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Blo ck Transforms in Progressive Image
Co ding
Trac D. Tran and Truong Q. Nguyen
1 Intro duction
Blo ck transform co ding and subband co ding have b een two dominant techniques in
existing image compression standards and implementations. Both metho ds actually
exhibit many similarities: relying on a certain transform to convert the input image
to a more decorrelated representation, then utilizing the same basic building blo cks
such as bit allo cator, quantizer, and entropy co der to achieve compression.
Blo ck transform co ders enjoyed success rst due to their low complexity in im-
plementation and their reasonable p erformance. The most p opular blo ck transform
co der leads to the current image compression standard JPEG [1] which utilizes the
8 8 Discrete Cosine Transform DCT at its transformation stage. At high bit
rates 1 bpp and up , JPEG o ers almost visually lossless reconstruction image
quality. However, when more compression is needed i.e., at lower bit rates , an-
noying blo cking artifacts showup b ecause of two reasons: i the DCT bases are
short, non-overlapp ed, and have discontinuities at the ends; ii JPEG pro cesses
each image blo ck indep endently. So, inter-blo ck correlation has b een completely
abandoned.
The development of the lapp ed orthogonal transform [2] and its generalized ver-
sion GenLOT [3, 4] helps solve the blo cking problem to a certain extent by b or-
rowing pixels from the adjacent blo cks to pro duce the transform co ecients of the
current blo ck. Lapp ed transform outp erforms DCT on two counts: i from the anal-
ysis viewp oint, it takes into accountinter-blo ck correlation, hence, provides b etter
energy compaction that leads to more ecient entropy co ding of the co ecients;
ii from the synthesis viewp oint, its basis functions decay asymptotically to zero
at the ends, reducing blo cking discontinuities drastically.However, earlier lapp ed-
transform-based image co ders [2,3,5]have not utilized global information to their
full advantage: the quantization and the entropy co ding of transform co ecients
are still indep endent from blo ck to blo ck.
Recently, subband co ding has emerged as the leading standardization candidate
in future image compression systems thanks to the development of the discrete
wavelet transform. Wavelet representation with implicit overlapping and variable-
length basis functions pro duces smo other and more p erceptually pleasant recon-
structed images. Moreover, wavelet's multiresolution characteristics have created
an intuitive foundation on which simple, yet sophisticated, metho ds of enco ding
the transform co ecients are develop ed. Exploiting the relationship between the
parent and the o spring co ecients in a wavelet tree, progressivewavelet co ders
[6, 7, 9] can e ectively order the co ecients by bit planes and transmit more sig-
ni cant bits rst. This co ding scheme results in an emb edded bit stream along
with many other advantages such as exact bit rate control and near-idemp otency
1. Blo ckTransforms in Progressive Image Co ding 2
p erfect idemp otency is obtained when the transform maps integers to integers . In
these subband co ders, global information is taken into account fully.
From a frequency domain p oint of view, the wavelet transform simply provides an
o ctave-band representation of signals. The dyadic wavelet transform is analogous
to a non-uniform-band lapp ed transform. It can suciently decorrelate smo oth
images; however, it has problems with images with well-lo calized high frequency
comp onents, leading to low energy compaction. In this app endix, we shall demon-
strate that the emb edded framework is not only limited to the wavelet transform;
it can b e utilized with uniform-band lapp ed transforms as well. In fact, a judicious
choice of appropriately-optimized lapp ed transform coupled with several levels of
wavelet decomp osition of the DC band can provide much ner frequency sp ectrum
partitioning, leading to signi cant improvementover currentwavelet co ders. This
app endix also attempts to shed some lightonto a deep er understanding of wavelets,
lapp ed transforms, their relation, and their p erformance in image compression from
amultirate lter bank p ersp ective.
2 The wavelet transform and progressive image
transmission
Progressive image transmission is p erfect for the recent explosion of the internet.
The wavelet-based progressive co ding approach rst intro duced by Shapiro [6] relies
on the fundamental idea that more imp ortant information de ned here as what
decreases a certain distortion measure the most should b e transmitted rst. As-
sume that the distortion measure is the mean-squared error MSE , the transform
is paraunitary, and transform co ecients c are transmitted one by one, it can b e
i;j
jc j
i;j
, where N is the total number proven that the mean squared error decreases by
N
of pixels [16]. Therefore, larger co ecients should b e transmitted rst. If one bit is
transmitted at a time, this approach can b e generalized to ranking the co ecients
by bit planes and the most signi cant bits are transmitted rst [8]. The progressive
transmission scheme results in an emb edded bit stream i.e., it can be truncated
at any p ointby the deco der to yield the b est corresp onding reconstructed image .
The algorithm can b e thought of as an elegant combination of a scalar quantizer
with p ower-of-two stepsizes and an entropy co der to enco de wavelet co ecients.
Emb edded algorithm relies on the hierachical co ecients' tree structure that we
called a wavelet tree, de ned as a set of wavelet co ecients from di erent scales
that b elong in the same spatial lo cality as demonstrated in Figure 1 a , where
the tree in the vertical direction is circled. All of the co ecients in the lowest
frequency band make up the DC band or the reference signal lo cated at the upp er
left corner . Besides these DC co ecients, in a wavelet tree of a particular direction,
eachlower-frequency parent node has four corresp onding higher-frequency o spring
nodes. All co ecients b elow a parentnode in the same spatial lo calityis de ned
as its descendents. Also, de ne a co ecient c to be signi cant with resp ect to
i;j
a given threshold T if jc j T , and insigni cant otherwise. Meaningful image
i;j
statistics have shown that if a co ecient is insigni cant, it is very likely that its
o spring and descendents are insigni cant as well. Exploiting this fact, the most
sophisticated emb edded wavelet co der SPIHT can output a single binary marker
1. Blo ckTransforms in Progressive Image Co ding 3
to representvery eciently a large, smo oth image area an insigni cant tree . For more details on the algorithm, the reader is refered to [7].
parent offspring descendents xx xx xxxx xxxxxxxx xx xxxx xxxxxxxx xxxx xxxxxxxx xx xxxx xxxxxxxx xxxxxxxx xxxxxxxx xx xx xxxxxxxx xx xx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx
(a) (b)
FIGURE 1. Wavelet and blo ck transform analogy.
Although the wavelet tree provides an elegant hierachical data structure which
facilitates quantization and entropy co ding of the co ecients, the eciency of the
co der still heavily dep ends on the transform's ability in generating insigni cant
trees. For non-smo oth images that contain a lot of texture, the wavelet transform is
not as ecient in signal decorrelation comparing to transforms with ner frequency
selectivity and sup erior energy compaction. Uniform-band lapp ed transforms hold
the edge in this area.
3 Wavelet and blo ck transform analogy
Instead of obtaining an o ctave-band signal decomp osition, one can have a ner
uniform-band partitioning as depicted in Figure 2 drawn for M = 8 . The ner
frequency partitioning compacts more signal energy into a fewer number of co ef-
cients and generates more insigni cant ones, leading to an enhancement in the
p erformance of the zerotree algorithm. However, uniform lter bank also has uni-
form downsampling all subbands now have the same size . A parent no de do es
not have four o spring no des as in the case of the wavelet representation. How
would one come up with a new tree structure that still takes full advantage of the
inter-scale correlation b etween blo ck-transform co ecients?
The ab ove question can be answered by investigating an analogy between the
wavelet and blo ck transform as illustrated in Figure 1. The parent, the o spring,
and the descendents in a wavelet tree cover the same spatial lo cality, and so are the
co ecients of a transform blo ck. In fact, a wavelet tree in an L-level decomp osition
L
is analogous to a 2 -channel transform's co ecient blo ck. The di erence lies at the
bases that generate these co ecients. It can be shown that a 1D L-level wavelet
1. Blo ckTransforms in Progressive Image Co ding 4
|H| DC band |H|
(a) (b)
π π ω π π π π ω 0 2 o o o π 0
MM 8 4 2
FIGURE 2. Frequency sp ectrum partitioning of a M -channel uniform-band transform
b dyadic wavelet transform.
decomp osition, if implemented as a lapp ed transform, has the following co ecient
matrix:
3 2
n n n
] h [ ] h [ ] h [n] h [
0 0 0 0
L 2 L 1
2 2 2
n n n
7 6
] h [ ] h [ ] h [n] h [
0 L 2 1 L 1 0 0
2 2 2
7 6
n n
7 6
h [n] h [ ] ] h [
0 0 L 2 1
2
2
7 6
n n
7 6
h [n] h [ ] h [ ]
0 0 1
L 2
2 2
7 6
7 6
.
.
P = : 1.1
7 6
L
.
7 6
7 6
h [n]
1
7 6
7 6
h [n]
1
7 6
5 4
h [n]
1
h [n]
1
From the co ecient matrix P ,we can observe several interesting and imp ortant
L
characteristics of the wavelet transform through the blo ck transform's prism:
The wavelet transform can be viewed as a lapp ed transform with lters of
L
variable lengths. For an L-level decomp osition, there are 2 lters.
Each basis function has linear phase; however, they do not share the same
center of symmetry.
The blo ck size is de ned by the length of the longest lter. If h [n] is longer
0
and has length N , the longest lter is on top, covering the DC comp onent,
0
L
and it has a length of 2 1 N 1 +1.For the biorthogonal wavelet pair
0
with h [n] of length 9 and h [n] of length 7 and L =3,the eight resulting
0 1
basis functions have lengths of 57; 49; 21; 21; 7; 7; 7; and 7:
For a 6-level decomp osition using the same 9 7 pair, the length of the longest
basis function grows to 505! The huge amountofoverlapp ed pixels explains
the smo othness of as well as the complete elimination of blo cking artifacts in
wavelet-based co ders' reconstructed images.
Each blo ck of lapp ed transform co ecients represents a spatial lo cality similarly
to a tree of wavelet co ecients. Let O i; j b e the set of co ordinates of all o spring
of the no de i; j inanM -channel blo ck transform 0 i; j M 1 , then O i; j
can b e represented as follows:
O i; j = f 2i; 2j ; 2i; 2j +1 ; 2i +1; 2j ; 2i +1; 2j +1 g: 1.2
1. Blo ckTransforms in Progressive Image Co ding 5
All 0; 0 co ecients from all transform blo cks form the DC band, which is similar
to the wavelet transform's reference signal, and each of these no des has only three
o springs: 0; 1 , 1; 0 , and 1; 1 : This is a straightforward generalization of the
structure rst prop osed in [10]. The only requirement here is that the number of
channel M has to b e a p ower of two. Figure 3 demonstrates through a simple rear-
rangement of the blo ck transform co ecients that the rede ned tree structure ab ove
do es p ossess a wavelet-likemultiscale representation. The quadtree grouping of the
co ecients is far from optimal in the rate-distortion sense; however, other parent-
o spring relationships for uniform-band transform such as the one mentioned in [6]
do not facilitate the further usage of various entropy co ders to increase the co ding
eciency.
FIGURE 3. Demonstration of the analogy b etween uniform-band transform and wavelet
representation.
4 Transform Design
A mere replacement of the wavelet transform bylow-complexity blo ck transforms is
not enough to comp ete with SPIHT as testi ed in [10,11]. We prop ose b elow several
novel criteria in designing high-p erformance lapp ed transforms. The overall cost
used for transform optimization is a combination of co ding gain, DC attenuation,
attenuation around the mirror frequencies, weighted stopband attenuation, and
unequal-length constraint on lter resp onses:
C = C + C + C + C + C :
overall 1 co ding gain 2 DC 3 mirror 4 weighted-stopband 5 unequal-length
1.3
The rst three cost functions are well-known criteria for image compression.
Among them, higher co ding gain correlates most consistently with higher ob jective
p erformance PSNR . Transforms with higher co ding gain compact more energy
into a fewer numb er of co ecients, and the more signi cant bits of those co ecients
always get transmitted rst. All designs in this app endix are obtained with a version
of the generalized co ding gain formula in [19]. Low DC leakage and high attenuation
near the mirror frequencies are not as essential to the co der's ob jective p erformance
as co ding gain. However, they do improve the visual quality of the reconstructed
image [5,17].
The ramp-weighted stopband attenuation cost is de ned as
1. Blo ckTransforms in Progressive Image Co ding 6
R
P
M 1
j! j! 2
C = jW e H e j d! ;
weighted-stopband k k
k =1
! 2
S
j!
where W e is a linear function starting with value one at the p eak of the
k
frequency resp onse decaying to zero at DC. The frequency weighting forces the
highband lters to pick up as little energy as p ossible, ensuring a high number of
insigni cant trees. This cost function also helps the optimization pro cess in obtain-
ing higher co ding gains.
The unequal-length constraint forces the tails of the high-frequency band-pass
lters' resp onses to havevery small values not necessarily zero es . The higher the
frequency band, the shorter the e ective length of the lter gets. This constraint
is added to minimize the ringing around strong image edges at low bit rates, a
typical characteristic of transforms with long lter lengths. Similar ideas have b een
presented in [20, 26, 27] where the lters have di erent lengths. However, these
metho ds restrict the parameter search space severely, leading to low co ding gains.
High-p erformance lapp ed transforms designed sp eci cally for progressive image
co ding are presented in Figure 4 c - d . Figure 4 a and b show the p opular
DCT and LOT for comparison purp oses. The frequency resp onse and the basis
functions of the 8-channel 40-tap GenLOT shown in Figure 4 c exemplify a well-
optimized lter bank: high co ding gain and low attenuation near DC for b est energy
compaction, smo othly decaying impulse resp onses for blo cking artifacts elimination,
and unequal-length lters for ringing artifacts suppression.
Figure 3 shows that there still exists correlation between DC co ecients. To
decorrelate the DC band even more, several levels of wavelet decomp osition can b e
used dep ending on the input image size. Besides the obvious increase in the co ding
eciency of DC co ecients thanks to a deep er co ecient trees, wavelets provide
variably longer bases for the signal's DC comp onent, leading to smo other recon-
structed images, i.e., blo cking artifacts are further reduced. Regularity ob jective
can be added in the transform design pro cess to pro duce M -band wavelets, and
awavelet-like iteration can b e carried out using uniform-band transforms as well.
The complete co der's diagram is depicted in Figure 5.
5 Co ding Results
The ob jective co ding results PSNR in dB for standard 512 512 Lena and Barbara
test images are tabulated in Table 1.1 where several di erent transforms are used:
DCT, 8-channel 8-tap lters, shown in Figure 4 a .
LOT 8-channel 16-tap lters, shown in Figure 4 b .
GenLOT, 8-channel 40-tap lters, shown in Figure 4 c .
LOT, 16-channel 32-tap lters, shown in Figure 4 d .
The blo ck transform co ders are compared to the b est progressivewavelet co der
SPIHT [7] and an earlier DCT-based emb edded co der [10]. All computed PSNR
quotes in dB are obtained from a real compressed bit stream with all overheads
included. The rate-distortion curves in Figure 6 and the tabulated co ding results
1. Blo ckTransforms in Progressive Image Co ding 7
DC Att. >= 310.6215 dB Mirr Att. >= 320.1639 dB Stopband Att. >= 9.9559 dB Cod. Gain = 8.8259 dB DC Att. >= 312.5597 dB Mirr Att. >= 314.1433 dB Stopband Att. >= 19.3814 dB Cod. Gain = 9.2189 dB
0 h0 0 h0
−5 h1 −5 h1
−10 −10 h2 h2
−15 −15 h3 h3
−20 −20 h4 h4 Magnitude Response (dB) Magnitude Response (dB) −25 −25 h5 h5
−30 −30 h6 h6 −35 −35 h7 h7 −40 −40 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Normalized Frequency Normalized Frequency (a) (c)
DC Att. >= 322.1021 dB Mirr Att. >= 314.1433 dB Stopband Att. >= 16.1804 dB Cod. Gain = 9.518 dB DC Att. >= 297.1536 dB Mirr Att. >= 303.5288 dB Stopband Att. >= 16.822 dB Cod. Gain = 9.7696 dB
0 h0 0 h0 h1 −5 h1 −5 h2 h3
−10 −10 h2 h4 h5 −15 −15 h6 h3 h7 −20 −20 h8 h4 h9 Magnitude Response (dB) Magnitude Response (dB) −25 −25 h10 h5 h11 −30 −30 h12 h6 −35 −35 h13 h14 h7 −40 −40 h15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Normalized Frequency Normalized Frequency
(c) (d)
FIGURE 4. Frequency and impulse resp onses of orthogonal transforms: a 8-channel 8-tap
DCT b 8-channel 16-tap LOT c 8-channel 40-tap GenLOT d 16-channel 32-tap LOT.
wl H 2 wl 0 H0 2 x[n] DC wl H (z) M H1 2 0 wl H1 2 Wavelet transform embedded H (z) M 1 bit plane coder compressed bit stream
H (z) M M-1
Block transform
FIGURE 5. Complete co der's diagram.
in Table 1.1 clearly demonstrate the sup eriority of well-optimized lapp ed trans-
forms over wavelets. For a smo oth image like Lena where the wavelet transform
can suciently decorrelate, SPIHT o ers a comparable p erformance. However, for
a highly-textured image like Barbara, the 8 40 GenLOT and the 16 32 LOT
co der can provide a PSNR gain of around 2 dB over a wide range of bit rates.
1. Blo ckTransforms in Progressive Image Co ding 8
Lena Progressive Transmission Coders Barbara Progressive Transmission Coders Comp. SPIHT Xiong et al 8 x 8 8 x 16 8 x 40 16 x 32 Comp. SPIHT Xiong et al 8 x 88 x 16 8 x 40 16 x 32 Ratio (9-7WL) (DCT) DCT LOT GenLOT LOT Ratio (9-7WL) (DCT) DCT LOT GenLOT GLBT 1:8 40.41 39.62 39.91 40.09 40.43 40.16 1:8 36.41 36.10 36.31 37.43 38.08 38.02 1:16 37.21 36.00 36.38 36.75 37.32 36.96 1:16 31.40 30.82 31.11 32.70 33.47 33.47 1:32 34.11 32.25 32.90 33.57 34.23 33.87 1:32 27.58 26.83 27.28 28.80 29.53 29.70 1:64 31.10 -- 29.67 30.48 31.16 30.85 1:64 24.86 -- 24.58 25.70 26.37 26.63 1:100 29.35 -- 27.80 28.61 29.31 28.98 1:100 23.76 -- 23.42 24.34 24.95 25.14 1:128 28.38 -- 26.91 27.61 28.35 27.99 1:128 23.35 -- 22.68 23.37 24.01 24.09
(a) (b)
TABLE 1.1. Co ding results of various progressive co ders a for Lena b for Barbara.
Unlike other blo ck transform co ders whose p erformance dramatically drops at very
high compression ratios, the new progressive co ders are consistent throughout as
illustrated in Figure 6. Lastly, b etter decorrelation of the DC band provides around
0:3 0:5 dB improvementover the earlier DCT emb edded co der in [10].
38 Barbara 37 Lena 8x40 GenLOT 8x40 GenLOT 36 36.5 16x32 LOT 16x32 LOT SPIHT SPIHT 36 34
35.5 32 35
34.5 30 PSNR (dB) PSNR (dB)
34 28 33.5
33 26
32.5 24 32 20 25 30 35 40 45 50 10 20 30 40 50 60 70 80 90 100 Compression Ratio Compression Ratio
(a) (b)
FIGURE 6. Rate-distortion curves of various progressive co ders a for Lena b for
Barbara.
Figure 7 - 9 con rm lapp ed transforms' sup eriority in reconstructed image quality
as well. Figure 7 shows reconstructed Barbara images at 1:32 byvarious blo ck trans-
forms. Comparing to JPEG, blo cking artifacts are already remarkably reduced in
the DCT-based co der in Figure 7 a . Blo cking is completely eliminated when DCT
is replaced by b etter lapp ed transforms as shown in Figure 7 c - d , and Figure 8.
A closer lo ok in Figure 9 a - c where only 256 256 image p ortions are shown
so that artifacts can b e more easily seen reveals that b esides blo cking elimination,
good lapp ed transform can preserve texture b eautifully the table cloth and the
clothes pattern while keeping the edges relatively clean. The absence of excessive
ringing considering the transform's long lters should not come across as a sur-
prise: a glimpse of the time resp onses of the GenLOT in Figure 4 c reveals that
the high-frequency bandpasses and the highpass lter are very carefully designed {
their lengths are essentially under 16-tap. Comparing to SPIHT, the reconstructed
images haveanoverall sharp er and more natural lo ok with more de ning edges and
more evenly reconstructed texture regions. Although the PSNR di erence is not as
striking in the Goldhill image, the improvement in p erceptual quality is rather sig-
1. Blo ckTransforms in Progressive Image Co ding 9
ni cant as shown in Figure 9 d - f . Even at 1:100, the reconstructed Goldhill image
in Figure 8 d is still visually pleasant: no blo cking and not much ringing. More ob-
jective and sub jectiveevaluation of blo ck-transform-based progressive co ding can
b e found at http://saigon.ece.wisc.edu/~wa veweb/ Coder/ index.h tml.
(a) (b)
(c) (d)
FIGURE 7. Barbara co ded at 1:32 by a 8 8 DCT b 8 16 LOT c 8 40 GenLOT
d 16 32 LOT.
As previously mentioned, the improvement over wavelets keys on the lapp ed
transform's ability to capture and separate lo calized signal comp onents in the fre-
quency domain. In the spatial domain, this corresp onds to images with directional
rep etitive texture patterns. To illustrate this p oint, the lapp ed-transform-based
1. Blo ckTransforms in Progressive Image Co ding 10
(a) (b)
(c) (d)
FIGURE 8. Goldhill co ded by the 8 40 GenLOT co der at a 1:16, 33.36 dB b 1:32,
30.79 dB c 1:64, 28.60 dB d 1:100, 27.40 dB.
co der is compared against the FBI Wavelet Scalar Quantization WSQ standard
[23]. When the original 768 768 gray-scale ngerprint image is shown in Figure
10 a is compressed at 1 : 13:6 43366 bytes by WSQ, Bradley et al rep orted a
PSNR of 36:05 dB. Using the 16 32 LOT in Figure 4 d , a PSNR of 37:87 dB can
be achieved at the same compression ratio. For the same PSNR, the LOT co der
can compress the image down to 1 : 19 where the reconstructed image is shown
in Figure 10 b . To put this in p ersp ective, the wavelet packet SFQ co der in [22]
rep orted a PSNR of only 37:30 dB at 1:13.6 compression ratio. At 1:18:036 32702
bytes , WSQ's reconstructed image as shown in Figure 10 c has a PSNR of 34:42
1. Blo ckTransforms in Progressive Image Co ding 11
(a) (b) (c)
(d) (e) (f)
FIGURE 9. Perceptual comparison b etween wavelet and blo ck transform emb edded co der.
Zo om-in p ortion of a original Barbara b SPIHT at 1:32 c 8 40 GenLOT emb edded
co der at 1:32 d original Goldhill e SPIHT at 1:32 c 8 40 GenLOT emb edded
co der at 1:32.
dB while the LOT co der pro duces 36:32 dB. At the same distortion, we can com-
press the image down to a compression ratio of 1:26 22685 bytes as shown in
Figure 10 d . Notice the high p erceptual image quality in Figure 10 b and d : no
visually disturbing blo cking and ringing artifacts.
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1. Blo ckTransforms in Progressive Image Co ding 12
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