International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
Wavelet Compression of ECG Signals Using
SPIHT Algorithm
Mohammad Pooyan, Ali Taheri, Morteza Moazami-Goudarzi, Iman Saboori
Some of the transformations used in transformational
compression methods are Fourier transform, discrete cosine
transform (DCT), Walsh transform, Karhunen-Loeve
transform (KLT), and wavelet transform.
The main idea in using transformation in compression
methods is to compact the energy of signal in much less
samples than in time domain, so we can discard small
transform coefficients (set them to zero). From this point of
view, we want to use a transformation that more compacts the
energy of signal in less samples in transform domain, which
are transform coefficients. It has been shown that wavelet
transform has a good localization property in time and
frequency domain and is exactly in the direction of transform
compression idea. Here, for ECG compression, we use
wavelet transform with SPIHT coding algorithm, modified for
1-D signals, for coding the wavelet coefficients.
Abstract— In this paper we present a novel approach for wavelet
compression of electrocardiogram (ECG) signals based on the set
partitioning in hierarchical trees (SPIHT) coding algorithm. SPIHT
algorithm has achieved prominent success in image compression.
Here we use a modified version of SPIHT for one dimensional
signals. We applied wavelet transform with SPIHT coding algorithm
on different records of MIT-BIH database. The results show the high
efficiency of this method in ECG compression.
Keywords— ECG compression, wavelet, SPIHT.
I. INTRODUCTION
LECTROCARDIOGRAM (ECG) signal is a very useful
source of information for physicians in diagnosing heart
abnormalities. With the increasing use of ECG in heart
diagnosis, such as 24 hour monitoring or in ambulatory
monitoring systems, the volume of ECG data that should be
stored or transmitted, has greatly increased. For example, a 3
channel, 24 hour ambulatory ECG, typically has storage
requirement of over 50 MB. Therefore we need to reduce the
data volume to decrease storage cost or make ECG signal
suitable and ready for transmission through common
communication channels such as phone line or mobile
channel. So, we need an effective data compression method.
The main goal of any compression technique is to achieve
maximum data reduction while preserving the significant
signal morphology features upon reconstruction. Data
compression methods have been mainly divided into two
major categories: 1) direct methods, in which actual signal
samples are analyzed (time domain), 2) transformational
methods, in which first apply a transform to the signal and do
spectral and energy distribution analysis of signals.
Examples of direct methods are: differential pulse code
modulation (DPCM), amplitude zone time epoch coding
(A TEC), turning point, coordinate reduction time encoding
system (CORTES), Fan algorithm, ASEC. Reference [1] is a
good review of some direct compression methods used in
ECG compression.
E
II. WAVELET TRANSFORM
A. Introduction
In wavelet transform, we use wavelets as transform basis.
Wavelet functions are functions generated from one single
function Z by scaling and translation:
1
t b
Za,b (t ) =
Z
(1)
a
a
The mother wavelet Z(t ) has to be zero integral,
(
)
¨ Z(t )dt = 0 . From (1) we see that high frequency wavelets
correspond to a < 1 or narrow width, while low frequency
wavelets correspond to a > 1 or wider width.
The basic idea of wavelet transform is to represent any
function f as a linear superposition of wavelets. Any such
superposition decomposes f to different scale levels, where
each level can be then further decomposed with a resolution
adapted to that level. One general way to do this is writing f
as the sum of wavelets Zm,n (t ) over m and n . This leads to
discrete wavelet transform (DWT):
f (t ) = cm,n Zm,n (t )
(2)
By introducing the multi-resolution analysis (MRA) idea by
Mallat [3], in discrete wavelet transform we really use two
functions: wavelet function Z(t ) and scaling function K(t ) . If
Manuscript received January 25, 2004.
M. Pooyan is with the Department of Electrical Engineering, Shahed
University, Tehran, Iran. (email: m pooyan@shahed.ac.ir).
A. Taheri and M. Moazami-Goudarzi are with the Faculty of Biomedical
Engineering, Amir Kabir University of Technology, Tehran, Iran (email:
ali.taherii@gmail.com, mmoazami@bme.aut.ac.ir).
I. Saboori is with the department of electrical Engineering, Amir Kabir
University of Technology, Tehran, Iran (email: iman saboori@yahoo.com).
we have a scaling function K(t ) L2 (\) , then the sequence of
subspaces spanned by its scalings and translations
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International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
j
can easily be cascaded in pyramidal filter bank structure
without need for phase compensation. As there are no
nontrivial orthonormal linear phase FIR filter with exact
reconstruction property, we can relax the orthonormal
property by using biorthogonal filters.
If the basis of a wavelet expansion is not orthogonal, we
can find another set of basis functions that is a dual for the
first function set and satisfies the orthogonality relation:
K j ,k (t ) = 2 2 K ( 2 j t k ) , i.e.:
Vj = span { K j ,k (t ), j, k ] }
(3)
constitute a MRA for L2 (\) .
K(t ) must satisfy the MRA condition:
K(t ) =
2 h(n )K(2t n )
(4)
n
for n ] . In this manner, we can span the difference
between spaces Vj (i.e. spaces Wj, so thatVj +1 = Vj Wj )
Kk (t ), Kl (t ) =
j
Zj ,k (t ) = 2 2 Z(2 j t k )
Z(t ) =
2 g(n )K(2t n )
use dual filters. In order to have exact reconstruction, we
impose [2]:
g(n ) = (1)n h(n + 1) and g(n ) = (1)n h(n + 1) (9)
In [2] some biorthogonal wavelet bases are derived. Here
we use the spline filters with symmetric filters h(n ) with
(5)
n
For orthogonal basis we have:
g(n ) = (1)n h(n + 1)
(6)
For computing the wavelet transform of a function
f (t ) L2 (\) we must find its projection on this set of
subspaces, so we must express it in each subspace as a linear
combination of expansion functions of that subspace [4]:
d
f (t ) =
k =d
c(k )Kk (t ) +
d
d
d ( j, k )Zj ,k (t )
(8)
We have similar dual functions for wavelet functions
( Z(t ) ). In reconstruction using filter bank algorithm, we must
by wavelet functions produced from mother wavelet:
Then we have:
¨ Kk (t )Kl (t )dt = E(k l )
length 9 and g(n ) with length 7. This wavelet basis is
commonly said “biorthogonal 9/7 tap filters”. It has been
shown [8] that this wavelet has the best performance for
wavelet ECG compression in comparison with orthogonal and
some other biorthogonal wavelets. It performs slightly better
overall than the other wavelets tested for the wavelet ECG
compression in the sense of CR-PRD plot. It has lowest PRD
in comparison with other seven wavelet bases. The filter
coefficients of biorthogonal 9/7 are given in Table I.
(7)
j = 0 k =d
where Kk (t ) = K(t k ) corresponds to the space V0 and
Zj ,k (t ) corresponds to wavelet spaces of scale j.
Table I. Coefficients of the Biorthogonal 9/7 Tap Filters
By using the idea of MRA, implementation of wavelet
decomposition can be performed using filter bank constructed
by a pyramidal structure of lowpass filters h(n ) and highpass
filters g(n ) [3, 4] where h(n ) and g(n ) are filter coefficients
and can be found for any scaling function using (4) and (5).
The reconstruction of signal from transform coefficients is
also done by a filter bank with filter coefficients that can be
obtained having h(n ) and g(n ) . If the wavelet bases are
orthonormal i.e.:
k vl
£
¦0
K j ,k (t ), Zj ,l (t ) = ¨ K j ,k (t )Zj ,l (t )dt = ¦
¤
¦¦ 1
k =l
¥
then we have exact reconstruction, i.e. from the transform
coefficients outputted from filter bank, we can exactly reach
the signal samples that were filter inputs.
n
0
1
2
3
4
h(n )
0.852699
0.377403
-0.11062
-0.023849
0.037829
g(n )
0.788485
0.418092
-0.04069
-0.064539
III. SPIHT CODING ALGORITHM
A. Embedded Coding
In this paper we use SPIHT coding algorithm for coding
wavelet transform coefficients of ECG signal. Set partitioning
in hierarchical trees (SPIHT) is an embedded coding
technique. In an embedded coding algorithm, all encodings of
the same signal at lower bit rates (than target rate) are
embedded at the beginning of the bit stream for the target bit
rate. So we can use any amount of bits received for decoding,
at a lower bit rate that can be achieved when using the whole
bit stream of the coded signal. Effectively, bits are ordered in
importance. This type of coding is especially useful for
progressive transmission and transmission over a noisy
channel. Using an embedded code, an encoder can terminate
the encoding process at any point, thereby allowing a target
rate or distortion parameter to be met exactly. Typically, some
target parameters, such as bit count, is monitored in the
encoding process and when the target is met, the encoding
simply stops. Similarly, given a bit stream, the decoder can
cease decoding at any point and can produce reconstruction
B. Biorthogonal Wavelet Basis
Many signals we use are mostly smooth (except for sharp
slopes). For example images have regions of low gray level
difference, 1-D signals have smooth parts between some
peaks. So, it seems appropriate that an exact reconstruction
subband coding scheme for signal compression should
correspond to an orthonormal basis with reasonably smooth
mother wavelet. For having fast computation, the length of
filter must be short, but short filter leads to less smoothness
and we must do a tradeoff between them. On the other hand, it
is desired that FIR filter to be linear phase, since such filters
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International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
corresponding to all lower-rate encodings.
Embedded coding is similar in spirit to binary finite
precision representations of real numbers. All real numbers
can be represented by a string of binary digits. For each digit
added to the right, more precision is added. Yet, encoding can
cease at any time and provide the best representation of the
real number achievable within the framework of the binary
digit representation. Similarly, the embedded coder can cease
at any time and provide the best representation of the signal
achievable within its framework.
Embedded zerotrees of wavelet (E W), introduced by J. M.
Shapiro [5] is an embedded coding algorithm for image
compression. It works on discrete wavelet transform
coefficients of an image. It is very effective and
computationally simple technique for image compression.
SPIHT algorithm introduced for image compression in [6] is a
refinement to E W and uses its principles of operation. These
principles are partial ordering of transform coefficients by
magnitude with a set partitioning sorting algorithm, ordered
bit plane transmission and exploitation of self-similarity
across different scales of an image wavelet transform. The
partial ordering is done by comparing the transform
coefficients magnitudes with a set of octavely decreasing
thresholds. In fact, in this algorithm, a transmission priority is
assigned to each coefficient to be transmitted. Using these
rules, the encoder always transmits the most significant bit to
the decoder. SPIHT has even better performance than E W
in image compression. In [7], SPIHT algorithm is modified
for 1-D signals and used for ECG compression and we call it
1D-SPIHT.
We can observe in Fig. 1 that according to this relationship,
one coefficient in the lowest frequency subband (i.e. xL5) has
no descendent in terms of orientation trees.
In order to exploit the self similarity during the 1D-SPIHT
coding process, several oriented trees are taken from a wavelet
transformed signal. Every tree is rooted as the corresponding
top-most lowpass subband. The 1D-SPIHT algorithm assumes
that each coefficient xi is a good predictor of the coefficients
which are represented by the subtree rooted by xi, i.e. D(i). the
overall procedure is controlled by an attribute, which gives
information on the significance of the coefficients. A
coefficient of the wavelet transformed signal is significant
with respect to a threshold k if its magnitude is larger than 2k.
Otherwise it s called insignificant with respect to the threshold
k. It can be described as:
¦£¦ 1, if x i p 2k
Sk (x i ) = ¤
¦¦ 0, otherwise
¥
where Sk (x i ) denotes the significance of xi with respect to a
threshold k.
In the 1D-SPIHT, the wavelet coefficients are classified in
three sets, namely the list of insignificant points (LIP) which
contains the coordinate of those coefficients that are
insignificant with respect to the current threshold k, the list of
significant points (LSP) which contains the coordinates of
those coefficients that are significant with respect to k, and the
list of insignificant sets (LIS) which contains the coordinates
of the roots of insignificant subtrees. In addition, the contents
of LIS are classified in types A and B, which represent the
D(i) and L(i) cases, respectively. We use 22 steps to depict the
overall 1D-SPIHT coding process as follows:
(0)
B. Details of SPIHT Algorithm
1-D SPIHT is one of the excellent embedded coding systems
that deserve great attention. In the 1-D SPIHT, the original
signal is first decomposed into several subbands by 1D-DWT.
Then a special tree structure called the spatial orientation tree
is defined according to the similarity among coefficients
across subbands. Such an orientation tree represents the
parent-offspring relationship among these quadrate mirror
filters ( MF) decomposition subbands. We use arrows in Fig.
1 to illustrate the parent-offspring relationship defined in the
1D-SPIHT and a 5-level 1D-DWT is assumed. Each black dot
in this figure denotes a wavelet coefficient.
In Fig. 1, the subbands are arranged from lowpass subbands
to highpass subbands. Let xi be any wavelet coefficient and i
denote its corresponding coordinate. There are three important
definitions in the 1D-SPIHT parent-offspring relationship as
shown in Fig. 2:
1) O(i): offspring O(i) represents the set of the 2 coefficients
(as pointed by arrows) of next higher subband from
coefficient xi.
2) D(i): the descendent D(i) of coefficient xi is the set
containing all offspring in all later subbands.
3) L(i): a set defined by L(i) = D(i) - O(i)
Initialization
k = ¡¡ log2 max x i °° ,
¢
±
i
0 b i b K , where K denotes the number of DWT
coefficients.
LSP= and LIP=H, where H is a set of all roots
coordinates in the top-most lowpass subband.
Add all elements i H with D(i ) v to LIS as
type-A entries.
Compute
and
output
(1)
(2)
(3)
(4)
Sorting pass
For each i LIP
Output Sk (x i ) ;
If Sk (x i ) = 1 , then
move i to LSP and output the sign of xi;
(5)
For each i LIS
(6)
If i is of type-A then
(7)
Output Sk (D(i )) ;
(8)
If Sk (D(i )) = 1 then
(9)
For each j O(i )
(10)
Output Sk (x j ) ;
221
(11)
If Sk (x j ) = 1 then
(12)
Add j to LSP and output the sign of xj;
else append j to LIP;
International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
XL5
XH5
XH4
XH3
XH2
XH1
Fig 1. A 5-level 1D-DWT spatial orientation tree of the SPIHT.
(13)
(14)
(15)
(16)
(17)
(18)
(19)
if L(i ) v then
move i to the end of LIS as an entry of
type-B and go to step (5);
else remove i from LIS
if i is of type-B then
output Sk (L(i )) ;
if Sk (L(i )) = 1 then
append each j O(i ) to LIS as an entry of
type-A and remove i from LIS;
coefficients) for encoding them. The termination of encoding
algorithm is specified by a threshold value determined in the
program; changing this threshold, gives different CRs. The
output of the algorithm is a bit stream (0 and 1). This bit
stream is used for reconstructing signal after compression.
From it and by going through inverse of SPIHT algorithm, we
compute a vector of 1024 wavelet coefficients and using
inverse wavelet transform, we make the reconstructed 1024
sample frame of ECG signal.
(20) Refinement pass
For each i LSP except those included in the
sorting pass, output the kth bit of | xi |;
If k = 0 then end; else k = k 1 and go to step (1).
(21)
I. RESULTS AND DISCUSSION
A. Simulation Results
The ECG signals used in the simulation are from MIT-BIH
arrhythmia database. This database includes different shapes
of ECG signals. The records used are 100, 101, 102, 103, 104,
105, 106, 107, 118, 119, 200, 201, 202, 203, 205, 207, 208,
209, 210, 212, 213, 214, 215, 217 and 219 (25 records). The
distortion between original signal and reconstructed signal is
measured by percent root mean square difference (PRD):
In the 1D-SPIHT, wavelet coefficients are arranged in a
parent-offspring orientation tree in order to exploit the spatial
self-similarity property of wavelet coefficients across
subbands. The property implies that if a node coefficient is
insignificant with respect to a given threshold, probably all
nodes descending from that are insignificant too.
N
C. Proposed Compression Method
We first apply wavelet transform to the ECG signal. For
faster computations, we divide ECG signal to contiguous nonoverlapping frames of 1024 samples and we use each frame
for encoding separately. We apply wavelet transform to the
frames of ECG signal to 6 levels of decomposition. The
wavelet used is biorthogonal 9/7 (Table I). We can assume
that each wavelet coefficient is represented by a fixed-point
binary format, so we treat it as an integer, because SPIHT
algorithm works on integer values. Therefore we apply SPIHT
algorithm to these integers (produced from wavelet
PRD =
[ x (i) xˆ(i) ]2
i =1
N
× 100%
(10)
[ x (i) ]
2
i =1
where x (i ) is the i th sample of the original signal, xˆ(i ) is the
i th sample of reconstructed signal and N is the number of
samples of a frame of signal; here N = 1024 . Although PRD
does not account for differences between morphology of two
signals and may not report shape distortions, is used widely in
signal compression literature as a standard measurement,
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International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
reconstructed signal begins to distort unacceptably. For
example, comparison for record 117 in Fig. 8, shows that for
CR about 40, the reconstructed signal is composed of some
flat and some sharp parts, and many details of signal, clearly is
lost.
because it s easy to compute and compare. Compression ratio
(CR) is computed from the ratio of original file size (in bits) to
the length of output bit stream. The CR-PRD diagram for 25
xi
D(i)
II. SUMMARY AND CONCLUSION
In this paper we applied wavelet transform to the ECG
signal and encoded the wavelet coefficients with SPIHT
algorithm. The results show the high efficiency of this method
O(i)
CR-PRD Diagram for 25 Records from MIT-BIH Database
12
10
L(i)
PRD (%)
8
6
4
Fig 2. The definition of parent-offspring relationship in 1-D SPIHT
2
different records is plotted in Fig. 3. In Fig. 4, the average
PRD vs. average CR for all tested records is plotted. The
standard deviation of PRDs is also shown on the Fig. 4. The
original signal, reconstructed signal and error between them
for three records with the corresponding CR and PRD values
are shown in Fig. 5, 6 and 7. In Fig. 8, the result of
compressing record 117 with three different CRs is shown.
0
4
6
8
10
12
14
16
18
20
CR
Fig. 3. CR-PRD results for 25 records from MIT-BIH database.
Average CR-PRD for 25 Records with Standard Deviation
14
12
B. Discussion
Fig. 3 shows that PRD slightly increases by increasing CR.
It also reveals that results for all tested records are in an
acceptable range. This means the usefulness of the
compression method for different ECG records. Here, it
should be mentioned that in ECG compression, not only we
deal with normal ECG signals, but also we mostly deal with
ECG signals with arrhythmia that in general have not a simple
and common pattern. The usefulness of an ECG compression
method can be assessed by its performance in compressing
different shapes and patterns of ECG signal. As the tested
records from MIT-BIH database have different morphologies,
it s a very good test for the compression method. Since the
compression results are in a close range, we can see high
performance of the method. Fig. 4 depicts the close results for
all records and efficiency of the method for all shapes of ECG,
though by increasing the CR, the PRDs change in a wider
range. It shows that standard deviation of PRD for all CRs is
about 1%. From plots of reconstructed signals, we see that for
a CR around 20, almost all of important details and features of
the shape of the signal are preserved. Other simulations
showed that by increasing the CR over 20, the shape of the
10
PRD (%)
8
6
4
2
0
4
6
8
10
12
14
16
18
20
22
CR
Fig. 4. Average result for tested records. Standard deviation of PRD values is
also plotted.
Table II. Compression performance for record 117 from
MIT-BIH database with different compression methods
Compression method
CR
PRD (%)
A TEC [1]
6.8
10.0
Djohan [9]
8
3.9
Hilton [8]
8
2.6
LPC [10]
11.6
5.3
Proposed method
21.4
3.1
223
24
International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
record 117, original signal
record 117, CR=15.6535
record 117, CR=21.445
record 117, CR=43.8857
Original Signal MIT-BIH Record 119
Reconstructed Signal, CR=22.1405
PRD=5.2165 %
error
Fig. 8. Results of reconstructing 1024 samples of record 117 with three
different CRs.
Fig. 5. ECG compression using bi 9/7 wavelet and SPIHT algorithm for record
119 from MIT-BIH database. Top figure is original signal, the middle is
reconstructed signal and bottom signal is error. CR=22.1, PRD=5.2%
for ECG compression. By this method, we achieved the CR
about 20 with a very good reconstruction quality. In Table II
is given the CR and PRD values for some other compression
methods. It shows that SPIHT method has very better results.
SPIHT is a very computationally simple algorithm and is easy
to implement, in comparison with many complex coding
methods. It s also an embedded coding algorithm that makes it
useful for transmission purposes. Although we can achieve
higher CRs by utilizing some lossless arithmetic coding (such
as run-length coding which increases CR by about 5%), but
we lose the important feature of embedded coding; because
the arithmetic coding techniques work on the whole bit stream
and code it once, so in reconstructing signal, we should first
decode the received data for lossless coding to reach the bit
stream of SPIHT, losing the capability of progressive
decoding. Applying lossless coding can only helps in storage
applications.
Original Signal MIT-BIH Record 100
Reconstructed Signal, CR=18.6748
PRD=5.842 %
error
REFERENCES
Fig. 6. ECG compression using bi 9/7 wavelet and SPIHT algorithm for record
100 from MIT-BIH database. Top figure is original signal, the middle is
reconstructed signal and bottom signal is error. CR=18.6, PRD=5.8%.
[1]
Original Signal MIT-BIH Record 107
[2]
[3]
Reconstructed Signal, CR=23.4952
PRD=10.8134 %
[4]
[5]
error
[6]
[7]
[8]
Fig. 7. ECG compression using bi 9/7 wavelet and SPIHT algorithm for record
107 from MIT-BIH database. Top figure is original signal, the middle is
reconstructed signal and bottom signal is error. CR=23.5, PRD=10.8%.
224
S. M. S. Jalaleddine, C. G. Hutchens, R. D. Strattan, and W. A. Coberly,
ECG Data Compression Techniques- A Unified Approach , IEEE
Trans. Biomed. Eng., vol. 37,no. 4, pp. 329-343, Apr. 1990.
M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image
Coding Using Wavelet Transform”, IEEE Trans. Image Processing, vol.
1, no. 2, pp. 205-220, April 1992.
S. G. Mallat, “A Theory of Multiresolution signal decomposition: The
Wavelet Representation”, IEEE Trans. Pattern Anal. Mach. Intel., vol
11, no. 7, pp. 674-693, July 1989.
C. S. Burrus, R. A. Gopinath, H. Guo, Introduction to Wavelets and
Wavelet Transforms, Prentice-Hall, 1997.
J. M. Shapiro, “Embedded Image Coding Using erotrees of Wavelet
Coefficients”, IEEE Trans. Signal Processing, vol. 41, no. 12, pp. 34453462, Dec. 1993.
A. Said, W. A. Pearlman, “A New, Fast and Efficient Image Codec
Based on Set Partitioning in Hierarchical Tress”, IEEE Trans. Circ. Sys.
Vid. Tech., vol. 6, pp. 243-250, June 1996.
. Lu, D. Y. Kim, W. A. Pearlman, “Wavelet Compression of ECG
Signals by the Set Partitioning in Hierarchical Trees Algorithm”, IEEE
Trans. Biomed. Eng., vol. 47, no. 7, pp. 849-856, July 2000.
M. L. Hilton, “Wavelet and Wavelet Packet Compression of
Electrocardiograms”, IEEE Trans. Biomed. Eng., vol. 44, pp. 394-402,
May 1997.
International Journal of Signal Processing 1;4 © www.waset.org Fall 2005
[9]
A. Djohan, T. . Nguyen, W. J. Tompkins, “ECG Compression Using
Discrete Symmetrical Wavelet Transform”, Proc. IEEE Intl. Conf.
EMBS, 1995.
[10] A. Al-Shrouf, M. Abo- ahhad, S. M. Ahmed, “A novel compression
algorithm for electrocardiogram signals based on the linear prediction of
the wavelet coefficients”, Digital Signal Processing, vol. 13, no. 4, pp.
604-622, October 2003.
Ali Taheri was born in Tehran, Iran. He received the B.Sc. degree in electrical
engineering from Sharif University of Technology, Tehran, Iran in 2002. He is
currently a M.Sc. student at the Department of Biomedical Engineering, AmirKabir University of Technology, Tehran. His research interests include signal
processing, data compression, coding techniques, image analysis, and discrete
event systems and control systems design.
Morteza Moazami-Goudarzi was born in Boroujerd, Iran. He received the
B.Sc. degree in biomedical engineering from Shahed University, Tehran, Iran
in 2002. He is currently a M.Sc. student at the Department of Biomedical
Engineering, Amir-Kabir University of Technology, Tehran. His research
interests include biomedical signal processing, data compression, wavelets and
multiwavelets, and image analysis.
225
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