Computers and Biomedical Research 32, 449–469 (1999)
Article ID cbmr.1999.1523, available online at http://www.idealibrary.com on
Best Parameters Selection for Wavelet Packet-Based
Compression of Magnetic Resonance Images
A. N. Abu-Rezq, A. S. Tolba,1 G. A. Khuwaja, and S. G. Foda*
Physics Department, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait and;
*Electrical and Computer Engineering Department; Kuwait University, Kuwait
Received January 8, 1999
Transmission of compressed medical images is becoming a vital tool in telemedicine. Thus
new methods are needed for efficient image compression. This study discovers the best design
parameters for a data compression scheme applied to digital magnetic resonance (MR) images.
The proposed technique aims at reducing the transmission cost while preserving the diagnostic
information. By selecting the wavelet packet’s filters, decomposition level, and subbands that
are better adapted to the frequency characteristics of the image, one may achieve better image
representation in the sense of lower entropy or minimal distortion. Experimental results show
that the selection of the best parameters has a dramatic effect on the data compression rate of
MR images. In all cases, decomposition at three or four levels with the Coiflet 5 wavelet (Coif
5) results in better compression performance than the other wavelets. Image resolution is found
to have a remarkable effect on the compression rate. q 1999 Academic Press
Key Words: wavelet packets; best parameter selection; magnetic resonance imaging; compression; telemedicine.
1. INTRODUCTION
The unacceptability of artifacts when making medical diagnosis has limited the
wide use of image compression by the medical community. However, the imperative
for compression grows as medical centers use gigabytes or terabytes of disk space
per year. Transmission of medical images is becoming a vital tool in telemedicine.
Lossless compression is artifact free. Also high-quality lossy compression, with
its higher compression rate, could become useful for archiving images. Wavelet
packets provide true lossy compression without many of the artifacts of block-based
transform compressors. Reconstructed digital images must preserve the necessary
information content of the original images to avoid diagnostic errors. The performance of MR image compression algorithm is measured by its ability to minimize
entropy while preserving all clinically significant features. At high and medium
1
To whom correspondence should be addressed. E-mail: tolba@kuc01.kuniv.edu.kw.
449
0010-4809/99 $30.00
Copyright q 1999 by Academic Press
All rights of reproduction in any form reserved.
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ABU-REZQ ET AL.
bit rate (low and medium compression ratio), there is a strong correlation between
the image quality and these objective measures. At low bit rate (high compression),
the basis function characteristics have profound effects on the quality of the reconstructed image (1).
The best data reduction methods to date have been based on representations of
the images with multiresolutions in space and frequency domains. Most of the
spacial redundancy within images can be eliminated, thereby making the images
much easier to compress. However, no single algorithm can be expected to work
well for all classes of images. The sampling rates, the spectral composition, and
the pixel quantization influence the compressibility of the original image.
Wavelet-based data compression techniques are known to produce subjectively
better quality images than the standard Joint Photographic Experts Group (JPEG)
technique (2). The wavelet packets have proven to be an efficient tool for data
compression (2, 3, 9–19). The wavelet packets and the best basis algorithm offer
a library of orthonormal basis functions from which the optimal basis can be
selected to best match the characteristics of MR images. Wavelet packets are
particular linear combinations of wavelets. The best basis selection algorithm finds
a basis of adapted waveforms for a prescribed signal or a family of signals. Entropy
is suitable as a cost function for the minimization process in image compression.
A wavelet transform-based compression system operates in four main successive
steps as shown in Fig. 1. This research is concerned with the first two blocks. The
last two blocks are standard (8–12).
The paper is organized as follows: Section 2 gives a brief overview of data
compression techniques and their application to medical images. Section 3 gives
an introduction to wavelet packets and the selection of the best basis. Section 4
presents the comparisons and discussions of the performance of the different
wavelet bases. Finally, concluding remarks are given in Section 5.
2. OVERVIEW
In (1, 8), the linear phase 9/7 filter bank-based wavelet transform is used for
data compression of FBI fingerprints. Experiments indicate that the improvement
over the JPEG algorithm (the DCT in 8 3 8 blocks) increases with the compression
ratio. At high compression, the blocking effects in the JPEG output overwhelm
the signal so that it becomes impossible to follow the ridge lines.
In (3), simple Haar wavelets are shown to be enough for efficient data compression of image sequences from functional brain scans. Experiments on data compression showed that the 3-D Haar DWT is the most efficient method from the perspective of computational complexity, but the quality of the reconstructed images is
FIG. 1. Wavelet packet-based compression.
WAVELET PACKET-BASED COMPRESSION OF MRI
451
not measured. Experiments were performed on 3-D spatial volumes discretized at
64 3 64 3 8 voxel resolution (considered to be 8 slices of 64 3 64 pixel images).
A two-level 2-D discrete wavelet transform (DWT) is applied on the image slices
I[i,j]. Estimates Î[i,j ] were reconstructed with a fraction of the most significant
wavelet transform coefficients. Signal-to-noise ratios are computed as
SNR 5 10 log10
.I[ j, k].2
o
j,k
.I[ j, k] 2 Î[ j, k].2
o
j,k
[1]
with results averaged over all images and reported as mean decibles (dB). The
best SNR was 35.71 dB when keeping 0.8 of the transform coefficients.
In (4), results from a comparative study of different wavelet image coders using
a perception-based picture quality measure are presented. Experiments showed that
an excellent wavelet coder can result from a careful synthesis of existing techniques
of wavelet representation, quantization, and error-free encoding. The most effective
way to boost the overall performance of a wavelet coder is to exploit the dependency
of quantized coefficients, including zeros. The effect of variations between asymmetric orthogonal and symmetric biorthogonal wavelets is found to be noticeable,
but less significant when compared with the other two factors.
3. WAVELETS AND DATA COMPRESSION
3.1. Wavelet Transform Decomposition
Data compression is one of the most important applications of wavelet transform.
Wavelet transform can be generated from digital filter banks (1). Wavelet transforms
hierarchically decompose an input image into a series of successively lower resolution images and their associated detail images. Discrete wavelet transform of digital
images is implemented by a set of filters which are convolved with the image
rows and columns. An image is convolved with lowpass and highpass filters and
the odd samples of the filtered outputs are discarded resulting in downsampling
the image by a factor of 2. The wavelet decomposition results in an approximation
image (A) and three detail images in horizontal (HD), vertical (VD), and diagonal
(DD) directions. Decomposition into L levels of an original image results in a
downsampled image of resolution 2L with respect to the original image as well as
detail images.
3.2. Wavelet Packets Decomposition
Wavelet packets (WP) were introduced by Coifman et al. (5) as a generalized
family of multiresolution orthogonal or biorthogonal basis that include wavelets.
In the wavelet transform, only the lowpass filter is treated. This is based on the
assumption that the lower frequencies contain more information than the higher
frequencies. The main difference between wavelet transform and wavelet packets
decomposition is that, in wavelet packets, the basic two-channel filter bank can
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ABU-REZQ ET AL.
be iterated either over the lowpass or the highpass branch. Figure 2 shows the
wavelet packet decomposition tree. Image decomposition results in a quaternary
tree at three levels of decomposition. Wavelet packet basis can be generated by the
same quadrature filter pair that generates the wavelet. Coifman and Wickerhauser
developed an entropy-based algorithm for the best basis selection. The entropy is
used as a measure of energy compaction of a vector.
An entropy-based criterion is used to select the most suitable decomposition of
the MR image. This means that we look at each node of the decomposition tree
and quantify the information to be gained by performing each split.
MR images are analyzed using wavelet packets for splitting both the lower and
the higher bands into several bands at a time. A set of wavelet packets is obtained.
The following wavelet packet basis function {Wn}(n 5 0,. . ,`) is generated from
a given function W0:
W2n(l) 5 !2
W2n11(l) 5 !2
ok h(k)Wn(2l 2 k)
[2]
ok g(k)Wn(2l 2 k),
[3]
where the function W0 (l) can be identified with the scaling function f and W1 (l)
with the mother wavelet c. h(k) and g(k) are the coefficients of the lowpass and
highpass filters, respectively. Two 1-D wavelet packet basis functions are used to
obtain the 2-D wavelet basis function through the tensor product along the horizontal
and vertical directions. The corresponding 2-D filter coefficients can be expressed as:
hLL(k, l) 5 h(k)h(l)
[4]
hLH(k, l) 5 h(k)g(l)
[5]
FIG. 2. The wavelet packet 1-D decomposition binary tree at level 3.
WAVELET PACKET-BASED COMPRESSION OF MRI
453
hHL(k, l) 5 g(k)h(l)
[6]
hHH(k, l) 5 g(k)g(l)
[7]
3.3. Choice of the Best Filter Bank for Wavelet Packet Decomposition
The choice of filter bank which is used in wavelet decomposition is a critical
issue that affects image quality. A selected filter bank must result in perfect
reconstruction. To achieve the best compression rate of MR images, the best
filter and the best decomposition level must be chosen. Different orthonormal
and biorthonormal filters must be tested to get the filter with the highest energy
compaction capability. Experiments are performed to select the best filter and the
best decomposition level for the MR images.
To select the best wavelet type for data compression of MR images, one has
attempted to provide a partial answer by comparing the performance of different
wavelet functions with respect to the preserved energy as a function of the achieved
compression ratio. Ten different wavelet functions have been compared: the Haar
wavelet, the Duabechies wavelet of order 2 (db2), the Daubecies wavelet of order
8, the biorthogonal wavelet bior3.7 (one for decomposition and the other for
reconstruction), the biorthogonal wavelet bior5.5, the biorthogonal wavelet bior7.9,
the Coiflet (coif2), the Coiflet (coif5), the Symlet (sym2), and the Symlet (sym5).
The labels of the biorthogonal filter bior3.7 represent the lowpass analysis (or
highpass synthesis) filter length and the lowpass synthesis (or highpass analysis)
filter length, respectively. Coefficent details of these filters are available in (1).
3.4. Best-Basis Selection for Wavelet Packet Decomposition
The wavelet packet technique offers a family of different bases for the representation of a specific image. Since the ultimate goal of this research is to compress
medical images, the best basis should therefore be chosen such that it minimizes
the number of significantly nonzero coefficients in the resulting transformed image.
The best basis algorithm minimizes the entropy of the transform coefficients. The
algorithm takes the full wavelet decomposition tree according to the wavelet packets
method. Each node of the decomposition tree is assigned an entropy value.
Wavelet packet tree decomposition with orthogonal subbands is shown in Fig.
2. The original image can be represented by the direct sum of the leaves of any
subtree of the wavelet packet tree. Efficient wavelet packet implementation can
be done in O(N ld N ) which is equivalent to the time required by the Fast Fourier
transform. The use of an optimal subtree may reduce computation time. The search
for the best nonredundant representation of the data by the leaves of a subtree is
called best-basis selection [6]. Each subband is evaluated with a desired metric
such as entropy. Then a post-ordered search of the wavelet packets tree is conducted
in which a best-basis decision is made by comparing the quantitative value of each
node to the cumulative effects of the node’s descendant branch. Figure 3 shows
an example of a best-basis selection for a 2-D two-level decomposition wavelet
packet tree.
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ABU-REZQ ET AL.
FIG. 3. Original MRI and corresponding two-level wavelet packets’ decomposition.
The goal of applying the best basis algorithm is to optimally represent the MR
images by choosing a suitable orthonormal basis from a collection of basis functions.
The representation of the image is based on a basis function that minimizes some
cost function under the following conditions:
1. The cost function is small if only few coefficients are significant.
2. The cost function is additive.
The entropy will be used as a cost function and, for sequence xi is given by
m({xi}) 5 2o pn ln pn ,
[8]
n
where
pn 5
.xn.2
o .xi.2
i
Then, the best basis function relative to m for an image x is the basis chosen from
a set of possible basis functions for which the tramsformed image has the least
cost m. In this research, experiments were performed on a restricted subset of
rapidly computable basis functions.
In wavelet packet analysis, a signal is projected onto a number of orthogonal
subspaces Wi,n. The signal space V is decomposed into orthogonal subspaces such
that V 5 %i,nWi,n where i denotes the level of decomposition and n denotes the nth
subspace at that level. Each subspace Wi,n can be decomposed into two orthogonal
WAVELET PACKET-BASED COMPRESSION OF MRI
455
subspaces, namely W(i11),2n % W(i11),(2n11). The optimal wavelet packet representation of an MR image with respect to the cost function can then be obtained by
using the following algorithm (7):
1. Choose l as the maximal number of levels of decomposition.
2. While the level of decomposition is less than l, for each subspace Wi,n do
the following:
● Compute the cost function for the transform coefficients on that subspace
m(Wi,n).
● Decompose Wi,j into two orthogonal subspaces and compute their cost functions m1 and m2.
● If m . (m1 1 m2), then retain m1 and m2; otherwise retain m.
An original test MR image and its two-level wavelet packets decomposition are
shown in Fig. 3.
3.5. Coefficent Thresholding
Coefficient values below an automatically specificed global positive threshold
limit are forced to zero while retaining almost all of the energy of the original
image. The optimal threshold limit that corresponds to 99.6% of retained energy
is selected automatically. The recovered energy is calculated by using a secondorder vector norm as
E5
1 2
,Î,2
3 100,
|I|2
[9]
where I and Î are the decomposed image before and after thresholding, respectively.
4. EXPERIMENTAL RESULTS AND DISCUSSION
In this section we present some results from our comparative study of several
wavelet basis functions. Preliminary investigations are performed on only 4 images
to conduct the most suitable wavelet shape (see Table 1A). Another 20 MRI images
are then used to conduct signifince test on the the average compression rate.
The 20 images are decomposed using wavelet packets at 4 levels. Preliminary
investigations showed that decomposition at 4 levels is much better than decomposition at 3 levels. Automatic global thresholding technique is used to conduct the
best data compression ratio while keeping 99.6% of the energy of the reconstructed
images. In the experiments, 6 major types of wavelets (Haar, Coiflet, Biorthogonal,
Daubechies, Coiflet, and Symmlet) were examined. Each of the 6 types is used
with different parameters and different levels of decompositions. A total number
of 18 filter variants are constructed using all combinations of filter types, parameter,
and decomposition level. Each of the above 18 filter variants was applied to each
of four MR images, and the best basis was selected. Table 1B shows the results
of experiments performed on 20 in the case of the best- and worst-performing
wavelet shapes. The results show that the Coiflet 5 filter and the decomposition
level 4 is the best choice for the MR images.
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ABU-REZQ ET AL.
TABLE 1A
WP-Image Compression Results Averaged over Four Images with a 99.6% Square Recovery Norm
Wavelet
Haar
Db2
Db8
Bior3.7
Bior5.5
Coif2
Coif5
Sym5
Sym8
Bior9.7
Levels
% NZC
% NNZC
Threshold level
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
93.86
94.46
93.94
94.66
95.05
95.85
94.44
96.52
95.33
96.62
95.22
95.98
96.07
96.96
95.20
95.76
95.47
96.31
92.32
93.88
7.14
5.54
6.06
5.34
4.95
4.15
5.56
3.48
4.67
3.38
4.78
4.02
3.93
3.04
4.80
4.24
4.53
3.69
7.68
6.12
7.87
7.75
8.46
8.08
8.63
8.10
9.22
9.02
7.27
6.66
9.09
8.93
9.31
8.67
6.69
8.46
8.46
8.18
7.41
7.56
Note. NZC, Number of zero coefficients; NNZC, Number of non-zero coefficients.
Assuming random samples coming from random distributions, then the interval
(x1 2 x2) 6 ta/2 sp
!n1 1 n1
1
[10]
2
provides a realization of a confidence interval for the mean difference (m1 2 m2)
of nonzero coefficients provided with two different wavelets, with confidence
(1 2 a), where n1 and n2 are number of measurements, and s1 and s2 are the
standard deviations, and
s2p 5
(n1 2 1)s21 1 (n2 2 1) s22
n1 1 n 2 2 2
[11]
According to Table 1B, we are 95% confident that evidence suggests that the
average performance of the Coiflet-5 wavelet is better than the average performance
of the Haar wavelet by a factor of 227.52%.
4.1. Best Filter Bank and Best Basis Selection
In wavelet packet analysis, the choice of the filter bank is a critical issue that
affects image quality. Important characteristics of wavelet filter bank design include
457
WAVELET PACKET-BASED COMPRESSION OF MRI
TABLE 1B
The Number of Nonzero Coefficients (NZC) for the Differenet Wavelet Shapes (Retained Energy,
99.6%; Number of Decomposition Levels, 4)
Bior7.9
Image
THR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Mean
S
Cl (m)
m 2 m3
Cl (m 2 m3)
RNZC
30.36
29.40
25.86
28.85
28.96
25.71
30.26
30.71
30.51
28.34
29.68
29.98
30.90
30.65
35.36
30.88
25.12
31.22
30.65
33.37
Coiflet 4
NZC
THR
3.77
3.56
4.45
3.78
4.31
4.65
3.63
3.69
3.44
3.91
3.63
3.71
3.39
3.27
2.73
3.75
4.47
3.82
3.27
2.82
3.703
0.502
3.70 6 0.235
0.8445
0.85 6 0.208
29.5486
33.89
34.15
29.38
32.28
31.44
29.48
33.41
34.40
33.32
31.89
33.38
33.76
33.23
34.01
38.24
33.43
29.79
33.64
34.01
36.49
Coiflet 5
NZC
THR
3.32
2.97
3.63
3.31
3.25
3.31
3.04
2.96
2.86
3.35
2.86
3.13
2.87
2.63
2.20
2.92
3.83
3.18
2.63
2.37
3.031
0.396
3.03 6 0.185
0.1730
0.17 6 0.181
6.0532
34.1
33.82
29.88
32.21
31.63
29.21
32.82
33.54
33.74
31.19
32.39
32.8
33.5
33.57
38.56
33.18
28.9
34.66
33.57
36.52
Bior5.5
NZC
THR
3.13
2.78
3.49
3.25
3.27
3.23
2.80
2.74
2.63
2.88
2.69
3.02
2.31
2.65
2.14
2.72
3.33
3.21
2.65
2.24
2.858
0.378
2.86 6 0.177
0.000
—
0
28.06
25.8
22.95
25.93
27.81
24.52
25.79
26.06
25.79
25.01
27.6
26.64
26.35
25.4
30.6
25.91
25.77
27.56
25.4
28.56
Bior6.8
NZC
THR
3.36
2.84
3.99
3.35
5.25
5.57
2.83
2.78
2.77
3.31
3.19
3.34
2.57
2.52
2.07
2.74
5.70
3.26
2.52
2.23
3.310
1.049
3.31 6 0.491
0.452
0.452 6 0.37
15.7978
35.36
33.08
30.06
32.9
32.63
29.8
33.63
33.74
33.48
32.38
35.51
33.24
33.73
33.63
38.85
34.17
29.42
34.11
33.63
37.39
Haar
NZC
THR
3.36
2.94
3.80
3.60
3.35
3.95
3.02
3.01
2.82
3.36
2.94
2.90
2.85
2.72
2.37
2.79
3.57
3.25
2.72
2.26
3.079
0.445
3.08 6 0.208
0.2210
0.22 6 0.193
7.7327
28.00
25.50
24.00
26.13
27.50
25.50
25.63
24.93
25.87
25.50
25.69
26.25
25.88
25.75
27.75
25.75
24.75
27.25
25.75
26.5
NZC
09.14
09.19
11.22
09.75
09.09
10.58
09.42
09.75
09.17
10.02
09.09
09.65
09.02
08.40
07.55
09.35
10.93
09.36
08.40
08.13
9.361
0.896
9.36 6 0.419
6.5025
6.5 6 0.322
227.5192
Note. CI, confidential interval; S, standard deviation; RNZC, ratio of non-zero coefficients.
perfect reconstruction capability, finite-length and regularity requirement that iterated lowpass filters converge to continuous functions (19). Several wavelet filter
banks are compared in Table 1B. Eighteen variants of both orthogonal and biorothogonal filters are evaluated. The Coiflet 5 wavelet is proved to be superior to all
other wavelets. Figure 4 shows one example of the original and reconstructed
images in the cases of the best-performing wavelet (Coiflet 5), the Bior7.9 wavelet,
and the worst-performing wavelet (Haar). Figure 5 shows both the scaling function
and the mother wavelet of the best-performing wavelet (Coiflet 5). The wavelet
packets of the best-performing wavelet (Coiflet 5) are shown in Fig. 6.
4.2. Best Level Selection
Experiments were performed to obtain the most suitable number of decomposition levels. Both Table 2 and Fig. 7 show the percentage of zeroed coefficients
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ABU-REZQ ET AL.
A
FIG. 4. (A) (a) Original image, (b) reconstructed image, (c) WP decomposition at 4 levels using
Coiflet 5 wavelet, and (d) WP coefficients of the best decomposition tree. (B) (a) Original image, (b)
reconstructed image, (c) WP decomposition at 4 levels using Bior7.9 wavelet, and (d) WP coefficients
of the best decomposition tree. (C) (a) Original image, (b) reconstructed image, (c) WP decomposition
at 4 levels using Haar wavelet, and (d) WP coefficients of the best decomposition tree.
versus the number of levels. Although the increase in the number of zeroed coefficents when decomposing at five levels instead of four levels is significant, the
additional computational overhead for the case of five levels is many times that
needed for the case of four levels. Therefore, it is recommended to decompose
the MR images at only four levels.
Figure 8 shows the original, reconstructed MR images, and the distribution of
the percentage of zeroed coefficients, the retained energy E, the threshold limit
using the Coiflet 5 wavelet.
WAVELET PACKET-BASED COMPRESSION OF MRI
459
B
FIG. 4. Continued
4.3. Effect of the Resolution on the Compression Rate
To study the effect of original-image resolution on the compression performance,
experiments were performed on the Lena image. Table 3 and Fig. 9 show this effect.
4.4. Discussion
Preliminary investigations showed that the performance of wavelet packets is
much better than the performance of discrete wavelet transform. Experiments
performed on 20 MRI images showed that the Coiflet 5 gives the minimum number
of nonzero coefficients while retaining 99.6% of the signal energy. Figure 10 shows
the numbers of nonzero coefficents for the five best-performing wavelets compared
to the worst-performing wavelet (Haar wavelet).
Figure 11 shows the peak-signal-to-noise ratios (PSNR) for different wavelet
460
ABU-REZQ ET AL.
C
FIG. 4. Continued
FIG. 5. Scaling function and mother wavelet for the Coiflet 5 wavelet.
WAVELET PACKET-BASED COMPRESSION OF MRI
FIG. 6. Coiflet 5 wavelet packets from w0 to w8.
TABLE 2
Effect of Number of Decomposition Levels on the
Number of Zero Coefficents
Level
Haar
Bior7.9
Coiflet 5
1
2
3
4
5
66.86
79.60
82.04
82.40
82.45
70.52
85.19
89.75
91.99
93.13
71.92
87.95
91.86
92.80
93.28
FIG. 7. Decomposition levels versus zeroed coefficients.
461
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ABU-REZQ ET AL.
FIG. 8. Original, reconstructed MR images, and the distribution of the percentage of zeroed
coefficients, the retained energy E, and the threshold limit using a Coiflet 5 wavelet.
shapes for a fixed level of retained energy (99.6%). From Table 4 it is shown that
the PSNR for bior5.5 is higher than that for Coiflet 5, although the compression
efficency of the latter is better for an equal percentage of retained energy (99.6%).
To make a fair comparison between the Bior7.9 (according to Villasenor et al.
(15)) and Coiflet 5 wavelets, the Lena image is used. Table 3 shows approximately
TABLE 3
Relation between Resolution and
Compression Performance
Resolution
Bior7.9
Coiflet 5
128 3 128
256 3 256
512 3 512
90.23
94.84
98.38
91.72
95.70
98.30
WAVELET PACKET-BASED COMPRESSION OF MRI
463
FIG. 9. Image resolution vs number of zero coefficents.
equal performance in this specific case. Table 1B indicates better performance in
favor of the Coiflet 5 wavelet in the case of 20 MR test images. Table 5 gives a
comparison between both the Coiflet 5 wavelet and Bior7.9 used by Villasenor et
al. (15) for the case of wavelet packet decomposition at 4 levels (Fig. 12). The
PSNR value for the case of Bior7.9 is 32.59 dB which is much higher than the
corresponding value in the case of discrete wavelet decomposition (29.67 dB) as
reported in (15).
Figure 13b shows the percentage decrease in retained energy at various numbers
of zero coefficents in the case of decomposition using the Coiflet 5 wavelet at
four levels. Figures 13c and 13d show the reconstructed images using both the
Coiflet 5 and the Bior7.9 wavelets, respectively. There is no visible difference
between the two reconstructed images.
Figure 14 shows the original image together with the reconstructed images using
the best-performing Coiflet 5, the Bior7.9, and the worst-performing Haar wavelets.
The retained energy after image reconstruction is kept at 99.6% in all cases.
Although there is a large difference in the number of nonzero coefficents, there is
almost no visible difference. To visualize the difference between the reconstructed
FIG. 10. Number of nonzero coefficents versus the wavelet shape.
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FIG. 11. The peak signal-to-noise ratio for different wavelet shapes.
images based on different wavelet shapes, a section from the MRI images of Fig.
14 is zoomed out for 600% and the contours are extracted for each image. Figure
15 shows the results.
5. CONCLUSIONS AND FUTURE RESEARCH
The wavelet packets and the best basis selection algorithm are used for multiresolution representation of magnetic resonance images. Experiments were performed
on 20 test MR images. A comparative study of different wavelets is performed
using a percentage of retained image energy and PSNR for a specific number of
TABLE 4
PSNR of Different Wavelet Shapes
Image
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Mean
S
Bior5.5
Coif5
Bior7.9
Hear
32.1574
33.2404
33.3596
32.6978
33.2974
33.6107
33.1053
33.1494
33.225
32.955
32.9958
32.5149
33.0627
33.3747
32.7392
32.9782
33.5723
32.4282
33.3747
33.04
33.41
0.842
32.6263
33.4816
33.4583
33.1263
33.2255
33.4532
33.3781
33.3967
33.5203
33.2763
33.4391
32.9733
33.3213
33.7128
33.4499
33.2982
33.4794
32.8117
33.7128
33.6485
33.4
0.792
32.65
33.46
33.56
33.25
33.44
33.71
33.38
33.39
33.55
33.25
33.44
32.97
33.33
33.74
33.35
33.29
33.64
32.87
33.74
33.58
33.73
0.785
32.6423
33.4637
33.4298
33.0923
33.2357
33.4985
33.341
33.6126
33.4913
33.2445
33.4042
32.936
33.2933
33.683
33.417
33.2945
33.4464
32.8298
33.683
33.6476
33.69
0.802
Note. S, Standard deviation.
WAVELET PACKET-BASED COMPRESSION OF MRI
465
TABLE 5
Comparison between Bior7.9 and Coiflet 5
Wavelet shape
Bior7.9
Coiflet 5
Threshold
NZC
PSNR
45.25
47.46
98.38
98.30
32.5903
32.8671
nonzero coefficients. Several wavelet types were investigated to conduct the bestperforming wavelet for efficient data compression of magnetic resonance images.
The best filter and best decomposition level were selected on the basis of quality
measures such as the retained energy and peak signal-to-noise ratio. The experimental results presented in Section 4 indicate the following:
1. The wavelet shape has a significant impact on the performance of the compression scheme for a specific level of image quality.
2. The Coiflet wavelet 5 produced the highest compression rate of MR images
compared to all other wavelets.
FIG. 12. (a) Best decomposition using Coiflet 5 at four levels. (b) Best decomposition using
Bio7.9 at four levels.
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FIG. 13. (a) Original image. (b) Energy vs % zero coefficents. (c) Reconstructed images using
Coiflet 5. (d) Reconstruction using Bior7.9.
3. Decomposition for up to four levels resulted in the best compromise between
compression rate and computational overhead.
4. The performance of wavelet packets is superior to the wavelet transform
analysis in all cases.
5. The effect of image resolution on the compression rate is more noticeable
than the effect of the number of decomposition levels and wavelet shape.
6. The higher the resolution of the original image, the better is the compression performance.
WAVELET PACKET-BASED COMPRESSION OF MRI
467
FIG. 14. (a) Original MR image. (b) Reconstructed image using Coiflet 5. (c) Reconstructed
image using Bior7.9. (d) Reconstructed image using Haar.
7. The suitable wavelet for compression of a specific class of images (say MRI)
may not be suitable for other image classes.
A statistical significance test that is based on 20 MR images validates our
conclusions. Our future research will make use of the interframe relationships for
achieving higher compression rates.
ACKNOWLEDGMENTS
We acknowledge the support of Kuwait University, Kuwait. Also, the authors are grateful to Keith
A. Johnson and J. Alex Becker for giving us the permission to use the images of the “Whole
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ABU-REZQ ET AL.
FIG. 15. Visualization of reconstruction differences. The upper row shows (from left to right)
the original section, and the reconstructed sections using Coiflet 5, Bior7.9, and Haar wavelets,
respectively. The lower row of images show the contour maps of the images in the upper row.
Brain Atlas” (20). The authors thank the anonymous referees for their comments which contributed
significantly to the enhancement of the revised version of this paper.
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