Non-uniform Quantization of Detail Components in Wavelet
Transformed Image for Lossy JPEG2000 Compression
Madhur Srivastava
1
and Prasanta K. Panigrahi
2
1
Department of Biological and Environmental Engineering, Cornell University, Ithaca, New York, U.S.A.
2
Department of Physical Sciences, Indian Institute of Science Education and Research, Mohanpur, West Bengal, India
Keywords:
JPEG-2000 Standard, Discrete Wavelet Transform, Thresholding, Non-uniform Quantization, Image Com-
pression.
Abstract:
The paper introduces the idea of non-uniform quantization in the detail components of wavelet transformed
image. It argues that most of the coefficients of horizontal, vertical and diagonal components lie near to zeros
and the coefficients representing large differences are few at the extreme ends of histogram. Therefore, this
paper advocates need for variable step size quantization scheme which preserves the edge information at the
edge of histogram and removes redundancy with the minimal number of quantized values. To support the
idea, preliminary results are provided using a non-uniform quantization algorithm. We believe that successful
implementation of non-uniform quantization in detail components in JPEG-2000 still image standard will
improve image quality and compression efficiency with lesser number of quantized values.
1 INTRODUCTION
The emergence of JPEG (Joint Photographic Experts
Group)-2000 still image compression standard has led
to a different approach to image analysis of image
data compared to the previous JPEG standard (Mar-
cellin et al., 2000). It has higher compression effi-
ciency and better error resilience. Furthermore, it is
progressive by resolution and quality in comparison
to other still image compression standards (Skodras et
al., 2001). These new features have enabled the use of
JPEG-2000 in new areas like internet, color facsimile,
printing, scanning, digital photography, remote sens-
ing, mobile, medical imagery, and e-commerce (Sko-
dras et al., 2001). One of the major reasons for bet-
ter performance and the wide range of applications
of JPEG-2000 is due to the introduction of Discrete
Wavelet Transform (DWT) in the standard replacing
Discrete Cosine Transform. In addition, the quanti-
zation of DWT coefficients of image has led to rate-
distortion feature in the standard. Part-I of JPEG-
2000 image compression standard uses fixed-size uni-
form scalar deadzone quantization scheme. In Part-II,
deadzone with variable length is incorporated in uni-
form scalar quantization. Additionally, trellis coded
quantization (TCQ) scheme is combined in the Part-
II of JPEG compression standard (Marcellin et al.,
2002). However, there are some limitations on the
step size selection in the case of lossy image compres-
sion using non-invertible wavelets (see section 3).
This paper advocates the need for non-uniform
quantization scheme in the detail components of
DWT for lossy image compression to overcome the
current limitations, to improve image quality at same
size, and to improve the compression ratio at a par-
ticular image quality. The main motivation for advo-
cating non-uniform quantization is that in each detail
component, the majority of the coefficients lie near
to zero and the coefficients representing large differ-
ences are few at the extreme ends of histogram. This
is due to the fact that each detail component repre-
sents the high frequency coefficients and in an image,
high frequency coefficients only occur at the edges
which constitute extremely low percentage of the en-
tire image. Therefore, there is a need for a quantiza-
tion scheme that provides variable step size with big-
ger step size around the zero, and smaller step size
at the ends of the histogram plot of each horizontal,
vertical and diagonal component. On the other hand,
the approximation component is untouched with non-
uniform quantization. This is because in approxima-
tion component almost all the frequency coeffients
have substantial magnitude and contain high amount
of information, unlike detail components which have
high redundancy.
The paper is organized in the following way. Sec-
604
Srivastava M. and K. Panigrahi P. (2013).
Non-uniform Quantization of Detail Components in Wavelet Transformed Image for Lossy JPEG2000 Compression.
In Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods, pages 604-607
DOI: 10.5220/0004333706040607
Copyright
c
 SciTePress

tion 2 briefly describes discrete wavelet transform. In
section 3, uniform quantization scheme is given with
its limitations on non-invertible wavelets. Section 4
provides a prospective non-uniform quantization al-
gorithm. Preliminary results supporting the proposed
algorithm are given in section 6. Finally, section 7
concludes the paper stating the advantages of using
the proposed non-uniform quantization algorithm and
provides direction for future work.
2 DISCRETE WAVELET
TRANSFORM
In discrete wavelet transform the image is decom-
posed into four pieces at each level. It is identical to
the subbands spaced in the frequency domain. At first
level the original image is decomposed into four lev-
els which are labeled as LL, LH, HL and HH as shown
in fig. 1. Where LL subband is called the average
Figure 1: Representation of three level 2-D DWT.
Figure 2: Wavelet filter bank for one level image.
part. It is low pass filtered in both the directions and
it is most likely identical to the original image, hence
it is called approximation. LH is the difference of the
horizontal rows, HL gives the vertical differences and
HH gives the diagonal difference. The components
LH, HL and HH are called detailed components. The
approximation part is further decomposed until the
final coefficient is left. The decomposed image can
be reconstructed using a reconstruction filter. Fig. 2
shows the wavelet filter bank for one level image re-
construction. Since approximation part is identical to
original image hence it contains wavelet coefficients
of larger amplitudes. On the other hand in detailed
components, wavelet coefficients are smaller in am-
plitude and are close to zero.
3 UNIFORM SCALAR
QUANTIZATION
In Part-I of the standard, JPEG-2000 applies dead-
zone uniform scalar quantizater on the wavelet trans-
formed image. The quantization index q is calculated
using the following formula,
q = sign(y)⌊
|y|
∆
⌋ (1)
where ∆ is quantizer step size and y is the input to the
quantizer.
The uniform scalar quantizer with variable size of
deadzone modifies the above formula to the follow-
ing,
q =

0 |y| < −nz∆
sign(y)⌊
|y|+nz∆
∆
⌋ |y| ≥ −nz∆
(2)
It has to be noted from equation 2 that the fixed
width of step size is maintained, except for the dead-
zone region. However, in the case of non-invertible
wavelets there are some limitations regarding step
size and its binary representation (Marcellin et al.,
2002). Firstly, there has to be only one quantization
step size per subband. This constraints the step size
to be less than or equal to the quantization step size
of all the different regions of interest in the subband.
Secondly, the step size can acquire upto 12 significant
binary digits. Thirdly, the upper bound of step size
can quantize almost all the coefficients of subband to
zero when the step size is twice or more than the sub-
band magnitude. Lastly, the lower bound on the step
size restricts high accuracy coding as upto 21 and 30
fractional bits of HH coefficients are allowed to be
encoded for 8 and higher bits, respectively.
4 PROPOSED NON-UNIFORM
QUANTIZATION ALGORITHM
An algorithm is proposed for the quantization of each
detail component in the wavelet domain into variable
step sizes using mean and standard deviation. Starting
from the weighted mean of histogram plot, the algo-
rithm is recursively applied on the sub-ranges to find
the next threshold level and continuing it till ends of
the histogram are reached.
The method takes into account that majority of co-
efficients lie near to zero and coefficients representing
large differences are few at the extreme ends of his-
togram. Hence, the procedure provides for variable
step size with bigger block size around the mean, and
having smaller blocks at the ends of histogram plot
Non-uniformQuantizationofDetailComponentsinWaveletTransformedImageforLossyJPEG2000Compression
605

Figure 3: Left to Right: Original ’Lenna’ image; Reconstructed image at quantized level 2; Reconstructed image at quantized
level 4; Reconstructed image at quantized level 6; Reconstructed image at quantized level 8; Reconstructed image at quantized
level 10. DB9 wavelets are used in all forward and inverse wavelet transforms.
of each horizontal, vertical and diagonal components,
leaving approximation coefficients unchanged. The
algorithm is based on the fact that a number of dis-
tributions tends toward a delta function in the limit
of vanishing variance. In the following we systemati-
cally outline the algorithm.
1. n, the number of step sizes are taken as input.
2. Range R = [a,b]; initially a= min(histogram) and
b = max(histogram).
3. Find weighted mean (µ) of values ranging in R.
4. Initially, thresholds T
1
= µ and T
2
= µ+0.001 . For
T
2
, 0.001 is added to avoid the use of weighted
mean again. Any value can be taken such that it
doesnt move significantly from weighted mean.
5. Repeat steps 6-9 (n− 2)/2 times
6. Find weighted mean (µ
1
) and standard deviation
(σ
1
) of values ranging [a,T
1
] and weighted mean
(µ
2
) and standard deviation (σ
2
) of values ranging
[T
2
,b].
7. Thresholds t
1
and t
2
are calculated as t
1
=µ
1
- k
1
σ
1
and t
2
=µ
2
+ k
2
σ
2
where k
1
and k
2
are free param-
eters. These are used to increase or decrease the
block size.
8. Assign the values ranging [t
1
,T
1
] and [T
2
,t
2
] with
their respective weighted mean.
9. Assign T
1
=t
1
-0.001 and T
2
=t
2
+0.001 . The value
0.001 is added and subtracted to avoid the reuse
of t
1
and t
2
. Any value can be taken such that it
doesnt move significantly from t
1
and t
2
.
10. Finally, Take weighted mean of values ranging
[a,T
1
] and [T
2
,b] and assign the same to the re-
spected range.
The above algorithm was used to compare
Daubechies and Coiflet wavelet family on there effi-
cacy to provide effective image segmentation in (Sri-
vastava et al., 2011). Moreover, it is also applied in
(Srivastava et al., unpublished) to carry out wavelet
based image segmentation. However, no study has
been conducted of its effectiveness on lossy image
compression. The preliminary results shown in the
next section suggests that the above algorithm can be
useful in carrying out non-uniform quantization in de-
tail coefficients.
5 PRELIMINARY
EXPERIMENTAL RESULTS
The proposed algorithm is applied on the standard
test images from the USC-SIPI Image Database
(sipi.usc.edu/database). Peak Signal to Noise Ratio
(PSNR) and Mean Structural Similarity Index Mea-
sure (MSSIM) (Wang et al., 2004) are used to evalu-
ate the quality of reconstructed images. The PSNR is
computed using the following formula,
PSNR = 10log
10
(
I(x,y)
2
max
MSE
) (3)
MSE =
1
MN
M
∑
x=1
N
∑
y=1
(I(x, y) −
˜
I(x,y))
2
(4)
where M and N are dimensions of the image, x and y
are pixel locations, I is the input image, and
˜
I is the
reconstructed image. The MSSIM is calculated with
the help of following formulae,
SSIM(p,q) =
(2µ
p
µ
q
+ K
1
)(2σ
pq
+ K
2
)
(µ
2
p
+ µ
2
q
+ K
1
)(σ
2
p
+ σ
2
q
+ K
2
)
(5)
MSSIM(I,
˜
I) =
1
M
M
∑
i=1
SSIM(p
i
,q
i
) (6)
where µ is mean; σ is standard deviation; p and q are
window sizes of original and reconstructed images,
and the size of typical window is 8× 8; K
1
and K
2
are
the constants with K
1
= 0.01 and K
2
= 0.03; M is the
total number of windows.
Fig. 3 displays the reconstructed images at dif-
ferent quantized values after applying the proposed
algorithm on each detailed component, separately. In
addition to the fig. 3, table 1 shows the PSNR and
MSSIM of four test images with dimensions 512 ×
512 at quantized values ranging from 2 to 10. DB9 is
ICPRAM2013-InternationalConferenceonPatternRecognitionApplicationsandMethods
606

Table 1: PSNR and MSSIM of reconstructed gray-level test images using DB9 wavelets at different quantized values.
Image Name (Dimension) Number of Quantized Values PSNR (dB) MSSIM
Lenna (512 × 512) 2 37.7780645697 0.9982142147
4 41.8874641847 0.9992823393
6 43.8590907980 0.9994968461
8 44.2931530130 0.9995311420
10 44.3489820362 0.9995364013
Baboon (512 × 512) 2 26.8132232589 0.9595975231
4 31.5997241598 0.9869355763
6 32.9223693072 0.9905018802
8 33.0622299131 0.9908045725
10 33.0762467334 0.9908358964
Pepper (512 × 512) 2 35.1261558539 0.9968488432
4 38.0511064803 0.9982568820
6 40.6144660901 0.9990231546
8 40.9936699886 0.9991423921
10 41.0082447784 0.9991441908
House (512 × 512) 2 31.9886146645 0.9904885967
4 35.7235201624 0.9951268912
6 38.2730683421 0.9975855578
8 38.4299731739 0.9976918850
10 38.4349224964 0.9976960112
the wavelet used for taking DWT and inverse DWT.
As can be seen from fig.3, the images are not dis-
tinguishable visually. Results from table 1 substan-
tiates the claim. For ’Lenna’, minimum and maxi-
mum PSNR observed are 37.77 and 44.38, respec-
tively, with MSSIM being .99 at all quantized num-
bers. Overall, there is increase in PSNR and MSSIM
with the increase of the number of quantized values.
This increase may or may not be effectivefor practical
application. For example, ’Baboon’ will be best rep-
resented with 4 quantized levels because there is sub-
stantial increase of PSNR and MSSIM from quantized
level 2, but compared to quantized 6 to 10, the in-
crease of these higher levels are not effective enough
to be noticed visually.
6 CONCLUSIONS
This paper introduces the concept of non-uniform
quantization for detailed components in the JPEG-
2000 still image standard. We believe that further ex-
ploring this option might lead to the following pro-
gressive results in the current standard: (1) Improved
quality at same image size, (2) Better compression at
same quality, (3) Flexible number of quantized values
based on the actual statistics of wavelet tranformed
image, and (4) Variable step size compared to fixed
step size, maximizing the elimination of redundancy.
However, the results shown in the paper are pre-
liminary and suggestive. It will be interesting to ex-
amine the proposed quantization algorithm when em-
bedded in the JPEG-2000 standard. Factors which
will determine the success of proposed approach
will be actual compressed size, image quality, time
complexity and encoder complexity. Also, there is
wide scope for testing other non-uniform quantiza-
tion schemes, or perhaps creating new quantization
scheme customized to the standard.
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