A Multi-threshold Embedded Zerotree Wavelet Coder

Eui-Sung Kang

1

, Toshihisa Tanaka

2

, Tae-Hyung Lee

1

, and Sung-Jea Ko

1

1

Department of Electronics Engineering, Korea University

Anam-dong, Sungbuk-ku, Seoul 136-701 Korea

fmagasa, jeeni, sjkog@dali.korea.ac.kr

2

Department of Electrical and Electronic Engineering, Tokyo Institute of Technology

Ookayama, Meguro-ku, Tokyo 152-8552 Japan

tanaka@ide.titech.ac.jp

Abstract| In this paper, a novel and simple embed-

ded wavelet coder, called a multi-threshold embedded zerotree

wavelet coder, is proposed. The well-known embedded ze-

rotree wavelet (EZW) coder uses the successive approx-

imation quantization (SAQ) process and zerotree struc-

tures of wavelet coecients. The EZW coder scans itera-

tively whole wavelet coecients during the SAQ process,

which decreases the coding eciency considerably. In

the proposed scheme, we scan only signicant subbands

using the signicant test with the multi-threshold. The

multi-threshold obtained from the coecient with the

maximum magnitude in each subband is used to deter-

mine whether a subband has signicant coecients or

not. Since it is not necessary to encode the insignif-

icant subbands having no signicant coecients during

the SAQ process, we reduce signicantly the redundancy

generated by scanning higher subbands. Experimental

results show that the proposedmethod outperforms pop-

ular image coders such as EZW and JPEG.

I. Introduction

Wavelet-based image coding is a promising technique

to achieve ecient image coding due to being free from

blocking artifacts that block-based image coders pro-

duce [1], [2]. Moreover, this technique oers adapt-

ability to achieve scalability for multimedia and vi-

sual communications. Particularly, the embedded coder

can achieve exact rate control, since it generates a sin-

gle bitstream which can be truncated at any desired

rate [3], [4]. The major dierence between the embed-

ded coding and the traditional coding lies in the suc-

cessive approximation quantization where wavelet coef-

cients greater than a given threshold are assumed to

have equal importance and should be transmitted be-

fore coecients with smaller magnitudes. This SAQ

scheme adopted by the embedded coder can be eec-

tively applied to progressive image transmission. In the

EZW coder [3], an initial threshold is applied to whole

wavelet coecients. During the SAQ process, each coef-

cient is classied into one of four symbols POS, NEG,

ZTR, and IZ. These symbols are losslessly encoded us-

ing adaptive arithmetic coding [5]. This process is con-

tinued until stopping condition is met. Since the EZW

This work was supported by the Korea Science and Engineering

Foundation (KOSEF) under grant 985-0900-007-2.

coder scans whole wavelet coecients with respect to

the current threshold, some redundancy exists in the

symbols generated when higher subbands are scanned.

In this paper, we propose a novel wavelet coder, which

can eciently reduce the redundancy produced when

higher subbands are scanned. While the EZW coder

scans whole wavelet coecients during the SAQ pro-

cess, the proposed method scans only signicant sub-

bands according to the results of signicant test using

the multi-threshold. The multi-threshold is a set of coef-

cients with the maximummagnitude in each subband.

This multi-threshold is used to determine whether a

subband have signicant coecients or not. Since it

is not necessary to scan insignicant subbands having

no signicant coecients, our approach is not only com-

putationally ecient, but also obtains high coding ef-

ciency. Furthermore, our proposed embedded coder

can achieve the exact rate control and is utilized for

progressive image transmission.

This paper is organized as follows. In Section II, we

review the embedded zerotree coding and successive ap-

proximation quantization. In Section III, we demon-

strate our novel approach, a multi-threshold embedded

zerotree wavelet coder. In Section IV, we present exper-

imental results for our coder, and compare them with

other famous methods. Conclusions of this paper are

given in Section V.

II. Embedded wavelet image coding

The Shapiro's EZW coder exploits the self-similarity

of the wavelet transform image across dierent scales by

using a tree structure. The typical wavelet tree struc-

ture is dened recursively as shown in Fig. 1. We call

a coecient at a coarse scale a parent. All coecients

at the next ner scale at the same location and of sim-

ilar orientation are children. All coecients at all ner

scales at the same location and of similar orientation are

descendants. Note that coecients in the highest sub-

bands and LL subband can not have zerotree structure,

since they have no children. All the rest coecients have

four children. If a coecient is small in magnitude with

HL1

HH1

LH1

HL

HHLH

LH

LL HL

HH

2

22

3

33

3

Fig. 1. Wavelet tree structure of wavelet transform.

respect to a given threshold, then all of its descendants

of the similar orientation in the same spatial location

at all ner scales are likely to be small as well. The

EZW scheme uses eectively the SAQ and zerotrees.

The SAQ iteratively applies a sequence of thresholds

T

0

; : : : ; T

N1

to determine signicance of a coecient,

where T

i

= T

i1

=2. The initial threshold T

0

is cho-

sen so that jc

j

j < 2T

0

for all the wavelet coecients

c

j

. A coecient is called signicant if its amplitude is

greater than the threshold T

i

; otherwise insignicant.

When the parent is insignicant and its descendants

are insignicant, the tree is called zerotree and the tree

node is called zerotree root. A signicant map is de-

ned as the bitplane indicating whether the coecient

is signicant or not with respect to the current thresh-

old. The signicant map is encoded using four symbols,

POS, NEG, ZTR, and IZ as positive signicant, nega-

tive signicant, zerotree root, and isolated zero, respec-

tively, where isolated zero means an insignicant pixel

but there exists one more signicant descendants. This

is called dominant pass. Then, renement pass

1

is per-

formed for the coecients which have been contained

in the signicant map at earlier iterations. During the

i

th

renement pass, binary symbols (0 or 1) correspond-

ing to the i+ 1

th

most signicant bits of coecients are

produced. Those symbols generated in both dominant

pass and renement pass are encoded using the adaptive

arithmetic coder [5]. The encoder halves the threshold

and repeats another dominant and renement pass un-

til a target bitrate is met. More details are explained

in [3].

III. The Proposed Approach

As seen previously, if the current pixel is signicant,

the EZW method needs to scan its children. And if

1

In Shapiro's paper, this pass is called subordinate pass, but

we use more general term renement pass taking other similar

algorithms into consideration.

the current pixel is insignicant, this technique needs

to identify whether there are any signicant descen-

dants. Next, the proposed method using the multi-

threshold and zerotrees is presented. When we perform

a N -level wavelet decomposition, the multi-threshold,

M = ft

k

; k = 1; 2; : : : ; 3N + 1g is a set of the maxi-

mum magnitudes of coecients in each subband. Let

t

3i2

; t

3i1

, and t

3i

, respectively, represent the maxi-

mummagnitude in subbandsHH

i

; LH

i

, andHL

i

where

1 i N . And t

3N+1

corresponds to the maximum

magnitude of coecients in the subband LL

N

. Fig. 2

shows the multi-threshold , M = f10; 15; : : :; 50; 70g, of

a 3-level decomposed image. Next, the initial threshold

T

0

for the SAQ is selected such that

T

0

= 2

blog

2

Cc

(1)

where C is the largest magnitude of all the wavelet co-

ecients and bxc represents the largest integer which is

not greater than x. During each SAQ process, a binary

image is generated by applying this threshold to whole

wavelet coecients. In some subbands of the binary im-

age, all the coecients are equal to zero especially in the

rst several quantization processes. At the i

th

iteration,

a subband which satises t

k

< T

i

does not need to be

scanned. Here, we dene a signicant subband as a sub-

band that satises t

k

T and an insignicant subband

as a subband that satises t

k

< T . In Fig. 2, shaded

subbands indicate signicant subbands with respect to

the current threshold and the other subbands represent

insignicant subbands. To determine which subband is

signicant or insignicant with respect to the threshold

in the decoder, the multi-threshold information should

be transmitted to the decoder. To reduce overhead of

the multi-threshold information, we transmit n

k

which

satises

n

k

= blog

2

t

k

c (2)

instead of transmitting real coecient values.

In our approach, the zerotree coding is performed

like the EZW coder and compression algorithm with

reversible embedded wavelets (CREW) [6]. However,

we employ binary symbols, 0 for insignicant pixel and

1 for signicant one. For a signicant coecient, one

additional bit is needed to encode the sign of the co-

ecient. For a insignicant coecient, one additional

bit is also needed in order to indicate whether the co-

ecient is zerotree root or not. Like the CREW, POS,

NEG, IZ, and ZTR are replaced with 11, 10, 01, and

00, respectively. In our coder, however, the conversion

is performed to reduce the redundancy of symbols of the

subbands, called the marginal subbands, which are adja-

cent to insignicant subbands. Since all the coecients

t =101t =152

t =183

t =234

t =306

t =335

t =7010 t

=509

t =437t

=478

t =183

t =101t =152

t =234

t =306

t =335

t =7010 t

=509

t =478 t

=437

t =101t =152

t =18

t =234t =335

t =306

3

t =7010 t

=509

t =478 t

=437

(a) T

0

= 2

6

(b) T

1

= 2

5

(c) T

2

= 2

4

Fig. 2. Examples of the multi-threshold and signicant subbands. Shaded regions indicate signicant subbands and thick lines

indicate the marginal subbands .

in the marginal subbands have no signicant descen-

dants, we no longer need an additional bit to identify

whether those coecients have signicant descendants

or not. In the EZW and the CREW, if the current coef-

cient is signicant, its descendants should be scanned

and encoded. This fact implies that our method can of-

fer an ecient scanning which reduces this redundancy.

In addition, to encode coecients in the lowest sub-

band, we use only one bit for each coecients: 0 for

insignicant pixels and 1 for signicant ones, because

all the coecients in that subband are identied as pos-

itive.

Binary symbols generated in each pass are encoded

by adaptive arithmetic coding. Unlike the EZW, we

use the QM-coder [7] which is more computationally

ecient than the adaptive arithmetic coding [5] and

Q-coder [8]. And it can eectively exploit the strong

correlation of quantized wavelet coecients. To encode

symbols eciently, the QM-coder employs context mod-

eling [9], [10] which is the set of past sequence of sym-

bols on which the probability of the current symbol is

conditioned. For all quantized wavelet coecients as a

sequence of binary symbols x

1

; x

2

; : : : ; x

n

, the minimum

codelength l is given by

l = log

2

n

Y

i=1

p(x

i

jx

i1

; x

i2

; : : : ; x

1

): (3)

Here, to reduce the problem of estimating the sym-

bol distributions , we should estimate S, which is a

subsequence of x

1

; x

2

; : : : ; x

n

. During the dominant

pass of the EZW, the signicance of the parent coef-

cient and the previous one are used for context model-

ing of the current symbol x

i

, that is S = fx

i1

; x

P

g,

where x

P

is the parent symbol. Context modeling

for our coder is shown in Fig. 3: for a signicant

bit, S = fx

P

; x

N

; x

W

; x

E

; x

S

g, and for a sign bit,

x N

x Ex W

x S

x P

x i

x N

x Ex W

x S

x i

(a) Signicant bit (b) Sign bit

Fig. 3. Context modeling of the proposed coder. x

i

is the current

pixel and x

P

is its parent.

S = fx

N

; x

W

; x

E

; x

S

g. Note that x

N

and x

W

are sig-

nicance bits generated at the current threshold, and

x

E

and x

S

are those which are obtained at the previous

threshold. And context modeling for a renement bit

consists of only the previous pixel, i.e. S = fx

i1

g.

IV. Experimental Results

Coding simulations were performed for the 512 512

grayscale Lena image. We investigated the performance

of our proposed approach using the six-scale wavelet

decomposition with the 9/7 tap biorthogonal wavelet

lter [1]. We utilized the QM-coder to encode the sym-

bols obtained by the scanning process with the multi-

threshold. We compare the performance of the pro-

posed method with that of the EZW coder, the DCT-

based embedded image coder [11], the ACTCQ (arith-

metic and entropy constrained trellis quantization) [12],

and the JPEG coder [7]. The EZW coder and the DCT-

based embedded image coder are the embedded coders.

Note that the coding scheme in [7], [12] are not the em-

bedded coder. Fig. 4 shows the original image and the

decoded images obtained by our proposed method. It is

seen that the decoded images do not produce any block-

(a) Original image (b) at rate 0.5bpp (c) at rate 0.25bpp

Fig. 4. Original and decoded 512 512 Lena images.

ing artifacts and exhibit good visual quality. And Fig. 5

shows the rate-distortion performance for the original

image. At lower bit rates (below about 0.95bpp), our

method appears to produce better results than all the

candidates considered in our simulation. At higher bit

rates (around 1bpp), only ACTCQ give slightly better

performance than the proposed approach. This exper-

imental result shows that our coder outperforms those

popular image coders.

V. Conclusions

We have presented an image coding algorithm, a

multi-threshold embedded zerotree wavelet coder. This

approach utilizes eciently the multi-threshold and ze-

rotree structures of wavelet coecients. The proposed

method can reduce the redundancy produced when

higher subbands are iteratively scanned during the SAQ

process. We can also reduce symbol redundancy us-

ing the multi-threshold and binary representation of the

symbols generated by the SAQ process. Furthermore,

since our proposed method produces the fully embedded

bitstream like the other embedded coders, it can achieve

the exact rate control and can be easily applied to pro-

gressive image transmission. It was shown that the pro-

posed coder outperforms well-known image coders such

as the EZW coder, the JPEG, and the ACTCQ.

References

[1] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies,

\Imagecoding using wavelet transform," IEEE Trans. Image

Processing, vol. 1, pp. 205{220, Apr. 1992.

[2] J. Lu, V. R. Algazi, and R. R. Estes, Jr., \A comparative

study of wavelet image coders,"Optical Engineering, vol. 35,

Sept. 1996.

[3] J. M. Shapiro, \Embedded image coding using zerotrees of

wavelet coeceints," IEEE Trans. Signal Processing, vol. 41,

pp. 3445{3462, Dec. 1993.

[4] C. D. Creusere, \A new method of robust image compression

based on the embedded zerotree wavelet algorithm," IEEE

Trans. Image Processing, vol. 6, pp. 1436{1442, Oct. 1997.

[5] I. H. Witten, R. M. Neal, and J. G. Cleary, \Arithmetic cod-

ing for data compression,"Commun. ACM, vol. 30, pp. 520{

540, June 1987.

[6] A. Zandi, J. D. Allen, E. L. Schwarts, and M. Boliek,

\CREW: Compression with reversible embedded wavelets,"

in Proc. Data Compression Conference, (Snowbird, Utah),

pp. 212{221, Mar. 1995.

[7] W. B. Pennebakerand L. J. Mitchell, JPEG Still Image Data

Compression Standard. New York: Van Nostrand Reinhold,

1992.

[8] W. B. Pennebaker, J. L. Mitchell, G. G. Langdon, and R. B.

Arps, \An overview of the basic principles of the Q-coder

adaptive binary arithmetic coder," IBM J. Res. Develop.,

vol. 32, pp. 717{726, Nov. 1988.

[9] V. R. Algazi and R. R. Estes, Jr., \Analysis based coding of

image transform and subband coecients," in SPIE, Appli-

cations of Digital Image Processing XVII, pp. 11{21, 1995.

[10] C. Chrysas and A. Ortega, \Ecient context-based entropy

coding for lossy wavelet image compression," in Proc. IEEE

Data Compression Conf. '97, pp. 241{250, 1997.

[11] Z. Xiong, O. Guleryuz, and M. T. Orchard, \A DCT-based

embedded image coder," IEEE Signal Proc. Letters, vol. 3,

pp. 289{290, Nov. 1996.

[12] R. L. Joshi, V. J. Crump, and T. R. Fischer, \Image subband

coding using arithmetic coded trellis coded quantization,"

IEEE Trans. Circuits and Systems for Video Technology,

vol. 5, pp. 515{523, Dec. 1995.

20

22

24

26

28

30

32

34

36

38

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PSNR

[dB]

Rate[bpp]

ProposedShapiro[3]

JPEG[7]Xiong et al[11]Joshi et al[12]

20

22

24

26

28

30

32

34

36

38

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PSNR

[dB]

Rate[bpp]

ProposedShapiro[3]

JPEG[7]Xiong et al[11]Joshi et al[12]

Fig. 5. Comparison of coding performance.