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1
PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET CODERS
Shih-Hsuan Yang and Wu-Jie Liao
Department of Computer Science and Information Engineering
National Taipei University of Technology
1, Sec. 3, Chung-Hsiao E. Rd.
Email: shyang@ntut.edu.tw
ABSTRACT
In this paper, we analyze the design issues for the SPIHT (set partitioning in hierarchical
trees) coding, one of the most prestigious embedded-wavelet-based algorithms in the
literature. Equipped with the multiresolution decomposition, progressive scalar
quantization and adaptive arithmetic coding, SPIHT generates highly compact scalable
bitstreams suitable for real-time multimedia applications. The design parameters at each
stage of SPIHT greatly influence its performance in terms of compression efficiency and
computational complexity. We first evaluate two important classes of wavelet filters,
orthogonal and biorthogonal. Orthogonal filters are energy preserving while biorthogonal
linear-phase filters allow symmetric extension across boundary. We investigate the
benefits from energy compaction, energy conservation, and symmetric extension,
respectively. Second, the magnitude of biorthogonal wavelet coefficients may not
faithfully reflect their actual significance. We explore a scaling scheme in quantization
2
that minimizes the overall mean square error. The contribution of entropy coding is
measured at last.
1. INTRODUCTION
Compression lays the basis for the processing, transmission, and storage of multimedia
data. A picture is worth a thousand words, but full utilization of pictorial information is
impossible without compression. The most successful image coders adopt the
transform-coding structure shown in Fig. 1. A linear transformation such as the discrete
cosine transform (DCT) or discrete wavelet transform (DWT) converts the pixels into
uncorrelated and condensed transform coefficients. The quantizer adequately divides the
coefficient space into disjoint cells and the transform coefficients are reconstructed by a
representative value within the cell. Quantization is thus lossy in nature; under a specified
distortion requirement it aims to minimize the bit rate (or entropy) of the output symbols.
The entropy coder at the last stage finds the most economical binary representation for
the quantization symbol sequence. The baseline JPEG standard follows this structure that
combines DCT, perceptually weighted scalar quantization, and Huffman coding of
zig-zag scanned symbols. In 1993, Shapiro’s embedded zerotree wavelets (EZW) coding
[1] established a new transform-coding paradigm with DWT, successive scalar
quantization, and arithmetic coding. The essential novelty of EZW is the introduction of
the “zerotrees” (a group of insignificant wavelet coefficients pertaining to the same
spatial location and orientation). Of the various improvements of EZW, the set
3
partitioning in hierarchical trees (SPIHT) coding [2] is the most renowned. Because of its
excellent compression performance and implementation elegancy, SPIHT has become
one of the de facto standard coding algorithms in the image
coding/processing/transmission community.
Many researchers have investigated the design issues of EZW, SPIHT, and other
DWT-based image coders. Before the introduction of SPIHT, Villasenor et al. [3]
evaluated the compression efficiency of biorthogonal wavelet filters in terms of the
Holder regularity and the impulse and step response properties, where a simple adaptive
scalar quantization scheme with an optimized bit-allocation procedure was used as the
coding platform. Li et al. [4] examined several wavelet filters and extension methods for
EZW. Adams and Kossentini [5] evaluated a wide range of reversible DWT kernels
under the JPEG2000 framework. Unser and Blu [6] investigated the mathematical
properties of the Daubechies 9/7 and LeGall 5/3 wavelets pertaining to their compression
performance. Woods and Naveen [7] derived the optimal bit allocation for
non-orthogonal transforms. Moulin [8] derived a multiscale relaxation algorithm to
improve the coding performance of non-orthogonal wavelet coding. Liu and Moulin [9]
employed the mutual information to model the interscale and intrascale dependencies
between wavelet coefficients. In [10], Xiong et al. showed that DWT outperforms DCT
within 1 dB under the same embedded coding structure. The parent-child coding gain of
SPIHT was quantified in [11]. He and Mitra [12] presented a unified analysis framework
for the transform coding, where a new rate-distortion model in terms of the zeros of the
quantized coefficients was developed. Finally, more sophisticated quantization schemes
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such as vector quantization (VQ) and trellis-coded quantization (TCQ), have been
incorporated into SPIHT [13]-[15].
This paper comprehensively investigates the factors crucial to the performance of
SPIHT. Although SPIHT was implemented mostly with a conventional set of parameters
(e.g., Daubechies 9/7 wavelet with symmetric data extension), this setup may not be
appropriate for all applications. It is thus important to investigate how to attain the
desired performance with the available resources. Furthermore, this investigation offers
an insight into the modern wavelet coders. In the following sections, we first explore the
essential properties of wavelet transforms, including the orthogonality,
energy-compacting capability, and symmetry. Since a biorthogonal wavelet transform
distorts the magnitude of wavelet coefficients, we examine the effects of a scaling
scheme to the quantization efficiency. The effect of the arithmetic coding is presented at
last.
2. WAVELET TRANSFORMS AND SPIHT
2.1 The SPIHT Coding
The encoding process of SPIHT is summarized in Fig. 2. The DWT converts the pixels
into wavelet coefficients, which are organized as the spatial-orientation trees depicted in
Fig. 3. SPIHT adopts a two-pass scalar deadzone quantizer. The first pass, sorting pass,
identifies the significant coefficients with respect to a threshold and gives their sign. The
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positions of significant coefficients are recorded in the list of significant pixels (LSP).
Insignificant spatial-orientation trees (i.e., zerotrees) are recorded in the list of
insignificant sets (LIS), and the other isolated insignificant coefficients are indexed in the
list of insignificant pixels (LIP). The second pass, refinement pass, narrows the
quantization level by a half for all the entries in the LSP excluding those newly added in
the last iteration. The two passes are repeated with a halved threshold in the next iteration,
until a specified rate or distortion constraint has been reached. An optional arithmetic
coder can be used to generate even more compact bit streams. However, the arithmetic
coding incurs intensive computation and reduces error robustness for the otherwise
elegant SPIHT algorithm [2]. Without being otherwise specified, the coding results given
in this paper are exempt from the arithmetic coding.
2.2 Discrete wavelet transform (DWT)
Transformation plays an essential role in image processing. Transformation can be
regarded as an approximation of a signal with a new set of basis functions. The new
space (called the transform domain) manifests itself in the decorrelating capability,
space-frequency localization, energy compaction, and/or other properties desirable for
further processing. In contrast to the conventional Fourier analysis, the wavelet transform
reveals both transient and stationary characteristics of a signal under a multiresolution
framework [16]. For discrete-time signals, the wavelet transform can be realized with the
filter-bank structure shown in Fig. 4, where hB0B[n] and hB1 B[n] correspond to the scaling
(lowpass) coefficients and wavelet (highpass) coefficients, respectively. A p-level
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decomposition along the lowpass subbands creates p+1 subbands H1, H2, … , Hp and Lp,
where the Lp subband stands for the base (approximation) component and the H
subbands represent the details at various scales. Perfect reconstruction can be built from
the constituent subbands by upsampling and interpolative filtering with the synthesis
filters gB0 B[n] and gB1B[n]. For two-dimensional signals such as images, DWT is mostly
independently performed along rows and columns. Observe a two-level decomposition of
the Lena image shown in Fig. 3(a). The resulting DWT coefficients demonstrate two
important facts that support the zerotree coding. First, an overwhelming majority of
energy concentrates in the lowpass subbands. This property is termed “energy
compaction.” Second, there exists obvious correlation between parent and child nodes.
2.3 Properties of wavelet filters
The multiresolution analysis requires the lowpass filter hB0 B[n] to satisfy the following
scaling equation
2][0 =∑n
nh (1)
An orthonormal wavelet of length N+1 further satisfies
][)1(][ 01 nNhnh n −−= , ][][ 00 nhng −= , ][][ 11 nhng −= (2)
and the energy conservation condition
1][][][][ 21
20
21
20 ==== ∑∑∑∑
nnnnngngnhnh (3)
Biorthogonal wavelets have two sets of complementary bases that satisfy
7
][)1(][ 10 nhng n −−= , ][)1(][ 01 nhng n −−= . (4)
Orthonormal transformation implies norm preserving; effective quantization can thus be
directly applied to the transform coefficients. Several near-orthogonality measures for
biorthogonal wavelets have been proposed in the literature, mostly based on the
norm-preserving property [6], [17-19]. Let
,][200 ∑=
nnhw .][2
11 ∑=n
nhw (5)
It can be shown that wB0 B and wB1 B are the weighting factors of the mean-square quantization
error to the lowpass and highpass subbands, respectively [18]. In fact, wB0 B and wB1 B are
related to the Riesz constants upon considering the reconstruction error in the frequency
domain [6, 18]. In this paper, we adopt the near-orthogonality measure (NOM) defined in
[17]:
}.,max{NOM 10 ww= (6)
NOM upper bounds the multiplicative distortion to the quantization error introduced by
non-orthogonality. Clearly, orthogonal wavelets have the NOM value equal to 1. A larger
deviation of NOM from 1 indicates less orthogonality. Recently, a more elaborate model
for measuring orthogonality has been proposed in [19]. Their derivation was based on the
eigen-analysis of the discrete hyper-wavelet transform.
In this paper, we examine a wide variety of wavelet bases commonly referred to in
the image-coding community. According to the data type and orthogonality, these
wavelet bases naturally fall into three categories:
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A. Orthonormal wavelets: Haar (D2), D4, D6, D8.
B. Floating-point biorthogonal filters: 9/7D and 10/18.
C. Integer biorthogonal wavelets: 5/3, 9/7M, 5/11A, 5/11C, 13/7C, 13/7T, and
9/7WY.
Haar is the simplest nontrivial wavelet, which takes the sum and difference of input
samples for approximation and detail, respectively. D4, D6, and D8 are the Daubechies
orthonormal wavelets with compact support and maximal number of vanishing moments
[20]. The 9/7D wavelet is an odd-symmetric filter derived from an orthonormal mother
wavelet [21], while 10/18 is a longer even-symmetric filter [22]. Integer biorthogonal
wavelets [5] are fixed-point approximations to their parent real counterparts; they can be
implemented in the lifting framework without costly floating-point operations [5, 23, 24].
The 9/7WY wavelet [25] is a recently derived integer filter that is similar to 9/7D but
with much less computational burden. Moreover, transformation through integer wavelets
is reversible and suitable for a unified lossy and lossless codec [26]. The 5/3 and 9/7D
filters have been adopted in the JPEG-2000 standard [27].
The analysis filters hB0 B[n] and hB1 B[n] of the wavelets under study are listed in Table 1.
For biorthogonal wavelets, only half of the coefficients are given since the other half can
be deduced from symmetry. The NOM values of the biorthogonal filters are listed in
Table 2. The 9/7D and 9/7WY wavelets are much closer to orthogonality than the others.
For Category C wavelets, the scaling factor 2 in (1) is rescaled to 1 in Table 1 to
facilitate integer operations. We evaluated the time complexity of these wavelet
transforms on PC (Pentium 4). The results are given in Table 3, where the number
indicates the ratio of the required processing time with respect to the simplest Haar
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wavelet with convolution. It should be reminded that this complexity measure is
hardware dependent. No hardware acceleration such as the SIMD (single instruction
multiple data) technique was involved in our evaluation. Nevertheless, it is clear that the
convolution-based 10/18 filter is far more complex than the others even with the
floating-point support of the Pentium CPU. In contrast, the integer biorthogonal wavelets
are very attractive in practice.
3. OPTIMIZED WAVELET TRANSFORMS FOR THE SPIHT CODING
In this section, we investigate the parameters of the SPIHT coding crucial to compression
efficiency. Table 4 lists the coding performance of SPIHT at four bit rates (1/8, 1/4, 1/2,
and 1 bpp). Four 512×512 gray-level images Lena, Baboon, Pepper, and F16 (Fig. 5)
selected from the USC image databases [28] are tested. A 5-level DWT with each of the
aforementioned filters is employed for multiresolution decomposition. The visual quality
is objectively measured by the peak signal-to-noise ratio (PSNR). For an image X = (xB1 B,
x B2 B, …, x BM B) and its distorted version Y = (y B1 B, y B2 B, …, y BM B), the PSNR is computed as follows
.)(1
255log 10PSNR
1
2
2
10
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
∑=
M
iii yx
M
(7)
Considerable PSNR gap (up to 3dB) is observed when different wavelets are employed,
especially for smooth images such as Lena and Pepper. Among the examined filters, the
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9/7D, 9/7WY and 10/18 filters generally achieve the best PSNR performance. Filters of
shorter taps compromise on their compression efficiency. It is also noted that the
biorthogonal wavelets are substantially better than the orthogonal ones; the 5/3 filter
outperforms the much more complex D8 filter in many cases.
It was conjectured that energy compaction might be the dominant factor for the
compression performance of wavelet transform. We measure the energy compaction in
two ways, approximation spectral significance (in spectral domain) and reconstruction
error (in spatial domain), which are both listed in Table 5. The approximation spectral
significance is the percentage of the sum of squared coefficients within the approximation
subband (LL5) to the sum of squared coefficients of all subbands. The reconstruction
error is defined as the mean square error (MSE) when only the LL5 subband is decoded
(the other subbands are filled with 0). In the view of the SPIHT coding, the former
primarily affects the formation of zerotrees whereas the latter is related to the
quantization error. The 9/7D and 9/7WY wavelets possess the best approximation
spectral significance while the 10/18 wavelet possesses the least reconstruction error.
This partially explains the excellent performance of these two wavelets. The 9/7D and
9/7WY may also benefit from it’s being near orthonormal [6]. However, the coding
performance is not solely determined by energy compaction and orthogonality. For
example, higher approximation spectral significance of orthogonal wavelets does not
translate into better coding performance.
To further distinguish between orthogonal and biorthogonal wavelets, it is of interest
to know the advantage obtained by symmetric extension. Filtering with finite-length
sequences causes data expansion. A universal solution to this problem is the circular
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convolution together with periodic extension. However, this approach introduces
boundary artifacts; the aliasing components introduced by periodic extension makes
compression less efficient. To circumvent this difficulty, symmetric extension of input
samples have been developed for linear-phase filters [29]. Imposing the linear-phase
constraint, however, usually breaks the orthogonality of the filter. The only real-valued
orthogonal linear-phase wavelet with compact support is the trivial Haar filter. Good
linear-phase biorthogonal FIR wavelets (categories B and C) are thereby designed. Note
that the method of data extension should be in conformity with the type of filter’s
symmetry (Fig. 6). Similar consideration should be also borne in mind for
down-sampling and up-sampling. The results shown in Table 4 were obtained with the
best extension method, i.e., periodic extension for orthogonal wavelets and appropriate
symmetric extension for biorthogonal wavelets. In Table 6, the coding results with both
symmetric and periodic extension are given for comparison. Symmetric extension
provides a substantial edge over periodic extension at low rates for low-activity images
(Lena and Pepper). Compared to the orthogonal wavelets of similar length, the
distinguished 5/3, 9/7D (and 9/7WY) filters stand out not solely owing to symmetry.
Unser and Blu [6] attributed the success of these filters to the better approximation for
smooth regions of images [6].
The number of decomposition levels (p) is another factor that influences the
performance of wavelet-based image coders. The PSNR performance of SPIHT for p = 4
and p = 6 under a similar test environment of Table 4 are given in Tables 7 and 8,
respectively. When a very scarce bit budget is available, encoding down to the
higher-level nodes is beneficial because these nodes refer to a larger area. As a
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consequence, a nontrivial increase is observed from p = 4 to p = 5 at low bit rates.
Moving one level further (from p = 5 to p = 6) makes negligible contribution because the
LL6 subband contains few coefficients relative to the available bit budget.
4. QUANTIZATION AND ENTROPY COING OF SPIHT
Divide-and-conquer is the basic philosophy behind the transform coding. An adequate
transformation manages to remove the inter-pixel correlation and arrange the transform
coefficients in a prioritized order. Simple scalar quantization can thus be effortlessly
performed on the resulting transform coefficients. An optimal encoder produces the most
economical representation of wavelet coefficients in the order of their relative importance.
In SPIHT, the magnitude of wavelet coefficients is the prime indicator of their
significance. However, this indicator may not faithfully reflect the actual importance in
the rate-distortion sense, especially when non-orthogonal transformation is employed.
In this paper, we investigate an optimized quantization scheme by scaling subbands.
Adjusting the magnitude of wavelet coefficients may alter their quantized value and
encoding priority. We define the scaling factor K, by which the coefficients of highpass
filter hB1B[n] are multiplied. A smaller scaling factor implies an emphasis on lowpass
coefficients upon quantization and a better change of forming zerotrees. The best choice
of K makes a compromise between the quantization error and the number of bits required
for specifying the significance map. Table 9 gives the optimal scaling factor (using
exhaustive search) and the resulting PSNR gain (cf. Table 4). For a quantization scheme
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that does not take the parent-child correlation into consideration, scaling can provide
substantial coding gain [30]. However, for SPIHT the PSNR increase is negligible (less
than 0.21 dB) and insensitive to the scaling factor. Our conjecture is that the zerotree
coding of SPIHT has encoded the wavelet coefficients in the order of their relative
importance. Both the approximation and detail information is well preserved even with
the skewed coefficients.
The arithmetic coding can further squeeze the quantization symbols. However, the
arithmetic coding involves more intensive computation. Our simulation under the PC
environment indicates that the overall computation time approximately increases by 50%
with the arithmetic coding. The PSNR gain of the arithmetic coding for SPIHT is listed in
Table 10 (cf. Table 4). A nontrivial but limited PSNR gain (about 0.5 dB) is achieved and
this margin is relatively consistent across the filters.
5. CONCLUSION
We have investigated the factors crucial to the performance of the prestigious SPIHT
coding. We first explored the influence of wavelet filters, data extension types, and
decomposition levels. Second, a scaling scheme for restoring the energy distortion of
wavelet subbands was investigated. Finally, the effect of the entropy coding was
examined. A comprehensive evaluation in terms of PSNR and time complexity was made
at four bit rates for four test images. The results of this paper establish the guidelines for
implementing wavelet-based codecs.
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6. ACKNOWLEDGEMENT
This work was supported by the National Science Council, R. O. China, under the
contract number NSC 92-2218-E-027-016. The authors want to thank the anonymous
reviewers for their helpful comments.
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BIOGRAPHIES
Shih-Hsuan Yang received the B.S. degree in electrical engineering from the National
Taiwan University in 1987. He obtained the M.S. and Ph.D. degrees in electrical
engineering and computer science from the University of Michigan, Ann Arbor, in 1990
and 1994, respectively. Since 1994, he has been a faculty member of the National Taipei
University of Technology, Taiwan. He is currently an associate professor of Computer
Science and Information Engineering. His major research interests include image and
video coding, multimedia transmission, data hiding, and information theory.
Wu-Jie Liao was born in Yunlin, Taiwan, in 1979. He received the B.S. and M.S. degrees
from the National Taipei University of Technology in 2002 and 2004, respectively.
Currently he is working with the Primax Electronics Ltd., Taipei, Taiwan, for developing
multifunction peripherals.
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Table 1. Analysis filters of the wavelets under study. The negative indexes apply to categories B and C. An extra scaling factor of 2 is needed for Category C wavelets to conform to the scaling equation.
Filter Index Category & Name 0 1 (-1) 2 (-2) 3(-3) 4 (-4) 5 (–5) 6(-6) 7(-7) 8(-8)
h B0B[n] 0.7071 0.7071 Haar (D2) h B1B[n] 0.7071 -0.7071
h B0B[n] 0.48296 0.83652 0.22414 -0.12941 D4 h B1B[n] 0.12941 0.22414 -0.83652 0.48296 h B0B[n] 0.33267 0.80689 0.45988 -0.13501 -0.08544 0.03523 D6 h B1B[n] -0.03523 -0.08544 0.13501 0.45988 -0.80689 0.33267 h B0B[n] 0.23038 0.71485 0.63088 -0.02798 -0.18703 0.03084 0.03288 -0.01060
A
D8 h B1B[n] 0.01060 0.03288 -0.03084 -0.18703 0.02798 0.63088 -0.71485 0.23038 h B0B[n] 0.85267 0.37740 -0.11062 -0.02385 0.03783 9/7D h B1B[n] 0.78849 -0.41809 -0.04069 0.06454 h B0B[n] 0.75891 0.07679 -0.15753 8.2e-5 0.02885 B
10/18
h B1B[n] 0.62336 -0.16337 -0.08566 0.01377 0.03083 0.00253 -0.00945 2.7e-6 0.00095h B0B[n] 3/4 1/4 -1/8 5/3
h B1B[n] 1 -1/2 h B0B[n] 23/32 1/4 -1/8 0 1/64 9/7M h B1B[n] 1 -9/16 0 1/16 h B0B[n] 3/4 1/4 -1/8 5/11A h B1B[n] 63/64 -67/128 0 7/256 1/128 -1/256 h B0B[n] 3/4 1/4 -1/8 5/11C h B1B[n] 31/32 -35/64 0 7/128 1/64 -1/128 h B0B[n] 41/64 5/16 -31/256 -1/16 7/128 0 -1/256 13/7C h B1B[n] 1 -9/16 0 1/16 h B0B[n] 87/128 9/32 -63/512 -1/32 9/256 0 -1/512 13/7T h B1B[n] 1 -9/16 0 1/16 h B0B[n] 19/32 43/160 -12/160 -3/160 9/320
C
9/7WY h B1B[n] 9/8 -19/32 -1/16 3/32
Table 2: NOM (near-orthogonality measure) of the biorthogonal wavelets under study
Table 3. Relative time complexity of the wavelet transforms under study. D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY
1.00 1.92 2.92 4.03 3.94 7.58 1.03 1.17 1.39 1.40 1.21 1.26 1.61
9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY 1.040 1.215 1.438 1.347 1.438 1.438 1.310 1.300 1.021
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Table 4. SPIHT’s compression performance with various filters for (a) Lena (b) Baboon (c) Pepper (d) F16 (5-level decomposition).
(a)
(b)
(c)
(d)
Table 5. (1) Approximation spectral significance (1P
stP row with each image) (2)
reconstruction error (2P
ndP row with each image).
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY97.8 97.4 97.3 97.9 98.0 97.4 95.7 96.7 95.7 95.7 96.7 96.7 98.0
Lena 830 839 822 701 744 659 786 762 776 772 734 743 744 98.9 99.0 98.9 99.0 99.2 98.7 98.0 98.5 98.0 98.0 98.6 98.6 99.2
Baboon 960 922 908 897 886 876 901 895 899 899 885 888 886 94.7 94.7 94.1 95.4 95.7 94.3 92.4 94.0 92.5 92.5 93.6 93.8 95.8
Pepper 1823 1665 1693 1435 1605 1410 1692 1652 1678 1675 1588 1612 1603 98.8 98.8 98.7 98.8 99.0 98.3 97.8 98.3 97.8 97.8 98.3 98.3 99.0
F16 997 928 893 866 889 863 915 907 914 917 886 892 889
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 27.53 28.99 29.39 29.50 30.53 30.69 30.05 30.32 30.22 30.25 30.51 30.50 30.520.25 30.21 31.88 32.37 32.54 33.59 33.76 32.95 33.36 33.16 33.27 33.53 33.50 33.580.5 33.50 35.27 35.75 35.87 36.75 36.88 36.09 36.56 36.32 36.43 36.72 36.70 36.741.0 37.45 38.94 39.27 39.36 39.92 39.95 39.30 39.60 39.47 39.51 39.77 39.74 39.93
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 20.97 21.28 21.37 21.41 21.49 21.60 21.29 21.26 21.26 21.21 21.45 21.42 21.490.25 22.14 22.55 22.64 22.69 22.88 22.97 22.53 22.56 22.52 22.46 22.77 22.72 22.890.5 24.09 24.60 24.79 24.86 25.12 25.13 24.58 24.72 24.63 24.62 24.92 24.88 25.141.0 27.31 27.97 28.21 28.30 28.62 28.60 28.00 28.19 28.09 28.09 28.40 28.36 28.63
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 25.82 27.38 27.68 27.54 29.10 29.11 28.70 28.77 28.80 28.79 28.95 28.94 29.080.25 28.78 30.30 30.53 30.51 31.79 31.72 31.46 31.57 31.54 31.48 31.68 31.67 31.780.5 31.83 32.99 33.11 33.09 33.83 33.81 33.60 33.55 33.63 33.55 33.73 33.69 33.811.0 35.01 35.72 35.78 35.73 36.19 36.22 35.90 35.89 35.92 35.83 36.07 36.04 36.18
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 26.78 27.99 28.18 28.40 29.28 29.26 28.85 28.97 28.93 28.92 29.10 29.09 29.260.25 29.57 30.95 31.39 31.51 32.51 32.55 32.00 32.28 32.22 32.22 32.34 32.36 32.500.5 33.43 34.73 35.16 35.36 36.39 36.42 35.85 36.22 36.05 36.08 36.33 36.35 36.391.0 38.37 39.55 39.97 40.11 40.85 40.85 40.25 40.57 40.44 40.45 40.70 40.70 40.85
21
Table 6. Coding results for period/symmetric extension (a) Lena (b) Baboon (c) Pepper (d) F16.
(a)
(b)
(c)
(d)
bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY
0.125 30.06 /30.53
30.21 /30.69
29.61 /30.05
29.88 /30.32
29.73 /30.22
29.74 /30.25
30.07 /30.51
30.05 /30.50
30.05 /30.52
0.25 33.22 /33.59
33.27 /33.76
32.61 /32.95
33.00 /33.36
32.84 /33.16
32.93 /33.27
33.16 /33.53
33.14 /33.50
33.21 /33.58
0.5 36.53 /36.75
36.48 /36.88
35.91 /36.09
36.27 /36.56
36.12 /36.32
36.20 /36.43
36.42 /36.72
36.42 /36.70
36.52 /36.74
1.0 39.77 /39.92
39.74 /39.95
39.15 /39.30
39.45 /39.60
39.31 /39.47
39.35 /39.51
39.62 /39.77
39.60 /39.74
39.77 /39.93
bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY
0.125 21.43 /21.49
21.55 /21.60
21.23 /21.29
21.16 /21.26
21.20 /21.26
21.16 /21.21
21.40 /21.45
21.35 /21.42
21.43 /21.49
0.25 22.82 /22.88
22.90 /22.97
22.46 /22.53
22.46 /22.56
22.45 /22.52
22.42 /22.46
22.70 /22.77
22.67 /22.72
22.81 /22.89
0.5 25.02 /25.12
25.08 /25.13
24.49 /24.58
24.59 /24.72
24.53 /24.63
24.52 /24.62
24.82 /24.92
24.80 /24.88
25.04 /25.14
1.0 28.53 /28.62
28.53 /28.60
27.91 /28.00
28.09 /28.19
28.00 /28.09
28.01 /28.09
28.32 /28.40
28.28 /28.36
28.54 /28.63
bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY
0.125 28.63 /29.10
28.67 /29.11
28.22 /28.70
28.46 /28.77
28.33 /28.80
28.35 /28.79
28.67 /28.95
28.66 /28.94
28.62 /29.08
0.25 31.48 /31.79
31.34 /31.72
31.19 /31.46
31.24 /31.57
31.35 /31.54
31.30 /31.48
31.42 /31.68
31.41 /31.67
31.47 /31.78
0.5 33.59 /33.83
33.60 /33.81
33.52 /33.60
33.43 /33.55
33.55 /33.63
33.46 /33.55
33.60 /33.73
33.59 /33.69
33.56 /33.81
1.0 36.08 /36.19
36.08 /36.22
35.81 /35.90
35.80 /35.89
35.82 /35.92
35.72 /35.83
35.99 /36.07
35.95 /36.04
36.07 36.18
bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY
0.125 29.07 /29.28
29.09 /29.26
28.64 /28.85
28.72 /28.97
28.73 /28.93
28.70 /28.92
28.90 /29.10
28.90 /29.09
29.06 /29.26
0.25 32.32 /32.51
32.33 /32.55
31.78 /32.00
32.04 /32.28
32.04 /32.22
32.05 /32.22
32.18 /32.34
32.17 /32.36
32.30 /32.50
0.5 36.21 /36.39
36.21 /36.42
35.72 /35.85
36.00 /36.22
35.90 /36.05
35.91 /36.08
36.11 /36.33
36.16 /36.35
36.20 /36.39
1.0 40.70 /40.85
40.65 /40.85
40.12 /40.25
40.38 /40.57
40.30 /40.44
40.29 /40.45
40.55 /40.70
40.53 /40.70
40.69 /40.85
22
Table 7. SPIHT’s compression performance with 4-level (p = 4) decomposition. (a) Lena
(b) Baboon
(c)Pepper
(d) F16
Table 8. SPIHT’s compression performance with 6-level (p = 6) decomposition. (a) Lena
(b) Baboon
(c) Pepper
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 27.16 28.35 28.84 28.94 30.03 30.15 29.67 29.79 29.78 29.75 29.94 29.93 30.020.25 29.96 31.51 32.02 32.23 33.36 33.50 32.72 33.13 32.94 33.06 33.30 33.26 33.350.5 33.36 35.07 35.63 35.75 36.65 36.78 36.00 36.43 36.23 36.32 36.59 36.57 36.641.0 37.39 38.87 39.22 39.31 39.86 39.90 39.25 39.56 39.41 39.47 39.73 39.71 39.86
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 20.82 21.10 21.19 21.23 21.28 21.42 21.11 21.09 21.07 21.00 21.25 21.22 21.290.25 21.99 22.38 22.48 22.54 22.70 22.81 22.41 22.45 22.41 22.36 22.64 22.59 22.700.5 23.98 24.50 24.66 24.74 24.99 25.02 24.47 24.61 24.52 24.52 24.81 24.78 25.001.0 27.23 27.89 28.12 28.22 28.53 28.53 27.93 28.13 28.02 28.03 28.34 28.30 28.54
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 25.55 26.99 27.24 27.21 28.64 28.80 28.19 28.57 28.25 28.22 28.44 28.69 28.610.25 28.62 30.12 30.36 30.34 31.62 31.57 31.19 31.43 31.36 31.33 31.47 31.54 31.610.5 31.75 32.94 33.06 33.04 33.71 33.75 33.54 33.53 33.58 33.50 33.66 33.65 33.691.0 34.97 35.69 35.75 35.70 36.16 36.20 35.85 35.87 35.87 35.78 36.03 36.02 36.15
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 26.20 27.48 27.70 27.58 28.51 28.68 28.21 28.37 28.31 28.31 28.48 28.48 28.500.25 29.22 30.64 31.05 31.09 32.13 32.27 31.63 32.00 31.88 31.90 32.09 32.08 32.120.5 33.24 34.57 35.01 35.10 36.20 36.23 35.75 36.04 35.92 35.94 36.12 36.15 36.191.0 38.26 39.44 39.89 39.98 40.75 40.75 40.17 40.47 40.36 40.37 40.62 40.60 40.74
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 27.59 29.06 29.45 29.56 30.61 30.63 30.13 30.36 30.28 30.30 30.59 30.57 30.590.25 30.24 31.93 32.39 32.56 33.63 33.73 32.96 33.39 33.17 33.29 33.56 33.53 33.610.5 33.52 35.29 35.76 35.88 36.76 36.87 36.10 36.57 36.33 36.44 36.72 36.71 36.761.0 37.45 38.93 39.26 39.36 39.92 39.95 39.30 39.60 39.47 39.51 39.77 39.74 39.93
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 21.00 21.30 21.40 21.44 21.53 21.58 21.31 21.30 21.29 21.24 21.49 21.45 21.530.25 22.16 22.57 22.67 22.72 22.91 22.95 22.55 22.59 22.53 22.48 22.79 22.74 22.930.5 24.10 24.62 24.80 24.87 25.14 25.12 24.59 24.73 24.63 24.63 24.93 24.89 25.151.0 27.31 27.98 28.21 28.30 28.62 28.59 27.99 28.19 28.08 28.09 28.40 28.36 28.64
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 25.87 27.44 27.70 27.60 29.16 29.05 28.75 28.89 28.88 28.88 29.05 29.04 29.140.25 28.81 30.32 30.53 30.51 31.80 31.70 31.47 31.62 31.55 31.50 31.72 31.72 31.800.5 31.84 33.00 33.10 33.09 33.83 33.80 33.60 33.55 33.63 33.55 33.73 33.70 33.821.0 35.01 35.72 35.76 35.73 36.19 36.21 35.90 35.89 35.92 35.83 36.08 36.05 36.18
23
(d) F16
Table 9. Optimal scaling factor/performance gain relative to Table 4. (a) Lena
(b) Baboon
(c) Pepper
(d) F16
Table 10. PSNR gain of arithmetic coding relative to Table 4. (a) Lena
(b) Baboon
bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 26.88 28.10 28.30 28.52 29.38 29.19 28.91 29.04 28.98 28.98 29.15 29.14 29.360.25 29.61 30.99 31.42 31.55 32.55 32.52 32.02 32.31 32.23 32.22 32.36 32.40 32.550.5 33.46 34.76 35.17 35.36 36.41 36.41 35.85 36.23 36.06 36.09 36.33 36.35 36.411.0 38.40 39.56 39.96 40.11 40.86 40.83 40.24 40.57 40.44 40.45 40.70 40.69 40.85
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.9/0.11 0.9/0.02 1.0/0 0.9/0.02 1.1/0.05 1.1/0.03 1.1/0.03 1.1/0.01 1.0/0 1.0/0 0.9/0.02 1.0/0 1.1/0.070.25 0.8/0.03 0.9/0.03 1.0/0 0.9/0.04 1.1/0.05 1.0/0 1.1/0.02 1.1/0.02 1.2/0.04 1.2/0.06 1.0/0 1.1/0.01 1.1/0.050.5 0.8/0.03 0.9/0.03 0.9/0.03 0.9/0.05 1.1/0.05 1.0/0 1.2/0.08 1.0/0 1.2/0.06 1.1/0.01 1.0/0 1.0/0 1.1/0.061.0 0.8/0.01 0.9/0.01 1.0/0 0.9/0.01 1.0/0 1.0/0 1.3/0.11 1.0/0 1.3/0.09 1.3/0.06 0.8/0.02 0.8/0.02 1.0/0
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 1.1/0.03 1.1/0.03 1.1/0.03 1.1/0.03 1.1/0.02 1.0/0 0.9/0.03 1.0/0 0.9/0.04 0.9/0.04 1.0/0 1.0/0 1.1/0.020.25 0.9/0.06 0.9/0.02 0.9/0.07 0.9/0.06 0.9/0.06 0.9/0.06 0.9/0.07 0.8/0.02 0.9/0.07 0.8/0.08 0.9/0.02 0.8/0.01 0.9/0.060.5 1.1/0.04 1.1/0.03 1.0/0 1.0/0 0.9/0.03 0.8/0.12 0.8/0.16 0.8/0.09 0.8/0.15 0.8/0.12 0.8/0.13 0.8/0.13 0.9/0.011.0 1.1/0.02 1.0/0 1.0/0 1.0/0 1.0/0 1.2/0.08 1.3/0.13 1.2/0.03 1.3/0.1 1.3/0.07 1.2/0.06 1.2/0.06 1.0/0
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 1.3/0.17 1.3/0.12 0.8/0.19 0.8/0.24 1.0/0 0.9/0.04 1.3/0.05 0.9/0.15 1.3/0.08 1.3/0.07 0.9/0.12 0.9/0.12 1.0/00.25 1.1/0.08 0.8/0.13 0.8/0.18 1.3/0.19 1.0/0 1.0/0 1.0/0 0.9/0.04 0.9/0.02 0.9/0.03 0.9/0.05 0.9/0.05 1.0/00.5 0.7/0.05 0.8/0.12 0.8/0.17 0.8/0.18 0.9/0.09 0.9/0.08 0.9/0.02 0.9/0.14 0.9/0.05 0.8/0.07 0.9/0.09 0.9/0.11 0.9/0.111.0 1.1/0.04 0.8/0.07 1.2/0.11 1.2/0.13 1.1/0.09 1.1/0.06 1.4/0.05 1.1/0.06 1.3/0.03 1.3/0.04 1.1/0.05 1.1/0.06 1.1/0.09
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.8/0.04 0.8/0.1 0.9/0.21 0.9/0.09 1.0/0 0.9/0.09 1.0/0 1.0/0 0.9/0.03 1.0/0 0.9/0.13 0.9/0.09 1.0/00.25 1.3/0.11 0.8/0.05 0.8/0.17 0.8/0.21 1.0/0 0.9/0.04 1.0/0 1.0/0 1.0/0 1.0/0 0.9/0.11 0.9/0.05 1.0/00.5 1.2/0.11 0.8/0.1 0.8/0.16 0.8/0.16 0.9/0.01 1.0/0 1.0/0 1.0/0 1.1/0.04 1.1/0.06 1.0/0 1.0/0 1.0/01.0 0.8/0.02 0.9/0.08 0.9/0.01 0.9/0.05 1.0/0 1.0/0 1.2/0.12 1.3/0.12 1.3/0.13 1.3/0.13 1.3/0.04 1.3/0.03 1.0/0
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.56 0.39 0.42 0.44 0.28 0.42 0.42 0.39 0.40 0.43 0.43 0.40 0.270.25 0.47 0.47 0.44 0.45 0.35 0.34 0.38 0.42 0.45 0.46 0.43 0.41 0.350.5 0.51 0.48 0.44 0.45 0.35 0.32 0.45 0.33 0.43 0.40 0.31 0.32 0.361.0 0.55 0.47 0.46 0.46 0.42 0.42 0.44 0.40 0.41 0.38 0.39 0.40 0.41
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.15 0.15 0.14 0.16 0.20 0.14 0.18 0.18 0.18 0.19 0.19 0.17 0.190.25 0.30 0.34 0.38 0.40 0.39 0.32 0.27 0.29 0.29 0.31 0.31 0.32 0.380.5 0.36 0.36 0.34 0.34 0.40 0.40 0.40 0.36 0.40 0.41 0.38 0.39 0.391.0 0.53 0.49 0.48 0.49 0.56 0.54 0.47 0.53 0.48 0.48 0.52 0.53 0.56
24
(c) Pepper
(d) F16
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.55 0.44 0.49 0.59 0.45 0.48 0.54 0.65 0.56 0.54 0.62 0.59 0.450.25 0.51 0.46 0.49 0.50 0.34 0.34 0.44 0.44 0.44 0.45 0.44 0.42 0.340.5 0.51 0.37 0.40 0.41 0.41 0.39 0.35 0.45 0.40 0.39 0.40 0.43 0.431.0 0.56 0.49 0.51 0.53 0.58 0.55 0.47 0.49 0.48 0.45 0.52 0.52 0.58
D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.44 0.54 0.63 0.53 0.44 0.58 0.55 0.60 0.58 0.56 0.63 0.60 0.450.25 0.56 0.47 0.60 0.69 0.52 0.59 0.55 0.56 0.55 0.59 0.66 0.60 0.520.5 0.74 0.62 0.75 0.70 0.60 0.56 0.55 0.58 0.58 0.60 0.56 0.55 0.591.0 0.77 0.68 0.58 0.57 0.53 0.51 0.64 0.55 0.59 0.59 0.58 0.54 0.52
25
List of figure captions
Fig. 1. Framework of a transform coder.
Fig. 2. Encoding process of SPIHT.
Fig. 3. (a) Two-level wavelet transform (b) SPIHT’s spatial-orientation trees.
Fig. 4. Filter bank structure of wavelet transform.
Fig. 5. Test images: (a) Lena (b) Baboon (c) Pepper (d) F16.
Fig. 6. Extension types (a) periodic extension (b) odd-symmetric extension (c)
even-symmetric extension (d) anti-symmetric extension.
26
Transformation Quantization Entropy CodingImage
Fig. 1. Framework of a transform coder.
OriginalImage DWT
Sorting Pass RefinementPass
EntropyCoding
Bit Streams
SPIHTQuantization
Fig. 2. Encoding process of SPIHT.
*
(a) (b)
Fig. 3. (a) Two-level wavelet transform (b) SPIHT’s spatial-orientation trees.
h0
h1
D2
D2
Down-samplingby 2
x[n]
LowpassSubband (L1)
HighpassSubband (H1)
y[n]
U2
U2
Up-samplingby 2
g0
g1
+
Analysis System Synthesis System Fig. 4. Filter bank structure of wavelet transform.
27
(a) (b)
(c) (d)
Fig. 5. Test images: (a) Lena (b) Baboon (c) Pepper (d) F16.
(a) (b) (c) (d)
Fig. 6. Extension types (a) periodic extension (b) odd-symmetric extension (c) even-symmetric extension (d) anti-symmetric extension.
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