, pp. 572-588, September 1994.
[7] H. Jafarkhani and N. Farvardin, \A Scalable
Wavelet Image Coding Scheme Using Multi-
Stage Pruned Tree-Structured Vector Quantiza-
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[8] M. Vitterli, \Multirate Filter Banks for Subband
Figure 3: Wavelet Synthesis Filter Bank.
Coding," in Subband Image Coding, J. W. Woods,
Ed., Kluwer Academic Publishers, Norwell, MA,
pp. 43{100, 1991.
Figure 4: Two Equivalent Structures.
x1(k) g1((cid:0)N1) (cid:1)(cid:1)(cid:1) g1(0) (cid:1)(cid:1)(cid:1) g1(N1)
Figure 1: Two-Band Subband Analysis and Synthesis N21 <k< N(cid:0)2N1 y(2k(cid:0)N1) (cid:1)(cid:1)(cid:1) y(2k) (cid:1)(cid:1)(cid:1) y(2k+N1)
x2(k) g2((cid:0)N2) (cid:1)(cid:1)(cid:1) g2(0) (cid:1)(cid:1)(cid:1) g2(N2)
Structure. N42 <k< N(cid:0)4N2 y(4k(cid:0)N2) (cid:1)(cid:1)(cid:1) y(4k) (cid:1)(cid:1)(cid:1) y(4k+N2)
x3(k) g3((cid:0)N3) (cid:1)(cid:1)(cid:1) g3(0) (cid:1)(cid:1)(cid:1) g3(N3)
N83 <k< N(cid:0)8N3 y(8k(cid:0)N3) (cid:1)(cid:1)(cid:1) y(8k) (cid:1)(cid:1)(cid:1) y(8k+N3)
x4(k) g4((cid:0)N4) (cid:1)(cid:1)(cid:1) g4(0) (cid:1)(cid:1)(cid:1) g4(N4)
N84 <k< N(cid:0)8N4 y(8k(cid:0)N4) (cid:1)(cid:1)(cid:1) y(8k) (cid:1)(cid:1)(cid:1) y(8k+N4)
Table 1: Update Table for Coe(cid:14)cients Su(cid:14)ciently Far
from the Boundaries.
63 -34 49 10 7 13 -12 7
-31 23 14 -13 3 4 6 -1
15 14 3 -12 5 -7 3 9
Figure2: SubbandAnalysisandSynthesisFilterBanks. -9 -7 -14 8 4 -2 3 2
-5 9 -1 47 4 6 -2 2
3 0 -3 2 3 -2 0 4
2 -3 6 -4 3 6 3 6
5 11 5 6 0 3 -4 4
Table 2: Example of 3-level wavelet transform of an
image [4].
6