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In this paper, a new image compression algorithm is proposed based on the efficient construction of wavelet coefficient lower trees. The main contribution of the proposed lower-tree wavelet (LTW) encoder is the utilization of coefficient trees, not only as an efficient method of grouping coefficients, but also as a fast way of coding them. Thus, it presents state-of-the-art compression performance, whereas its complexity is lower than the one presented in other wavelet coders, like SPIHT and JPEG 2000. Fast execution is achieved by means of a simple two-pass coding and one-pass decoding algorithm. Moreover, its computation does not require additional lists or complex data structures, so there is no memory overhead. A formal description of the algorithm is provided, while reference software is also given. Numerical results show that our codec works faster than SPIHT and JPEG 2000 (up to three times faster than SPIHT and fifteen times faster than JPEG 2000), with similar coding efficiency
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006 1437
Low-Complexity Multiresolution Image Compression
Using Wavelet Lower Trees
Jose Oliver, Member, IEEE, and Manuel P. Malumbres, Member, IEEE
Abstract—In this paper, a new image compression algorithm is
proposed based on the efficient construction of wavelet coefficient
lower trees. The main contribution of the proposed lower-tree
wavelet (LTW) encoder is the utilization of coefficient trees, not
only as an efficient method of grouping coefficients, but also as a
fast way of coding them. Thus, it presents state-of-the-art com-
pression performance, whereas its complexity is lower than the
one presented in other wavelet coders, like SPIHT and JPEG 2000.
Fast execution is achieved by means of a simple two-pass coding
and one-pass decoding algorithm. Moreover, its computation does
not require additional lists or complex data structures, so there
is no memory overhead. A formal description of the algorithm is
provided, while reference software is also given. Numerical results
show that our codec works faster than SPIHT and JPEG 2000
(up to three times faster than SPIHT and fifteen times faster than
JPEG 2000), with similar coding efficiency.
Index Terms—Image compression, low complexity, tree-based
coding, wavelets.
I. INTRODUCTION
D
URING the last decade, several image compression
schemes emerged in order to overcome the known limi-
tations of block-based algorithms that use the discrete cosine
transform (DCT). Some of these alternative proposals were
based on more complex techniques, like Vector Quantization
[2] and Fractal image coding [3], whereas others simply pro-
posed the use of a different and more suitable mathematical
transform: the discrete wavelet transform (DWT) [4]. At that
time, there was a general idea: more efficient image coders
could only be achieved by means of sophisticated techniques
with high complexity. The embedded zero-tree wavelet coder
(EZW) [5] can be considered the first Wavelet image coder
that broke that trend. Since then, many wavelet coders were
proposed and finally, the DWT was included in the JPEG 2000
standard [6] due to its compression efficiency among other
interesting features (scalability, etc.).
All the wavelet-based image coders, and in general all the
transform-based coders, consist of two main stages. In the first
one, the image is transformed from spatial domain to another
one, in the case of the wavelet transform a combined spatial-fre-
quency domain, called wavelet domain. In the second pass, the
Manuscript received March 16, 2005; revised May 30, 2006; accepted May
30, 2006. This work was supported by the Spanish Ministry of Education and
Science under grant TIC2003-00339. A preliminary version of this paper was
first presented at the IEEE Data Compression Conference, Snowbird, UT, in
March 2003. This paper was recommended by Associate Editor H. Gharavi.
J. Oliver is with the Department of Computer Engineering (DISCA), Poly-
technic University of Valencia, 46022 Valencia, Spain (e-mail: joliver@disca.
upv.es).
M. P. Malumbres is with the Department of Physics and Computer Engi-
neering, Miguel Hernandez University, 03202 Elche, Spain (e-mail: mels@umh.
es).
Digital Object Identifier 10.1109/TCSVT.2006.883505
transform coefficients are quantized and encoded in an efficient
way to achieve high compression efficiency and other features.
The wavelet transform can be implemented as a regular
filter-bank, however several strategies have been proposed to
reduce the running time and memory requirements. This way, a
line-based processing is proposed in [7], whereas an alternative
wavelet transform method, called lifting scheme, is proposed in
[8]. The lifting transform provides in-place calculation of the
coefficients by overwriting the input samples, and reduces the
number of operations required to compute the DWT.
On the other hand, the coding pass is not usually improved
in terms of complexity and memory usage. When designing a
new wavelet image encoder, the most important factor to opti-
mize is usually the rate/distortion (R/D) performance, whereas
other features like embedded bit-stream, signal-to-noise ratio
(SNR) scalability, spatial scalability and error resilience are also
considered. In this paper, we propose an algorithm aiming to
achieve state-of-the-art coding efficiency, with very low execu-
tion time. Moreover, due to in-place processing of the coeffi-
cients, there is no memory overhead (it only needs memory to
store the source image). In addition, our algorithm is naturally
spatial scalable and it is possible to achieve SNR scalability.
The key idea of the proposed algorithm is the use of wavelet
coefficient trees as a fast method of efficiently grouping co-
efficients. Tree-based wavelet coders have been widely used
in the literature [5], [9], [10], [14], presenting good R/D per-
formance. However, their excellent opportunities for fast pro-
cessing of quantized coefficients have not been clearly shown
so far. Wavelet trees are a simple way of grouping coefficients,
reducing the total number of symbols to be coded, which in-
volves not only good compression performance but also fast
processing. Moreover, for a low-complexity implementation,
bit-plane coding, present in many wavelet coders [5], [6], [9],
[11], [12], must be avoided. This way, multiple scans of the
transform coefficients, which involves many memory accesses
and causes high cache miss rates, is not performed, at the ex-
pense of generating a non-(SNR)-embedded bitstream.
This paper is organized as follows. In Section II, we analyze
the complexity and coding efficiency of some important wavelet
image coders, focusing on the non-embedded proposals. In Sec-
tion III, the proposed tree-based algorithm is described. Sec-
tion IV describes some implementation details and optimization
considerations. Finally, in Section V, we compare our proposal
with other wavelet image coders using real implementations.
II. P
REVIOUS WAVELET IMAGE CODERS AND
THEIR COMPLEXITY
One of the first efficient wavelet image coders reported in the
literature is EZW [5]. It is based on the construction of coeffi-
cient-trees and successive-approximations, which can be imple-
1051-8215/$20.00 © 2006 IEEE
1438 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006
mented with bit-plane coding. Due to its successive-approxima-
tion nature, it is SNR scalable, although at the expense of spa-
tial scalability. SPIHT [9] is an advanced version of this algo-
rithm, where coefcient-trees are processed in a more efcient
way. In this coder, coefcient-trees are partitioned depending
on the signicance of the coefcients belonging to each tree. If
a coefcient in a tree is higher than a threshold determined by
the current bit-plane, the tree is successively divided following
some established partitioning rules. Both EZW and SPIHT need
the computation of coefcient-trees to search for signicant co-
efcients and they take various iterations focusing on a dif-
ferent bit-plane, which involves high computational complexity.
A block-based version of SPIHT, called SPECK, is presented
in [11]. The main difference of this new version is that coef-
cients are grouped and partitioned using rectangular structures
instead of coefcient trees. A low-complexity implementation
of SPECK, called SBHP [12], was proposed in the framework
of JPEG 2000. In SBHP, an image is rst divided into blocks
(sized 64
64 or 128 128) and then, each block is processed
as in SPECK. To reduce complexity, Huffman coding is used
instead of arithmetic coding, which causes a decrease in coding
efciency.
In the nal JPEG 2000 standard [6], the proposed algorithm
(a modied version of EBCOT [13]) does not use coefcient-
trees, but it performs bit-plane coding in code-blocks, with three
passes per plane so that the most important information is en-
coded in rst place. In order to overcome the disadvantage of
not using coefcient-trees, it uses an iterative optimization al-
gorithm, based on the Lagrange multiplier method, along with
a large number of contexts. JPEG 2000 obtains both spatial
and SNR scalability by reordering the encoded image. Hence,
the nal bitstream is very versatile. However, this encoder is
complex, mainly due to the use of the time-consuming iterative
optimization algorithm, the introduction of a reordering stage,
and the use of bit-plane coding with many contexts. In addi-
tion, a larger bitstream than the one actually encoded is usually
generated.
Many times, wavelet image encoders have features that are
not always needed, but which make them both CPU and memory
intensive. Fast processing is often preferable. Low complexity
may be desirable for high-resolution digital camera shooting,
where high delays could be annoying and even unacceptable
(even more for modern digital cameras, which tend to increase
their resolution). In general, image editing for large images (es-
pecially in GIS applications) cannot be easily tackled with the
complexity of previous encoders.
Since iterative methods and bit-plane coding must be avoided
to reduce complexity, very fast coding can only be achieved
through simpler non-SNR-embedded techniques (like in base-
line JPEG). In these encoders, an image is encoded at a constant
quality after applying a uniform quantization to the coefcients.
A. Non-Embedded Coders
One of the rst tree-based non-embedded coders is SFQ [10].
This wavelet encoder uses space quantization by means of a
tree-pruning algorithm, which modies the shape of the trees
by pruning their branches, whereas a scalar quantization is ap-
plied for frequency quantization. Although it achieves higher
compression than SPIHT, the iterative tree pruning stage makes
it about ve times slower than SPIHT.
Non-embedded coding was rst proposed to reduce com-
plexity in tree-based wavelet coding in [14], where a non-em-
bedded version of SPIHT was proposed. In this modied
SPIHT, once a coefcient is found to be signicant, all the
signicant bits are encoded, avoiding the renement passes (see
[9] for details). Note that a non-embedded version of SPECK
and SBHP is also possible with the same modications. Al-
though these non-embedded versions are faster than the original
ones, neither multiple image scan nor bit-plane processing of
the sorting passes (used to nd signicant coefcients) is
avoided, and hence, the complexity problem still remains. Note
that these modications of SPIHT and SPECK are neither SNR
nor resolution scalable.
Besides the JPEG standard, other DCT-based non-embedded
image coders have been proposed. In particular, the intra mode
of the H.264 standard [16]. However, due to the use of time-con-
suming prediction techniques on the coding side, the coding/de-
coding processes are very asymmetric, and the resulting encoder
is very slow.
III. M
ULTIRESOLUTION IMAGE CODING
USING LOWER TREES
For the most part, digital images are represented with a set
of pixels,
. The encoder proposed in this paper is applied to
a set of coefcients
resulting from a dyadic decomposition
, with . The most commonly used decomposi-
tion for image compression is the hierarchical wavelet subband
transform [4], thus an element
is called transform co-
efcient. In a wavelet transform, we call
, , and
the subbands resulting from the rst level of the image decom-
position, corresponding to horizontal, vertical and diagonal fre-
quencies. The rest of the transform is computed with a recursive
wavelet decomposition on the remaining low frequency sub-
band, until a desired decomposition level (N) is achieved (
is the remaining low frequency subband).
In Section II, we mentioned that one of the main drawbacks
in previous wavelet-based image encoders is their high com-
plexity. Many times, that is due to time-consuming iterative
methods, and bit-plane coding used to provide a fully embedded
bitstream. Although embedding is a nice feature in an image
coder, it is not always needed and other alternatives, like spa-
tial scalability, may be more valuable depending on the nal
application. In this section, we propose a tree-based coding al-
gorithm that is able to encode the wavelet coefcients without
performing an image scan per bit plane.
Tree-based wavelet image encoders are proved to efciently
store the transform coefcients, achieving good performance re-
sults. However, in the algorithm proposed in this paper, a tree-
based structure is introduced, not only to remove redundancy
among subbands, but also as a simple and fast way of grouping
coefcients.
As in the rest of tree-based encoders, coefcients can be log-
ically arranged as trees, as shown in Fig. 1. In this gure, we
observe that the coefcients in the wavelet subbands (except the
leaves) have always four direct descendants (i.e., four descen-
dants at a distance of one), while the rest of descendants can be
recursively obtained from the direct descendants. On the other
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006 1439
Fig. 1. Coefcient-trees in the proposed algorithm.
hand, in the subband, three out of every 2 2 coefcients
have four direct descendants, and the remaining coefcient in
the 2
2 block has no descendant.
In our proposal, the quantization process is performed with
two strategies: one coarser and another ner. The ner one con-
sists in applying a scalar uniform quantization to the coef-
cients, and it can be applied with the normalization factor in
the lifting transform (if the normalization factor in the lifting
scheme is
, and the quantization factor is , only a multiplica-
tion by
is needed). The coarser one is based on removing
bit planes from the least signicant part of the coefcients, and it
is performed while the algorithm is applied. Related to this bit
plane quantization, we dene
as the number of least
signicant bits to be removed.
We say that a coefcient
is signicant if it is different
from zero after discarding the least signicant
bits, in
other words, if
. In our proposal, signicant co-
efcients are encoded by using arithmetic coding, with a symbol
indicating the number of bits needed to encode that coefcient.
Then, the signicant bits and sign are binary encoded (in other
words, raw encoded).
Regarding the insignicant coefcients, a coefcient is
called lower-tree root if this coefcient and all its descendants
are insignicant (i.e., lower than
). The set formed
by all these coefcients is a lower-tree. We use the symbol
to point out that a coefcient is the root of a
lower-tree. The rest of coefcients in the lower-tree are labeled
as
, but are not encoded. On the
other hand, if a coefcient is lower than
, but it does not
form or belong to a lower-tree because it has at least one sig-
nicant descendant, it is considered
.
A. Lower-Tree Encoder Algorithm
Once we have dened the basic concepts to understand the
algorithm, we are ready to describe the coding process. It is
a two-pass algorithm. During the rst pass, the wavelet coef-
cients are properly labeled according to their signicance and
the lower-trees are formed. In the second image pass, the coef-
cient values are coded by using the labels computed in the rst
pass.
The coding algorithm is presented in Algorithm 1. Let us de-
scribe this algorithm. At the encoder initialization, the number
of bits needed to represent the highest coefcient
is calculated. This value and the parameter are
output to the decoder. Afterwards, we initialize an adaptive
arithmetic encoder that is used to encode the number of bits
required by the signicant coefcients, and the
and
symbols.
In the rst image pass, the lower-tree labeling process is
performed in a recursive way, by building the lower-trees
from leaves to root. In the rst level subbands, coef-
cients are scanned in 2
2 blocks and, if the four coef-
cients are insignicant (i.e., lower than
), they
are considered part of the same lower-tree, being labeled
as
. Then, when scanning
higher level subbands, if a 2
2 block has four in-
signicant coefcients, and all their direct descendants are
, the coefcients in that block are
also labeled as
, increasing the
size of the lower-tree.
However, when at least one coefcient in the block is signif-
icant, the lower-tree cannot continue growing. In that case, an
insignicant coefcient in the block is labeled as
if all
its descendants are
, otherwise an
insignicant coefcient is labeled as
.
In the second pass, all the subbands are explored from
the Nth level to the rst one, and all their coefcients are
scanned in medium-sized blocks (to take advantage of
data locality). For each coefcient in a subband, if it is
a lower-tree root or an isolated lower, the corresponding
or symbol is encoded.
On the other hand, if a coefcient has been labeled as
no output is needed because
this coefcient is already represented by the lower-tree to
which it belongs.
A signicant coefcient is coded as follows. A symbol
indicating the number of bits required to represent that co-
efcient is arithmetically coded, and the signicant bits and
sign are raw coded. However, two types of numeric symbols
are used depending on the direct descendants of that coef-
cient. (a) A regular numeric symbol
, which simply
shows the number of bits needed to encode a coefcient, (b)
and a special
numeric symbol ,
which not only indicates the number of bits of the coef-
cient but also the fact that its descendants are labeled as
, and thus they belong to a
lower-tree not yet codied. This type of symbol is able to
represent efciently some special lower-trees, in which the root
coefcient is signicant and the rest of coefcients are insignif-
icant. Note that the number of symbols needed to represent both
sets of numeric symbols is
, therefore
the arithmetic encoder must be initialized to handle at least
this amount of symbols, along with two additional symbols:
the
and symbols. Observe
that the rst rplanes bits and the most signicant non-zero bit
are not encoded (the decoder can deduce the most signicant
non-zero bit through the arithmetic symbol that indicates the
number of bits required to encode this coefcient).
An important difference between our tree-based wavelet al-
gorithm and others like [5] and [9] is how the coefcient tree
building process is detailed. Our algorithm includes a simple
1440 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006
and efcient recursive method (within the E2 stage) to deter-
mine if a coefcient has signicant descendants in order to form
coefcient trees. However, EZW and SPIHT leave it as an open
and implementation dependent aspect, which increases dras-
tically the algorithm complexity if it is not carefully solved
(for example, if searches for signicant descendants are inde-
pendently calculated for every coefcient whenever they are
needed). Anyway, an efcient implementation for EZW and
SPIHT would increase their memory consumption due to the
need to store the maximum descendant for every coefcient ob-
tained during a pre-search stage.
B. Lower-Tree Decoder Algorithm
The decoding algorithm performs the reverse process in only
one pass, since symbols are directly decoded from the incoming
bitstream. All the subbands must be scanned in the order used
in the encoder. The granularity of this scan is 2
2 coefcient
blocks, thus coefcients that share the same parent (i.e., sibling
coefcients) are handled together. Note that all the coefcient
blocks have ascendant except those in the
subband.
When an insignicant coefcient is decoded, we set its value
to 0, since its order of magnitude is unknown (we only know
that it is lower than
). Later, insignicant coefcients in
a lower-tree are automatically propagated, since when the parent
of four sibling coefcients has been set to 0, all the descendant
coefcients are also assigned a value of 0, so that lower-trees are
recursively generated. However, if an isolated lower coefcient
has been decoded, it must not be propagated as a lower-tree.
Hence, a different value must be assigned. For this case, we keep
this coefcient as
until its 2 2 direct
descendants are scanned. At that moment, we can safely update
its value to 0 without risk of unwanted propagations, because no
more direct descendants of this coefcient will be scanned.
C. Encoder Features
The proposed algorithm is resolution scalable due to the se-
lected scanning order and the nature of the wavelet transform.
This way, the rst subband that the decoder attains is the
,
which is a low-resolution scaled version of the original image.
Then, the decoder progressively receives the remaining sub-
bands, from lower frequency subbands to higher ones, which
are used as a complement to the low-resolution image to recur-
sively double its size, which is known as Mallat decomposition
[17]. Spatial and SNR scalability are closely related features.
Spatial resolution allows us to have different resolution images
of the same image. Through interpolation techniques, all these
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006 1441
Fig. 2. A 2-level wavelet transform of an 8 8 example image.
images could be resized to the original size, so that the larger
an image is, the closer to the original it gets. Therefore, this al-
gorithm could also be used for SNR scalability purposes. Other
features, like the denition of Regions Of Interest (ROI), can be
implemented in this coder by using a specic
param-
eter for those coefcients in the trees representing that special
area.
As in most non-embedded encoders (like baseline JPEG,
H.264 intra mode, non-embedded SPIHT/SPECK, etc.), pre-
cise rate control is not feasible in the proposed algorithm
because it is usually achieved with bit-plane coding (like in
SPIHT) or iterative methods (like in JPEG 2000 or SFQ),
which signicantly increase the complexity of the encoder.
Nevertheless, rate control is still possible based on a fast anal-
ysis of the transform image features by extracting one or more
numerical parameters that are employed to determine the quan-
tization parameters needed to approximate a bit rate (see [15]
for details).
D. A Simple Example
Fig. 2 shows a small 8
8 image that has been transformed
using a 2-level DWT. In this section, we show the result of ap-
plying our tree-based encoder using this sample image. These
wavelet coefcients have been scalar quantized, and the selected
parameter is 2. Regarding the maxplane parameter, it
can be easily computed as
.
When the coarser quantization is applied using
,
the values within the interval
are absolutely quan-
tized. Thus, all the signicant coefcients can be represented by
using from 3 to 6 bits, and hence, the symbol set needed to repre-
sent the signicance map is
.
In this symbol set, special
numeric symbols
are marked with a superscript
,an symbol represents
an isolated lower coefcient, and an
indicates a regular
lower symbol (the root of a lower-tree). Coefcients be-
longing to a previously encoded lower-tree (those labeled as
) are not encoded and, in our
example, are represented with a star
. Fig. 3 shows the
symbols resulting from applying our algorithm to the example
image, and if we scan the subbands from the highest level (N)
to the lowest one (1), in 2
2 blocks, from left to right and top
to bottom, the resulting bitstream is the one illustrated in Fig. 4.
Note that the sign is not necessary for the LL subband since its
coefcients are always positive.
Fig. 3. Symbols resulting from applying our algorithm to the example image.
IV. I
MPLEMENTATION
CONSIDERATIONS
Implementation details and further adjustments may improve
the performance of a compression algorithm. In this section, we
give a guide for a successful implementation of the LTW algo-
rithm. All the improvements introduced in this section should
preserve fast processing.
Context coding has been widely used to improve the R/D per-
formance in image compression. Although high-order context
modeling presents high complexity, simpler context coding can
be efciently employed without noticeable increase in execution
time. We propose the use of two contexts based on the signi-
cance of the left and the upper coefcients, thus if they both are
insignicant or close to insignicant, a different model is used
for coding. Adding a few more models to establish more signi-
cance levels (and thus more number of contexts) would improve
compression efciency. However, it would slow down the algo-
rithm, mainly due to the context formation evaluation and the
higher memory usage.
Recall that
indicates the number of bits needed to
represent the highest coefcient in the wavelet decomposition.
This value (along with
) determines the number of sym-
bols needed for the arithmetic encoder. However, coefcients in
different subbands tend to be different in magnitude order, so
this parameter can be specically set for every subband level. In
this manner, the arithmetic encoder is initialized exactly with the
number of symbols needed in every subband, which increases
the coding efciency.
Related to the coarser quantization, consider the case in which
three coefcients in a block are insignicant, and the fourth
value is very close to insignicant. In this case, we can consider
that the entire block is insignicant and all its coefcients can
be labeled as
. The slight error in-
troduced is compensated by the saving in bit budget.
In general, each of these R/D improvements causes a slight
increase in PSNR (from 0.05 to 0.1 dB, depending on the source
image and the target bitrate).
V. C
OMPARISON WITH OTHER WAVELET CODERS USING
REAL IMPLEMENTATIONS
We have implemented the lower-tree wavelet (LTW) coding
and decoding algorithms in order to test their performance.
They have been implemented using standard C++ language
(without using assembly language or platform-dependant
features), and the simulation tests have been performed on a
regular Personal Computer (with a 500 MHz Pentium Celeron
1442 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006
Fig. 4. Example image encoded using the proposed tree-based algorithm.
TABLE I
PSNR (dB) W
ITH
DIFFERENT BIT RATES AND
CODERS
USING
LENA (512
512)
processor with 256 KB L2 cache), generating image les that
contain the compressed images, including the le headers
required for self-containing decompression. The reader can
easily perform new tests by using the LTW implementation
available at http://www.disca.upv.es/joliver/LTW.
In order to compare our algorithm with other wavelet en-
coders, we have selected the classical Lena and Barbara im-
ages (monochrome, 8 bpp, 512
512), the VisTex texture data-
base (monochrome, 8 bpp, 512
512) from the MIT media
lab, and the Café and Woman images (monochrome, 8 bpp,
2560
2048) from the JPEG 2000 test bed. The widely used
Lena image (from the USC) allows us to compare it with prac-
tically all the published algorithms, since results are commonly
expressed using this image. The VisTex set allows testing the
behavior of the coders when coding textures. Finally, the Café
and Woman images are less blurred and more complex than
Lena and represent pictures taken with a 5-Mpixel-high de-
nition digital camera.
Table I provides R/D performance when using the Lena
image. In general, we see that LTW achieves similar or better
results than the other coders, including the JPEG 2000 stan-
dard, whose results have been obtained using the reference
software Jasper [18], an ofcial implementation included in
the ISO/IEC 15444-5 standard. For SPIHT, we have used the
implementation provided by the authors in the original paper.
For H.264 intra mode coding, results have been obtained with
the JM 9.6 reference software. For this coder, the PSNR result
for each bitrate has been interpolated from the nearest bitrates,
since the quantization granularity is very low, and an exact
bitrate cannot be achieved.
PSNR results for the rest of images are shown in Table II.
In this only SPIHT and Jasper have been compared to LTW,
since the compiled versions of the rest of coders have not been
TABLE II
PSNR (dB) W
ITH DIFFERENT
BIT
RATES AND
CODERS FOR
CAFÉ,W
OMAN,
B
ARBARA,
AND VisTex D
ATABASE
released, or results are not published for these images. We can
observe that R/D performance for Café is still higher using
our algorithm, although Jasper performs similar to LTW. For
Woman, we see that our algorithm exceeds in performance
both SPIHT and Jasper. In the case of the textures from the
VisTex database, we present the average PSNR in this Table,
which shows that, in general, LTW works better than JPEG
2000 and SPIHT for coding of textures. Only a few images
from the set (and only at high-bit rates) exhibit better coding
results for JPEG 2000 (details of the results for each image
in the set are available at http://www.disca.upv.es/joliver/csvt/
textures.pdf). However, at high bit rates, Jasper encodes high-
frequency images, like Barbara, better than LTW. Two reasons
may explain it. First, when high-frequency images are encoded
at high bit rates, many coefcients in high-frequency subbands
are signicant, and hence our algorithm is not able to build
large lower-trees. Second, recall that JPEG 2000 includes many
contexts, and it causes higher performance for high-frequency
images. In our experiments, we have observed that if more
than two contexts are used in LTW, the R/D performance for
Barbara is close to the results shown with Jasper, but at the
cost of higher execution time.
On the other hand, the reader can perform a subjective evalu-
ation of the images Lena and Barbara encoded at 0.125 bpp with
SPIHT, JPEG 2000 and LTW by using Fig. 5, which shows a de-
tail of the face of Lena and another detail of Barbaras checked
trousers. In the rst group of pictures, if we look carefully at
Lenas lips, eyes and nose, we can observe that LTW offers
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006 1443
Fig. 5. Details of the Lena and Barbara images (at 0.125 bpp) for a subjective evaluation, using (a) SPIHT, (b) JPEG 2000, and (c) LTW.
TABLE III
E
XECUTION TIME COMPARISON FOR LENA AND CAFÉ (TIME IN MILLION OF
CPU CYCLES)
the best results, achieving better denition and sharpness than
SPIHT and especially than JPEG 2000. However, if we ana-
lyze a highly detailed image such as Barbara, and we focus on a
high-frequency area (e.g., the trousers in the Barbara image) we
can clearly see that JPEG 2000 gives the best results, and SPIHT,
which decompresses an image full of blurred areas, produces the
worst results. Both results are consistent with the PSNR results,
and conrm that tree-based algorithms work better with mod-
erate to low detailed images, while JPEG 2000 is a better option
in detailed images, due to the reasons aforementioned.
The main advantage of the LTW algorithm is its lower com-
plexity. Table III shows that our algorithm greatly outperforms
SPIHT and Jasper in terms of execution time.
1
For medium sized
images (512
512), our encoder is from 3.25 to 11 times faster
than Jasper, whereas LTW decoder executes from 1.5 to 2.5
times faster than Jasper decoder, depending on the rate. In the
case of SPIHT, our encoder is from 2 to 2.5 times faster, and
the decoding process is from 1.7 to 2.1 times faster. With larger
images, like Café (2560
2048), the advantage is greater. LTW
encoder is from 5 to 16 times faster than Jasper and the decoder
is from 1.5 to 2.5 times faster. With respect to SPIHT, our al-
gorithm encodes Café from 1.7 to 3 times faster, and decodes it
from 1.5 to 2.5 times faster.
Note that in these tables we have only evaluated the coding
and decoding processes, and not the transform stage, since the
wavelet transform used is the same in all the cases; the popular
Daubechies 9/7 biorthogonal wavelet lter. Other wavelet trans-
forms, like Daubechies 23/25, have shown better compression
performance. However, this improvement is achieved with more
lter taps, and thus increasing the execution time of the DWT.
1
Measuring the complexity of algorithms is a hard issue. Execution time
of their implementation is largely dependant of the optimization level. This
way, there are commercial implementations of JPEG 2000 not included in the
ISO/IEC 15444-5 that are faster than Jasper, however they are usually imple-
mented by using platform dependant pieces of code (in assembly language) and
multimedia SIMD instructions. In our tests, SPIHT, JPEG 2000 and LTW im-
plementations are, as far as possible, written and compiled under the same con-
ditions, using plain C/C++ language and MS Visual C++6.0 for all them.
1444 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 11, NOVEMBER 2006
Other non-embedded encoders, like SFQ and H.264 intra
mode, are much slower than LTW. In particular, SFQ coding is
more than 10 times slower than LTW, while the H.264 encoder
(JM 9.6 implementation) is more than 50 times slower than our
proposal, due to the time-consuming predictive algorithm used
in this encoder.
Besides compression performance and complexity, a third
major issue usually considered in an image coder is the memory
usage. Our wavelet encoder and decoder are able to perform
in-place processing of the wavelet coefcients, and thus they
do not need to handle additional lists or memory-consuming
structure. This way, only 21 Mbytes are needed to encode the
Café image using LTW (note that 20 Mbytes are needed to store
the image in memory using
integer type), whereas SPIHT and
Jasper require 42 and 64 Mbytes, respectively.
2
VI. C
ONCLUSION
In this paper, we have presented a new wavelet image en-
coder based on the construction and efcient coding of wavelet
lower-trees (LTW). Its compression performance is within the
state-of-the-art, achieving similar results as other popular algo-
rithms (SPIHT is improved in 0.20.4 dB, and JPEG 2000 with
Lena in 0.35 dB as mean value).
However, the main contribution of this algorithm is its lower
complexity. Depending on the image size and bitrate, it is able
to encode an image up to 15 times faster than Jasper and three
times faster than SPIHT. Thus, it can be stated that the LTW
coder is one of the fastest efcient image coders that can be
reported in the literature.
Therefore, due to its lower complexity, its high symmetry,
its simply design, and the lack of memory overhead, we think
that the LTW is a good candidate for real-time interactive multi-
media communications, allowing implementations both in hard-
ware and in software.
2
Results obtained with the Windows XP task manager, peak memory usage
column.
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