signal analysis and processing in biomedicine analysis and processing in...overview o mathematical...
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Signal Analysis and Processing in Biomedicine
Prof. dr. Srdjan Stankovic
Prof. dr. Irena Orovic
LECTURE NOTES
Development and development of this material is partly supported by Project
Studies in Bioengineering and Medical Informatics (BioEMIS)
For subject: Biomedical signal and image processing
Izrada i štampanje ovog nastavnog materijala je djelimicno pomognuta od
strane Projekta
Studije u bioinženjeringu i medicinskoj informatici
Za predmet: Digitalna obrada biomedicinskih signala i slika
TEMPUS-1-2012-1-UK-TEMPUS-JPCR
The authors are very thankful for support
Podgorica – January - 2016
TEMPUS BioEMIS Edition, 2016 2
Overview
O Mathematical transforms in biomedical signal
processing
O Computer tomography
O Compressive sensing
O Compressive sensing in biomedical imaging
O ECG signals
O Hermite transform in ECG signals analysis
O Detection of swallowing sounds – appl.
Dysphagia
O Telemedicine
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Mathematical transforms in biomedical signal
processing
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Fourier transform
time frequency
( ) ( ) j tF f t e dt
1( ) ( )
2
j tf t F e d
inverse:
• Some useful Properties:
( ) ( ) ( ) ( ) ( ) ( )j t j t j tf t g t e dt f t e dt g t e dt F G
Linearity
00( ) ( )
j tj tf t t e dt e F
Time shift
00( ) ( )
j t j te f t e dt F
Frequency shift
*{ ( ) ( )} ( ) ( ) ( ) ( )FT f t g t FT f g t d F G
Convolution
*{ ( ) ( )} ( ) ( )FT f t g t F G TEMPUS BioEMIS Edition, 2016 5
Discrete Fourier transform
0 50 1000
20
40
60
0 50 100-3
-2
-1
0
1
2
Noisy signal
1 2 ...
N-1
f(n)
0
21
0
( ) ( )N j nk
N
n
DFT k f n e
21
0
1( ) ( )
N j nkN
k
f n DFT k eN
time frequency TEMPUS BioEMIS Edition, 2016 6
Fourier transform
( )( ) j tx t Ae -Frequency modulated signal
Ideal time-frequency representation of
x(t) should concentrate energy along
the instantaneous frequency of the
signal. It is defined as:
2( , ) 2 ( - '( )),ITF t A t
ω=’(t)
is instantaneous frequency
where
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( , ) ( ) ( ) jSTFT t w x t e d
• Short Time-Fourier Transform
window function
1
( , ) ( , )m
M
xm
STFT t STFT t
STFT is linear
transform 1
( ) ( )M
mm
x t x t
Time-Frequency Representation
Multicomponent signals:
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S-method • S-method: Distribution with the auto-terms being the same as in the
Wigner distribution, but with reduced or completely removed cross-
terms: 1( , ) ( ) ( , ) *( , ) ,SM t P STFT t STFT t d
*
2 *
1
( , ) ( ) ( , ) ( , )
( , ) 2 Re ( , ) ( , )
d
d
d
L
i L
L
i
SM k n P i STFT k i n STFT k i n
STFT k n STFT k i n STFT k i n
Where 𝑃(𝜃 is finite frequency domain window. Discrete form of the S-
method is:
In order to avoid the presence of cross-terms, the
value of Ld should be less than half of the distance
between two auto-terms
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Wavelet transform Continuous wavelet transform:
O Wavelets can be used as basis for representing functions;
O Wavelets-functions formed by scaling and translation of basis (mother) functions in the time domain;
O It satisfies the following conditions:
Area under the curve is equal
to zero
The function is square
integrable
( ) 0t dt
2( )t dt
( ) ( )i i
i
y t a t
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Wavelet transform
O Continuous wavelet transform of the signal is given by:
,( , ) ( ) ( )a bW a b t f t dt
,
1( ) ( ) ( , )a b
a b
f t t W a b dadbC
Inverse form:
2( )
C d
-Fourier transform of ( )t𝛹(𝜔
C is positive and finite
• Wavelet is defined by
formula:
,
1( )a b
t bt
aa
- a, b - two real numbers used for
scaling and translation in time
basis function is shrunk in time
basis function is spread
0 1a
1a
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Wavelet families
O Daubeshies
O Haar
O Biorthogonal
O Mexican Hat
O Symlets
O Morlet
O Coiflets
O Meyer
Haar
Daubechies Morlet Symlet
Biorthogonal wavelet
function
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Discrete wavelets
0ma a 0 0
mb nb a• Discretization is done
by using:
,
m, n Z, b0>0
/ 2, 0 0 0( ) ( )m m
m n t a a t nb Discrete wavelet
function
/ 2, 0 0 0( ) ( )d m m
m nW a f t a t nb dt
Discrete wavelet
transformation
• Using a0=2 and b0=1 dyadic sampling is obtained
/ 2, ( ) 2 (2 )m m
m n t t n / 2
, 2 ( ) (2 )d m mm nW f t t n dt
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O Decomposition is applied to rows, then to columns
Standard wavelet decomposition
Li - low-frequency sub-band
Hi - high-frequency sub-band
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Quincunx decomposition O Decomposition uses only low frequency sub-band of different level
O Low frequency sub-band is then divided into low and high frequency part
ORIGINAL
IMAGEL1 H1
L2
H1
H2
H1
H2
H1
H2
H3
L4
H4
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Pyramidal wavelet decomposition
Uniform wavelet decomposition
LLLL LLHL HLLL HLHL
LLLH LLHH HLLH HLHH
LHLL
LHLH
LHHL
LHHH
HHLL HHHL
HHLH HHHH
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Hermite functions
2 2/2 /20 14 4
1 2
1 2( ) , ( ) ,
12( ) ( ) ( ), 2.
x x
p p p
xx e x e
px x x x p
p p
Hermite functions of
different orders
Recursive realization of
Hermite functions:
The Hermite functions provides good
localization and the compact support
in both time and frequency domain.
The i-th order Hermite function is
defined as:
2 2/2( 1) ( )( ) .
2 !
i t i t
i ii
e d et
dti
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Hermite expansion ( ) (1)
( ) (1) ,X P X
x i X iP
1,...,i P
( ) ( ) ( )f i X i x i
First step –removing the
baseline
Baseline substraction from the original
signal 1
0( ) ( )
N
p pp
f i c i
Decomposition using N Hermite functions
( ) ( ) ( )p pc i f i i di
Hermite expansion cofficients:
11
1( ) ( ) ( )
M pp m mM
mc i x f x
M
Gauss-Hermite quadrature technique can
be used to calculate the Hermite expansion
coefficients
mx
- a zeros of an M-th order Hermite
polynomial
1 21
( )( )
( ( ))
p mpmM
M m
xx
x
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Hermite transform
O The expansion using N Hermite functions can be written
in matrix form. First, we define the Hermite transform
matrix W (of size ):
0 0 0
2 2 2
1 1 1
1 1 1
2 2 2
1 1 1
1 1 1
2 2 2
1 1 1
(1) (2) ( )
( (1)) ( (2)) ( ( ))
(1) (2) ( )1
( (1)) ( (2)) ( ( ))
(1) (2) ( )
( (1)) ( (2)) ( ( ))
N N N
N N N
N N N
N N N
M
M
M
MN
M
M
W
If the vector of Hermite
coefficients is: 0 1 1, ,...,T
Nc c c c
and vector of M signal samples is:
[ (1), (2),..., ( )]Tf f f Mf
then we have
c WfFor a signal of length M, the complete set
of discrete Hermite functions consists of
exactly N=M function. In some
applications, a smaller number of Hermite
functions N<M can be used TEMPUS BioEMIS Edition, 2016 20
O Having in mind the Gauss-Hermite approximation, the
inverse matrix contains N Hermite functions, given
by:
O Now, the Hermite expansion for the case of discrete signal
can be defined as follows:
1W
Hermite transform
0 0 0
1 1 1 1
1 1 1
(1) (2) ( )
(1) (2) ( )
(1) (2) ( )N N N
M
M
M
Ψ W
1 f W c Ψc
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Computer tomography
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Computer Tomography O Clinical Computed Tomography (CT) was introduced
in 1971 - limited to axial imaging of the brain in neuroradiology
O It developed into a versatile 3D whole body imaging modality for a wide range of applications in for example O oncology,
O vascular radiology,
O cardiology,
O traumatology and i
O interventional radiology.
O Computed tomography can be used for diagnosis and follow-up studies of patients, planning of radiotherapy treatment, screening of healthy subpopulations with specific risk factors. TEMPUS BioEMIS Edition, 2016 23
O CT scanning is suitable for 3D imaging of the brain,
cardiac, musculoskeletal, and whole body CT imaging.
O The images can be viewed as colored 3D rendered
images, although radiologists prefer black and white 2D
images
Computer Tomography
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O Acquisition is achieved using:
O hundreds of detector elements along the detector arc (800-900 detector elements),
O using rotation of the x ray tube around the patient, taking about 1000 angular measurements
O using tens or even hundreds of detector rows aligned next to each other along the axis of rotation
Computer Tomography
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Image reconstructions from projections
O Image reconstruction based on projections has
important applications in various fields (e.g., in
medicine when dealing with computer
tomography, which is widely used in
everyday diagnosis).
O Consider an object in space, which can be
described by the function f(x,y). The projection
of function f(x,y) along an arbitrary direction
(defined by an angle ) can be defined as
follows:
( ) ( , ) , cos sinAB
p u f x y dl u x y TEMPUS BioEMIS Edition, 2016 26
Image reconstructions from projections
The previous equation can be written as follows:
( ) ( , ) ( cos sin )p u f x y x y u dxdy
The Fourier transform of the projection is given by:
( ) ( ) j uP p u e du
Furthermore, the two-dimensional Fourier transform of f(x,y) is defined as:
( )( , ) ( , ) x yj x y
x yF f x y e dxdy
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Image reconstructions from projections
As a special case, we can observe ( , )x yF for ωy=0
0 0
( ,0) ( , )
( ) ( )
x
x
j xx
j x
F f x y e dxdy
p x e dx P
To conclude: the Fourier transform of a
two-dimensional object along the axis
ωy=0 is equal to the Fourier transform
along the projection angle =0
In the rotated coordinate system, we have:
cos sin
sin cos
u x
l y
( cos sin )
( ) ( ) ( , )
( , ) ( , ),
j u j u
j x y
P p u e du f u l e dudl
f x y e dxdy F
cossin
( , ) ( , )x
y
x yF F
where TEMPUS BioEMIS Edition, 2016 28
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O Therefore, we can conclude:
O If we have the object projections, then we can determine their Fourier transforms.
O The Fourier transform of a projection represents the transform coefficients along the projection line of the object.
O By varying the projection angle from 0 to 180 we obtain the Fourier transform along all the lines (e.g., we get the Fourier transform of the entire object), but in the polar coordinate system.
O To use the well-known FFT algorithm, we have to switch from polar to rectangular coordinate system. Then, the image of the object is obtained by calculating the inverse Fourier transform.
O The transformation from the polar to the rectangular coordinate system can be done by using the nearest neighbor principle, or by using some other more accurate algorithms that are based on the interpolations.
Image reconstructions from projections
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Compressive sensing
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Compressive sensing O Compressive sensing approaches have been
intensively developed to overcome the limits of traditional sampling theory by applying a concept of compression during the sensing procedure.
O Compressive sensing aims to provide the possibility to acquire much smaller amount of data, but still achieving the same quality (or almost the same) of the final representation as if the physical phenomenon is sensed according to the conventional sampling theory.
O In that sense, significant efforts have been done toward the development of methods that would allow to sample data in the compressed form using much lower number of samples.
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Compressive sensing O Compressive sensing opens the possibility to simplify
very expensive devices and apparatus for data recording,
imaging, sensing (for instance MRI scanners, PET
scanners for computed tomography, high resolution
cameras, etc.).
O Furthermore, the data acquisition time can be
significantly reduced, and in some applications even to
almost 10 or 20% of the current needs.
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WE WILL NEED 6 TIMES LESS MEASUREMENTS
FOR THE SAME IMAGE QUALITY
IT MEANS 6
TIMES LOWER
EXPOSURE OF
PATIENTS
TO INVASIVE
METHODS TEMPUS BioEMIS Edition, 2016 34
Compressive sensing
O If the samples acquisition process is linear, than
the problem of data reconstruction from acquired
measurements can be done by solving a linear
system of equations.
O Measurement process can be modelled by the
measurement matrix .
O Signal f with N samples can be represented as a
signal reconstruction problem using a set of M
measurements obtained by using as follows:
O where y represents the acquired measurements.
Φf = y
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CS conditions O CS relies on the following conditions:
Sparsity – related to the signal nature;
O Signal needs to have concise representation when
expressed in a proper basis (K<<N)
Incoherence – related to the sensing modality; It
should provide a linearly independent
measurements (matrix rows)
Random undersampling is crucial
Restriced Isometry Property – is important for
preserving signal isometry by selecting an
appropriate transformation TEMPUS BioEMIS Edition, 2016 36
Standard approach for signal sampling and
its compressive sensing alternative
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CS problem formulation
O The method of solving the undetermined system of
equations , by searching for the
sparsest solution can be described as:
y=ΦΨx Ax
0min subject to x y Ax
0x l0 - norm
• We need to search over all possible sparse vectors x with
K entries, where the subset of K-positions of entries are
from the set {1,…,N}. The total number of possible K-
position subsets is N
K
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O A more efficient approach uses the near optimal solution
based on the l1-norm, defined as:
1min subject to x y Ax
CS problem formulation
• In real applications, we deal with noisy signals.
• Thus, the previous relation should be modified to include
the influence of noise:
1 2min subject to x y Ax
2e
L2-norm cannot be used because the minimization problem
solution in this case is reduced to minimum energy solution,
which means that all missing samples are zeros TEMPUS BioEMIS Edition, 2016 39
Signal - linear combination of the
orthonormal basis vectors
Summary of CS problem formulation
1
( ) ( ), : .N
i ii
f t x t or
f =Ψx𝐟
y=ΦfSet of random measurements:
random measurement
matrix
transform
matrix
transform
domain vector
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CS reconstruction algorithms O I group: convex optimizations such as
O Basis pursuit
O Dantzig selector,
O Greedy algorithms:
O Matching pursuit,
O Orthogonal matching pursuit;
O Iterative thresholding (hard and soft versions)
O Hybrid methods:
O Compressive sampling matching pursuit
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f
Transform matrix
Measurement matrix
y Measurement vector
Greedy algorithms – Orthogonal Matching Pursuit
(OMP)
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Iterative hard thresholding O The IHT algorithm is an iterative method
where Hk is the hard thresholding operator that sets all but the k largest (in magnitude) elements in a vector to zero.
O IHT is a simple reconstruction algorithm and it can recover sparse and approximately sparse vectors with near optimal accuracy.
O 1) The step size µ has to be chosen appropriately to avoid instability of the method
O 2) IHT has only a linear rate of convergence
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y – measurements
M - number of measurements
N – signal length
T - Threshold
• DFT domain is assumed as sparsity domain
• Apply threshold to initial DFT components
(determine the frequency support)
• Perform reconstruction using identified support
Single-Iteration Reconstruction
Threshold based Algorithm for components
100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
Initial FT and Threshold
100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
Reconstructed FT
100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
Initial FT
100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
Reconstructed FT
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Variance of DFT values
at the non-signal and
signal positions can be
calculated as:
2
1
( )var{ }
( 1)i
K
k k i
k
M N MF A
N
y=ΦΨx=Θx Is a CS matrix (could be DFT,
DCT, HT) TEMPUS BioEMIS Edition, 2016 45
In each iteration we
need to remove the
influence of
previously detected
components and to
update the value of
threshold
Iterative threshold based solution for detection of
desired components
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Compressive sensing in biomedical imaging
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Compressed sensing in CT
O Due to its powerful ability of reconstructing signal or image from highly undersampled data, CS has attracted tremendous attention in medical imaging community.
O The first attempt to apply CS to medical imaging was done by Lustig et al., in which they successfully reconstructed MR images with 40% of the data.
O There are also research efforts on compressed sensing CT because in principle a CS based algorithm can produce good reconstructions using fewer samples (projections), it has the potential to reduce the radiation exposure to the patient.
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Recovering of highly undersampled signal
512*512 Shepp-Logan Undersampled by 22 radial lines
Normal Reconstruction
CS reconstruction algorithm TEMPUS BioEMIS Edition, 2016 49
O The most popular type of method that has
been studied is the Total Variation (TV) based
compressed sensing methods.
O TV based methods use total variation as a
sparsity transform.
Compressed sensing in CT
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Total Variation minimization O Natural images are not sparse – neither in space
nor in the frequency domain
O Instead of the l1 norm minimization, Total Variation
(TV) minimization is commonly used
O Having in mind that the image gradient is sparse,
TV is, in fact, l1 norm minimization of the image
gradient
O Introduced by Rudin, Osher and Fatemi, for
solving the inverse problems
TEMPUS BioEMIS Edition, 2016 51
O Inverse problems:
O signal X, measurements y (modeled by applying the operator A – measurement matrix)
O Noisy measurements: y=AX+n
O Linear inversion (if A is linear), deconvolution problem
O Standard approach to linear inversion problems:
O Definition of the objective function
O Solution of the minimization function according to X
Total Variation minimization
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y nAX
Total Variation minimization
X - signal to be estimated
n – aditive noise
A - matrix that models measurement
process Defining the objective
function:
( ) ( , ) ( )F d R yX AX X
– difference measure between signals y and X
- it can be mean square error
R(X) – regularization function
λ – regularization parameter, λ>0
The value of the X should be such that the relation AX
corresponds to the vector y
2
2( , )
ld y AX y AX
d(y, AX)
TEMPUS BioEMIS Edition, 2016 53
O Solution X=A-1y is not applicable if A is not an
invertible matrix
O The goal of R(X) is to avoid such situations
O If the signal X is sparse, the function R(X)
corresponds to the l1 norm, i.e. objective
function is defined as:
Total Variation minimization
2 1
2( )
l lF X y AX X
TEMPUS BioEMIS Edition, 2016 54
O The regularization function R(X)
can be defined as TV norm:
O The is the gradient operator:
O TV norm in the discrete form:
Total Variation minimization
1
( )l
R X X
,
( 1, ) ( , )
( , 1) ( , )i j
i j i j
i j i j
X XX
X X𝛻
2 2
,
( ) ( ) ( ) ,h v
i j
TV X XX( 1, ) ( , )
( , 1) ( , )
h
v
i j i j
i j i j
X
X
X X
X X
- Row and column differences TEMPUS BioEMIS Edition, 2016 55
O Special attention is paid to the Iterative shrinkage/thresholding (IST) algorithms
O Used for solving multi-dimensional optimization problems.
O Problem of these algorithms: slow convergence
O Convergence of the IST algorithms is speed up by introducing the two step IST algorithm (two-step Iterative shrinkage/thresholding, TwIST)
Iterative shrinkage/thresholding algorithms
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Two-step Iterative shrinkage/thresholding, TwIST
O Finds solution X in two-steps:
O Where is defined as:
O Denoising and regularization functions are described by
following relations:
1 0
1 1
( ),
(1 ) ( ) ( ), 1.t t t t t
X X
X X X X
( ) ( ( ))T X X A y AX
21 1( ) arg min ( )
2reg
X
X X
( )reg iiD X X
Denoising operator
Regularization function for TV l1 problems TEMPUS BioEMIS Edition, 2016 57
O Optimal choice for the parameters α and β:
O There is a large number of methods for solving TV minimization problems: time marching scheme, fixed point iteration metod, majorization-minimization approach
O TV-L2 problem can be defined as:
Two-step Iterative shrinkage/thresholding, TwIST
2
1
1 21,
max( )1 m
k
k
1
1
0 ( ) ,
max( )
T
i m
m
k
A A
2 2min
2ii
D
X
X AX y
ɛ - constant
y measurements
X –image to be obtained
A-CS matrix TEMPUS BioEMIS Edition, 2016 58
Image reconstruction examples
Original
Estimate
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250
50 100 150 200 250
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100
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250
50 100 150 200 250
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100
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Estimate
50 100 150 200 250
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250
Estimate
50 100 150 200 250
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50 100 150 200 250
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Reconstructed images
DFT masks
Original image
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Image reconstruction – more examples
Reconstructe
d image
Reconstructe
d image
Reconstructe
d image
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ECG signals
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ECG signals -basics
O ECG signal represents the activity of the heart.
O The propagation of electrical waves resulted from cardiac activity can be measured as a difference of potentials between two points on the human body
O In the case of multi-channel ECG, the signals are measured between different pairs of electrodes on the human body.
O ECG signal characteristics can be described using 7 specific parts.
O P,Q,R,S and T peaks
O 2 iso-electric parts that separate P and T peak from the large QRS complex.
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ECG signals -basics
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ECG-signal basics O The heart rhythm is of sinus type characterized among other features
by the presence of P-complex before every QRS complex.
O The heart impulse rate is characterized by heart agitation in average in the frequency of 60–100 beats per minute (bpm) for adults, although it might be normal, especially for athletes, to have resting bpm as low as 30.
O Deviations of the rate beyond this range can be treated as abnormal case. However, the limit frequencies should be taken individually.
O For the time duration of PR, QRS and QT parts in the cardiac cycle, a reasonable rule is to consider that the interval QT is less than a half of the distance between two successive QRT complexes. That is, QT should be less than 1/2 of the RR interval.
O P complex (P wave) is normally 0.04–0.11 s in duration. Its deviation from the normal wave shape or its disappearing means a pathological case.
O The normal duration of ST segment is 0.02–0.12 s. Any drop in the duration of the ST segment suggests ischemic, whilst its shift above the cycle-axis suggests a heart attack.
O The normal T-complex is about half of the P-complex time. When it is inverted (except for aVR lead), it typically indicates a heart attack.
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Recorded evolution of ECG waveform showing the
pathological disturbances after the heart attack in ST, T and
Q parts
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Finding of proper heartbeats on the example of
R and P points. The signal and the markings of
characteristic points are drawn by the simulation
software.
TEMPUS BioEMIS Edition, 2016 66
• The signal split into 0.25-mV high and 30-second wide spans
• Inside these spans all minima and maxima are counted. Based on their
numbers the span of the signal is estimated and the mV spans containing
possible R and S points are acquired.
• On the basis of these spans the probable P and T points are found.
• The last part is the search for Q point and ST segment with the use of least
squares approach.
• A training algorithm can be used to obtain the ECG signal pattern
The algorithm is based on statistical analysis of probabilities of the existence of
characteristic points of ECG signal in the given millivolt (mV) and time frames.
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ECG signals and QRS complexes
O ECG signals represent the records of electrical activity of the heart over a period of time
O QRS complexes are the most characteristic waves in ECG signals
O QRS complexes are crucial in different stages of medical diagnosis and treatment of heart deceases
O Important research efforts have been made in the development of recognition, classification and compression algorithms for QRS complexes
O Effective storage, automatic detection of anomalies and automatic diagnosis based on the processing of QRS complexes are the state-of-the-art aims in modern biomedicine
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QRS complexes in ECG signals
-0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.8
-0.6
-0.4
-0.2
0
0.2
t [ms]
Am
plit
ude m
V
First 10 miliseconds of a real ECG
signal
Red parts denote intervals of QRS
complexes
PhysioNet: MIT-BIH ECG Compression Test
Database,
http://www.physionet.org/physiobank/database/cdb
Q
R
S
0 1 2 3 4 5 6 7 8 9 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
t [ms]
Am
plitu
de
mV
TEMPUS BioEMIS Edition, 2016 69
QRS complexes and the Hermite transform
O The p-th order Hermite function and the functions of the p-1 and p-2 orders and can be related with the following recursive formula:
O
𝜓0(𝑡, 𝜎 =1
𝜎 𝜋𝑒−
𝑡2
2𝜎2 , 𝜓1(𝑡, 𝜎 =2
𝜎 𝜋
𝑡
𝜎𝑒−
𝑡2
2𝜎2 ,
𝜓𝑝(𝑡, 𝜎 =𝑡
𝜎
2
𝑝 𝜓𝑝−1(𝑡, 𝜎 −
𝑝−1
𝑝 𝜓𝑝−2(𝑡, 𝜎 .
O Hermite tranform is defined by:
O 𝑓(𝑡 = 𝑐𝑝𝜓𝑝(𝑡, 𝜎 𝑀
𝑝=0, while the Hermite coefficients are:
O 𝑐𝑝 =1
𝑀
𝜓𝑝(𝑡𝑚,𝜎
𝜓𝑀−1(𝑡𝑚,𝜎 2 𝑓(𝑡𝑚 𝑀
𝑚=1, 𝑝 = 0,1, . . . , 𝑀 − 1
TEMPUS BioEMIS Edition, 2016 70
O Visual similarity between QRS
complexes as signals with
compact time support and
Hermite bass functions is obvious
O Thus, Hermite transform is often
incorporated in classification and
compression algorithms for QRS
complexes.
O A QRS complex can be modeled
as:
O s(𝑡 = 𝑐𝑝𝜓𝑝(𝑡, 𝜎 𝐾
𝑝=0
O Here, K is the number of non-zero
Hermite coefficients needed for
the successful representation of
QRS complex
0 10 20 30 40 500
0.5
1
0(tn)
0 10 20 30 40 50-1
0
1
1(tn)
0 10 20 30 40 50-1
0
1
2(tn)
0 10 20 30 40 50-1
0
1
tn
3(tn)
First four Hermite basis
functions
Only few Hermite coefficients are
needed for the representation of
QRS complexes
TEMPUS BioEMIS Edition, 2016 71
QRS complexes and the Hermite transform: Representation and the CS scenario
TEMPUS BioEMIS Edition, 2016 72
The influence of the Hermite scaling factor
O Hermite transform of the
QRS complex:
O first row – the selected QRS
complex,
O second row – Hermite
coefficients of the signal with
σ = 1 ⋅ 𝛥𝑡 ,
O third row – Hermite
coefficients of the signal with
σ = 5.7 ⋅ 𝛥𝑡 .
O Automatic procedure of
scaling factor optimization?
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.5
0
0.5
time [s]
Am
pli
tud
e [m
V]
0 5 10 15 20 25 30 35 40 450
0.5
1
p
|cp|
0 5 10 15 20 25 30 35 40 450
0.5
1
p
|cp|
TEMPUS BioEMIS Edition, 2016 73
A compression procedure O Use a suitably chosen scaling factor
O Aproximate the signal with K most significant HT
coefficients, such that the relative error:
𝐸 =‖𝐟 −𝐟‖2
‖𝐟‖2 < 10%
where 𝐟 is the inverse HT of the K most significant
coefficients, and f is the vector of the original signal
O Our experiments show that only 4-5 Hermite coeffcients
are needed to be stored in order to preserve a medically
acceptable error level.
O Very important in order to generate databases for every
patient with heart desease, in order to make efficient
diagnosis and anticipation of future problems TEMPUS BioEMIS Edition, 2016 74
A compression procedure O Original and approximated QRS complex
O Only 4 HT coefficients are used, signal optimally resampled
-0.05 0 0.05
-1
-0.5
0
0.5
1
(a)
Amplitude [mV]
nT [s]
Original QRS complex Hermite transform of the original QRS complex
0 5 10 15 20 25
-200
-100
0
100
200
(b)
p
-0.05 0 0.05
-1
-0.5
0
0.5
1Amplitude [mV]
(c)
Signal resampled by using optimal and m
tmT [s]
0 5 10 15 20 25-400
-300
-200
-100
0
Hermite transform of the resampled signal
p
(d)
TEMPUS BioEMIS Edition, 2016 75
Challenges
O Automatic optimization of the scaling factor using concentration measures
O Apply concentration measures in order to determine a threshold for the automatic recognition and/or classification of QRS complexes in ECG signals
O Use the Hermite transform as the core in expert systems and classification algorithms
O Results verification in consultation with cardiologists
TEMPUS BioEMIS Edition, 2016 76
Detection of swallowing sounds – appl. Dysphagia
TEMPUS BioEMIS Edition, 2016 77
Scope of the approach
O Innovative approach for the analysis of one-dimensional biomedical signals that combines the Hermite projection method with time-frequency analysis
O A two-step approach to characterize vibrations of various origins in swallowing accelerometry signals
O First, by using time-frequency analysis we obtain the energy distribution of signal frequency content in time.
O Second, by using fast Hermite projections we characterize whether the analyzed time-frequency regions are associated with swallowing or some other phenomena (vocalization, noise, bursts, etc.).
O The numerical analysis of the proposed scheme clearly shows that by using a few Hermite functions, vibrations of various origins are distinguishable.
TEMPUS BioEMIS Edition, 2016 78
1. SWALLOWING DIFFICULTIES
O Deglutition, or swallowing, is a well-defined, complex process of transporting food or liquid from the mouth to the stomach.
O Swallowing consists of four phases: oral preparatory, oral, pharyngeal, and esophageal.
O Patients suffering from dysphagia (swallowing difficulty), usually deviate from the well-defined pattern of healthy swallowing.
O Dysphagia is a common problem encountered in the rehabilitation of stroke patients, head injured patients, and others with paralyzing neurological diseases
O Today's dysphagia management relies heavily on the videofluoroscopic swallowing study (VFSS).
O VFSS is accepted as the gold standard, but requires expensive X-ray equipment and expertise from speech-language pathologists and radiologists.
TEMPUS BioEMIS Edition, 2016 79
2. Cervical auscultation and Swallowing accelerometry
O Cervical auscultation is a promising non-invasive tool for the
assessment of swallowing disorders, adopted by dysphagia
clinicians.
O Cervical auscultation approach involves the examination of
swallowing signals acquired via a stethoscope or other acoustic
and/or vibration sensors during deglutition.
O Swallowing accelerometry, a technique that involves an
accelerometer placed on the neck to monitor vibrations
associated with swallowing activities, has been used to detect
aspiration in several studies
O Nevertheless, the presence of various vibrations not associated
with swallowing can severely contaminate swallowing
accelerometry signals
O Vocalizations either voluntarily or involuntarily can have an
adverse effect on these signals, whose presence masks the
observed swallowing signals. TEMPUS BioEMIS Edition, 2016 80
3. Data O The sample data considered in this paper were
gathered over a three month period from a public science centre in Toronto, Ontario, Canada. A dual-axis accelerometer (ADXL322, Analog Devices) was attached to the participant’s neck using double-sided tape in order to monitor vibrations associated with swallowing.
O Data were band-pass filtered in hardware with a pass band of 0.1-3000 Hz and sampled at 10kHz using a custom LabVIEW program running on a laptop computer. Data were saved for subsequent off-line analysis
TEMPUS BioEMIS Edition, 2016 81
During data collection, participants were
cued to perform three types of swallows
involving saliva and water swallows. The
entire data collection session lasted 15
minutes per participant. The participants
were instructed not to vocalize.
Nonetheless, approximately one quarter of
all recordings contained either voluntary or
involuntary vocalizations.
TEMPUS BioEMIS Edition, 2016 82
Time-Frequency Analysis
O Signals considered in this paper mostly contain
swallowing vibrations.
O Some recordings also contain vibrations associated
with vocalization (e.g., speech, cough, laughter) and
various burst components produced by the equipment
and noise.
O The most dominant vibrations are those produced by
swallowing and vocalization.
O Fortunately, vibrations associated with different
phenomena provide unique time-frequency signatures.
TEMPUS BioEMIS Edition, 2016 83
O Thus, time-frequency representations are crucial for the analysis
and classification of these signals. Also, having in mind
multicomponent nature of these signals, time-frequency
representations without cross-terms should be used.
Time-Frequency Analysis
1500 2000 2500
150
200
250
300
350
1000 1500 2000
200
250
300
350
Time-frequency regions of spectrogram
for voiced sound (left), for swallowing
sound (right) TEMPUS BioEMIS Edition, 2016 84
Hermite projection method applied to time-frequency regions
O To understand differences between swallowing vibrations and other miscellaneous vibrations (speech, laugh, cough, etc.) we observe the structures in the time-frequency regions. Here, the spectrogram, as the simplest time-frequency distribution, is used:
O A certain pre-processing in classification procedure is applied, since the signal could be corrupted by noise: the time-frequency mask is defined to remove the noise influence and locate the significant signal components:
O where the threshold value ξ, i.e., the energy floor is obtained as:
22
( , ) ( , ) ( ) ( ) jSPEC t STFT t x t w e d
1 for ( , )( , )
0
SPEC tL t
otherwise
log10(max( ( , ))),10
SPEC tt
TEMPUS BioEMIS Edition, 2016 85
O Filtered spectrogram is given by:
O The energy vector is calculated to determine the time intervals
containing vibration activities:
O Consequently, the time support vector is obtained as (threshold
β is used to remove the remaining noise):
O The time-frequency regions for classification are extracted from
for
O The fast Hermite projection method is applied to these regions
Hermite projection method applied to time-frequency regions
( , ) ( , ) ( , )filtSPEC t L t SPEC t
( ) ( , )filtE t SPEC t
1 for ( )( ) ,
0f
E tE t
otherwise
( , )filtSPEC t arg ( ) 1 .ft E t
TEMPUS BioEMIS Edition, 2016 86
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
50
100
150
200
250
300
2 1 3 4 5 6 7 8 9 10 11 12 13
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
10
20
30
40
50
60Filtered time-frequency representation and assigned regions
(upper row), time support vector (bottom row) TEMPUS BioEMIS Edition, 2016 87
O We use a small number of functions for reconstruction such
that:
O simple structure regions are reconstructed with a small error
O complex structure regions with a significantly larger error.
O The difference between the original and reconstructed regions
i.e. the error, which depends on the region’s structure of a
region, will be used for characterization. This difference is
measured by the mean squared error, as follows:
O where R(t,) denotes the original region, R’(t,) is the
reconstructed region
Hermite projection method applied to time-frequency regions
21( ( , ) '( , ))
t
MSE R t R tT
TEMPUS BioEMIS Edition, 2016 88
O It has been shown that the voiced
vocalization regions (speech, cough,
laughter, etc.) are more complex,
and thus have the higher MSE, in
comparison comparing to the
regions with containing swallowing
sounds vibrations.
O Moreover, there is a significant gap
between the values of MSE for
voiced vocalization and for
swallowing sounds vibrations.
O On the other hand, the MSE for
regions with swallowing sounds
vibrations is much higher than for
the regions with containing noise.
O The difference between the original
and reconstructed regions, given in
terms of MSE, is used as a
parameter for characterizing to
characterize a regions
No. Region
description MSE
1 Noise 0.086
2 Noise 0.01
3 Noise 0.22
4 Swallowing
vibrations
36.29
5 Burst 0.13
6 Burst 3.4
7 Vocalization 669
8 Vocalization 1057.2
9 Vocalization 762
10 Vocalization 505
11 Noise 0.88
12 Swallowing
vibrations
18.6
13 Noise 3
TEMPUS BioEMIS Edition, 2016 89
Vo
cali
zati
on
s
1. 2. 3. 4. 5. 6.
2200 2250 2300 2350 2400
200
250
300
2000 2050 2100 2150
200
250
300
1300135014001450
200
250
300
350 5250 5300 5350 5400
150
200
250
300
350
400
2800 2850 2900 2950
150
200
250
300
350
400
MSE=1999 MSE=988 MSE=1057,2 MSE=1050,1 MSE=2783,8 MSE=4583,8
7. 8. 9. 10. 11. 12.
4000 4050 4100 4150 4200
150
200
250
300
350
400
2750 2800 2850 2900 2950
150
200
250
300
350
400 3050 3100 3150 3200 3250
100
200
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400
400 500 600
200
250
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350 50 100 150 200
150
200
250
300
350
400
2350 2400 2450 2500
200
250
300
350
MSE=1439,1 MSE=3093,8 MSE=2219,4 MSE=1969 MSE=1983 MSE=1087,1
Sw
all
ow
ing
Vib
rati
on
s
13. 14. 15. 16. 17. 18.
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500
60
80
100
120
140
160
180
200
220
2500 2600 2700 2800 2900 3000 3100 3200 3300
100
150
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500 1000 1500 2000
50
100
150
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500 1000 1500 2000
50
100
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5700 5800 5900 6000 6100 6200 6300 6400 6500 6600
50
100
150
200
250
5300 5350 5400
150
200
250
300
350
MSE=20 MSE=16 MSE=33.6 MSE=32 MSE=16 MSE=25
The illustrations of several zoomed time-frequency regions
and corresponding MSEs TEMPUS BioEMIS Edition, 2016 90
Telemedicine
TEMPUS BioEMIS Edition, 2016 91
Telemedicine O Humans tend to live longer
O Increased pressure on healthcare systems worldwide to provide higher quality health care
O Telemedicine promotes the use of multimedia services and systems for increasing the availability of care for patients
O Telemedicine provides a way for patients to be examined and treated, while the health care provider and the patient are at different physical locations.
O Telemedicine technologies - services to patients using signals that can be acquired over distances.
O Signal and image transmission, storage and processing are the major components of telemedicine.
TEMPUS BioEMIS Edition, 2016 92
Telemedicine
O Telenurcing
O Telepharmacy
O Telerehabilitation
O Teleradiology
O Telecardiology
O Telesurgery
TEMPUS BioEMIS Edition, 2016 93
Telenursing
O Provide home care to older adults and/or other patient groups, which preferred to stay in the comfort of their own homes.
O Generally, the patients welcome the use of multimedia systems to communicate with a nurse about their physical and psychological conditions.
O An analysis of telenursing in the case of metered dose inhalers in a geriatric population have shown that multimedia systems can provide most of services, and only small percentage require on-site visits.
TEMPUS BioEMIS Edition, 2016 94
Telepharmacy O Assumes providing pharmaceutical care to
patients and medication dispensing from distance.
O The telepharmacy services adhere to all official
regulations and services as traditional pharmacies,
including verification of drugs before dispensing
and patient counseling.
O Telepharmacy services maintain the same services
as the traditional pharmacies and provide
additional value-added features.
O Specifically, it has been shown that the utility of
telepharmacy services for education via video was
superior to education provided via written
instructions on an package insert.
TEMPUS BioEMIS Edition, 2016 95
Telerehabilitation O Telerehabilitation was established in 1997 when the
National Institute on Disability and Rehabilitation Research (U.S. Department of Education) ranked it as one of the top priorities for a newly established Rehabilitation Engineering Research Center (RERC).
O It covers diverse fields of investigations (e.g., intelligent therapeutic robots and other health gadgets)
O The main efforts are made to:
O provide telecommunication techniques to support rehabilitation services at a distance,
O then to provide technology for monitoring and evaluating the rehabilitation progress,
O and finally to provide technology for therapeutic intervention at a distance
TEMPUS BioEMIS Edition, 2016 96
Telerehabilitation
A typical telerehabilitation system
TEMPUS BioEMIS Edition, 2016 97
Patient
Doctor
Special telecardio
device
Telecardiology O Telecardiology involves merging technology with
cardiology in order to provide a patient with a proper medical care.
O Telecardiology is currently a well-developed medical discipline involving many different aspects of cardiology: O acute coronary syndromes,
O congestive heart failure,
O sudden cardiac arrest, arrhythmias).
TEMPUS BioEMIS Edition, 2016 98
Telecardiology
O It becomes an essential tool for cardiologists.
O Patient consultations with cardiologists via multimedia systems are becoming extremely common: a cardiologist receives many signals and images in real time to assess the patient condition
O Telecardiology has the two major aims: O to reduce the healthcare cost.
O to evaluate the efficiency of telecardiac tools (e.g., wireless ECG) at variable distances.
By accomplishing these two aims, telecardiology will enhance the psychological well-being of patients in addition to bridging the gap between rural areas and hospitals.
TEMPUS BioEMIS Edition, 2016 99
Telesurgery O Dissemination of new surgical skills and
techniques across the wide spectrum of practicing
surgeons is often difficult and time consuming,
especially because the practicing surgeons can be
located very far from large teaching centers.
TEMPUS BioEMIS Edition, 2016 100
O It allows:
O dissemination of expertise,
O widespread patient care,
O cost savings, and
O improved community care
O Telesurgery seems to be a powerful method for
performing minimally invasive surgery
O Operators using a telesurgery platform can complete
maneuvers with delays up to 500 ms.
O The emulated surgery in animals can be effectively
executed using either ground or satellite, while
keeping the satellite bandwidth above 5 Mb/s.
Telesurgery
TEMPUS BioEMIS Edition, 2016 101
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Diagnostic Radiology Physics: A Handbook for Teachers and Students
O F. Zhang, G. Bi, and Y. Chen, “Tomography time-frequency
transform,” IEEE Transactions on Signal Processing, Vol. 50, No. 6,
June 2002, pp. 1289-1297
O M. Lustig, et al., "Sparse MRI: The application of compressed sensing
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O Emmanuel J. Candès, Justin Romberg, Member, IEEE, and Terence
Tao, IEEE Transactions on Information Theory, vol. 52, no. 2, Feb.
2006
O S. Stankovic, I. Orovic, LJ. Stankovic, "Single-Iteration Algorithm for
Compressive Sensing Reconstruction," 21st Telecommunications
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Reconstruction Method based on Analysis of Compressive Sensed
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O S. Stankovic, I. Orovic, E. Sejdic, "Multimedia Signals and Systems:
Basic and Advance Algorithms for Signal Processing,"Springer-
Verlag, New York, 2015
O M. Brajovic, I. Orovic, M. Dakovic, S. Stankovic, "Gradient-based signal
reconstruction algorithm in the Hermite transform domain," Electronic
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O S. Stankovic, LJ. Stankovic, I. Orovic, "Compressive sensing approach
in the Hermite transform domain," Mathematical Problems in
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http://dx.doi.org/10.1155/2015/286590 , 2015
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O M. S. Nikjoo, C.M. Steele, E. Sejdić, T. Chau,” Automatic
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O C. M. Steele, E. Sejdic, T. Chau, “Noninvasive Detection of Thin-
Liquid Aspiration Using Dual-Axis Swallowing Accelerometry” DOI
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O I. Orovic, S. Stankovic, T. Chau, C. M. Steele, E. Sejdic, "Time-
frequency analysis and Hermite projection method applied to
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O A. Szczepański, K. Saeed, A Mobile Device System for Early
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O Szczepański, A.; Saeed, K. Real-Time ECG signal feature
extraction for the proposition of abnormal beat detection—
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O
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