ROBUST RECONSTRUCTION FOR CS-BASED FETAL BEATS DETECTION

UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it
CompressionandBeyond
ROBUSTRECONSTRUCTIONFORCS-BASEDFETAL
BEATSDETECTION
Giulia Da Poian dapoian.giulia@spes.uniud.it
Riccardo Bernardini bernardini@uniud.it
Roberto Rinaldo rinaldo@uniud.it
University of Udine
Polytechnic Department of Engineering and Architecture 
Via delle Scienze 206, Udine, Italy
See also:
http://ieeexplore.ieee.org/document/7305770/
http://www.mdpi.com/1424-8220/17/1/9/htm

ROBUSTRECONSTRUCTIONFORCS-BASEDFETALBEATSDETECTION
• We propose a novel system for the compression
and analysis of Abdominal Fetal Electrocardiogram
using Compressive Sensing (CS) and Independent
Component Analysis (ICA) applied in the
compressed domain, and sparse representations in
a specific dictionary.
• We describe the proposed scheme and a robust
variant of the Smoothed-L0 reconstruction
algorithm. 
• The proposed modification significantly improves
the reconstruction quality, both for synthetic and
real-world ECG signals.
Roberto Rinaldo August 31st, 2016

Outline
• Overview of Sparse Representations
and Compressive Sensing (CS)
• Gaussian Dictionary for ECG
approximation and CS applied to
ECG signal
• Analysis of non invasive Fetal
Electrocardiogram (fECG) - adopted
methodologies
• Reconstruction algorithm
• Results

Sparse Representation
coefficients
basis, frame
Many
(blue)
Sparse representation of an image via a multiscale wavelet transform
Approximation of image
obtained by keeping only the
largest 10% of the wavelet
coefficients.
A signal is sparse if most of its coefficients are (approximately) zero
Signals can often be well-approximated as a linear combination
of just a few elements from a known basis or dictionary

The column vectors of a dictionary are
discrete time elementary signals called dictionary
atoms
Sparse Representation
Taking Advantage of Sparsity for:
audio/image/video
signal detection and classification
blind source separation
• compression
• denoising
• superresolution
To improve sparsity of composite signals, one has to construct a transform matrix
with the best basis
DICTIONARY: collection of elementary waveforms
or atoms or basis functions.
Example of an Overcomplete Dictionary 250
x 4267)

COMPRESSIVE SENSING (CS): signals that are sparse in some
domain, can be fully reconstructed using only few random
measurements.
• Asymmetrical: Most processing at decoder
• Universality: Random measurements can be used for signals sparse in any basis
Compressive Sensing
Systems adopting compressive sensing can:
achieve sub-Nyquist sampling rates
directly acquire compressed
representations of signals
process signals and solve inference
problems in a reduced-dimensionality
domain with small or no penalties
Random Sensing
Matrix
Measurements

CompressiveSensing-Reconstruction
Since M<N there are infinitely many solutions
How to solve the undetermined system of
equation to recover the original signal from the
measurements vector?
Signal reconstruction algorithm aims to find
signal’s sparse coefficient vector
NP- hard
l0 norm minimization:
can reconstruct the signal exactly
with high probability using only
M=k+1 measurements
l1 norm minimization:
can exactly recover k-sparse signals
using M=c * k log(N/K) measurements

GaussianDictionary
Traditional approaches use analytical
sparsifying transform (e.g. DWT, DCT, …) to
sparse represent the ECG signals
Limited Compression Ratios
Limited Reconstruction Quality
Exploit the sparsity of the ECG
signal
Study the morphology of the PQRST
cycle
Scale (Shape)
parameter
Shift parameter
Symmetric waves (Q, R and S) can be
approximated by one Gaussian function
Asymmetric waves (P or T) require 2-3
functions

GaussianDictionary
Dictionary atom
Coefficient
Real ECG reconstruction examples from M=62
measurements (CR=75%)
using a sparse binary sensing matrix
Using Gaussian Dictionary
Reconstruction quality PRD=5.48%
Using Wavelets
Reconstruction quality PRD= 28.19%
the RR measure metric is used, which is calculated from
the differences between matched reference RR and test RRd
intervals. This metric is denoted here by RRmeas (ms)
RRmeas =
v
u
u
t 1
I 1
I 1X
i=1
(RRi RRd
i )2, (18)
where I is the total number of fetal QRS complexes in
the reference. These measures correspond to the Physionet
Challenge scores and were obtained with the same code used
by the Challenge scorer.
Since we are applying a compression technique (CS), recon-
struction quality is evaluated using the PRD metric, defined
as
PRD(%) =
sP
n(x(n) ˆx(n))2
P
n x(n)2
⇥ 100, (19)
where, x(n) and ˆx(n) are the original (after baseline wander
removal and notch filtering) and reconstructed signals, respec-
tively. This value is computed for each reconstructed segment
of every channel and then the average value is calculated.
According to [38], reconstructions with PRD values between
0% and 2% are qualified to have “very good” quality, while
values between 2% and 9% are categorized as “good”.
Finally, we consider the total time required by the algorithm
for beat classification, including reconstruction of all the 4
channels, in order to asses the possibility to implement the
proposed framework in a real-time application. The average
time required by the algorithm, for a 1 minute long signal, is
approximately 3.7 s. The reconstruction program is written in
Matlab, running on an Intel Core i7 processor, equipped with
16 GB memory.
RESUL
FROM
THE S
Record
r01
r04
r07
r08
r10
Conce
Challe
A, i
Challe
record
have a
the di
a min
and a
predic
value
positiv
for the
respec
To
the ab
differe
compr
results
percentage root-mean-square difference

GaussianDictionary
Proposed Gaussian Dictionary
is independent from the training set
dose not require any pre-processing
Increases the compression of:
25% with respect to CS with DWT
7% with respect to BSBL-BO
Bounded-block-Optimized Block Sparse Bayesian Learning (BSBL-BO) 
Orthogonal Matching Pursuit (OMP) 
Basis Pursuit Denoising (BPDN)

Abdominal fetalECG
Heart defects are among the most common birth defects and leading cause of
birth defect-related deaths
Noninvasive FECG monitoring makes use of electrodes placed on the
mother's abdomen
Recorded signals are a mixture of Maternal ECG, Fetal ECG and noise
(Respiration, EMG …)
Fetal QRSMaternal QRS

CSofabdominalfetalECG
Fetal ECG recorded on the abdomen has a low SNR
5-1000 times smaller in intensity than in the adult
Less sparse than adult ECG, for example in the Wavelet domain
Reconstruction does not have to affect the interdependence relation among
the multichannel recordings
Current CS algorithms generally fail in this
application
Gaussian Dictionary can be used to increase
the performance of CS applied to fetal ECG

ICAintheCSdomain
Independent Component Analysis ICA for the separation of mixed signals
(source signals are independent and have non-gaussian distributions)
We propose to perform ICA directly on the
compressed measurements
Compressed
Independent
Components
4 x m
Original mixture signal
(4 channels)
Reconstructed ICs from
ICA applied in CS domain
ICs from ICA applied on
original mixture
Abdominal
Signals
4 x N
Independent
Components
4 x N
Mixing Matrix
4 x 4

BeatsClassification
Information about time and frequency location of maternal and fetal QRS
complex
The first part of the dictionary is for Maternal ECG approximation
The second part of the dictionary for Fetal ECG approximation
Classification is based on the atoms activated during reconstruction
Decomposition of IC signal in the dictionary
part related to maternal approximation
Decomposition of IC signal in the dictionary
part related to fetal approximation

BeatsClassification
Detected Maternal QRS of the first
ICA signal
Detected Fetal QRS of the second ICA
signal
One more example of the proposed detection method: in red fetal beats and in green maternal
beats (on one of the 4 original signals)

Reconstructionalgorithm
• Reconstruction of the independent components is done using a modified version of the
SL0 reconstruction algorithm, introducing regularization for better immunity against
noise (D has D atoms)
random variables [2], sparse binary matrices, where has
only d non-zero randomly selected entries in each column,
have been proposed to reduce the computational cost [8]. In
this case, calculating x takes only O(dN) operations, with
a significant saving when d ⌧ N.
The SL0 algorithm proposed in [6] solves the problem in
Eq. (1) by approximating the l0-norm with a continuous func-
tion, and optimizing the resulting cost function to provide a
smooth measure of sparsity. Indeed, the l0-norm can be ap-
proximated using Gaussian functions, for small values [6],
as in
||s||S,0 , D
DX
i=1
exp( s2
i /2 2
). (2)
Thus, the minimization of the l0-norm is approximately
equivalent to maximize F (s) =
P
i exp( s2
i /2 2
). This
enables to replace the l0-norm minimization with a convex
problem, and maximize F (s) using a steepest ascent algo-
rithm. The parameter controls the trade-off between the
smoothness of the objective function and the accuracy of the
approximation of the l0-norm.
The algorithm proposed in [6] consists of two nested it-
erations, and the external loop is responsible to gradually de-
crease the value. Note that, when is sufficiently large,
exp( s2
/2 2
) ⇡ 1 s2
/2 2
, and the maximization of F (s)
ill-conditioned, th
and results in po
SL0 proposed in [
optimization prob
As in the SL0
using (2), and the
erations. The inte
feasible set {s| k
˜s = s µ k and p
min
ˆs
k ˆs
Using the Lagrang
rewritten as
min
ˆs
where is the reg
ˆs = ˜s
As for the SL0
is equal to the l2
Solving the proble
• SL0: approximate the L0 norm with the smooth function
• Problem (in the noisy case:
min
s
||s||0 s.t. As = y, A = D
F (s) =
DX
i=1
exp( s2
i /2 2
)
• Iterate decreasing ! to approach the L0 norm, and use a gradient based steepest ascend
procedure to maximize
• At each iteration, project back to the feasible set via
s s AT
(AAT
) 1
(As y)
||y As||2  ✏ )

• SL0 requires that matrix AAT is invertible, and this can be problematic with large M
(low compression ratio)
• λSL0: approximate the L0 norm with the SL0 smooth function
• Iterate decreasing ! to approach the L0 norm, and use a gradient based steepest ascend
procedure to maximize
• At each iteration, project back to the feasible set
F (s)
es, where has
in each column,
onal cost [8]. In
operations, with
es the problem in
continuous func-
tion to provide a
norm can be ap-
mall values [6],
2
). (2)
is approximately
s2
i /2 2
). This
n with a convex
pest ascent algo-
-off between the
e accuracy of the
of two nested it-
ill-conditioned, then application of A†
amplifies the error
and results in poor reconstruction, even using the Robust
SL0 proposed in [9]. Introducing a regularization term in the
optimization problem enables a stable recovery of x = Ds.
As in the SL0 algorithm, we approximate the l0-norm by
using (2), and the algorithm again consists in two nested it-
erations. The internal loop seeks the maximum of F in the
feasible set {s| k y As k2 ✏}. At each step we compute
˜s = s µ k and project ˜s by solving
min
ˆs
k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5)
Using the Lagrangian function of Eq. (5), the problem can be
rewritten as
min
ˆs
k Aˆs y k2
2 + k ˆs ˜s k2
2, (6)
where is the regularization parameter. The solution is
ˆs = ˜s AT
(AAT
+ IM ) 1
(A˜s y). (7)
As for the SL0 algorithm, for large values, the solution
ces, where has
s in each column,
ational cost [8]. In
) operations, with
ves the problem in
a continuous func-
ction to provide a
0-norm can be ap-
small values [6],
2 2
). (2)
m is approximately
( s2
i /2 2
). This
ion with a convex
epest ascent algo-
e-off between the
he accuracy of the
s of two nested it-
le to gradually de-
sufficiently large,
imization of F (s)
ill-conditioned, then application of A amplifies the error
min
ˆs
k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5)
rewritten as
min
ˆs
k Aˆs y k2
2 + k ˆs ˜s k2
2, (6)
ˆs = ˜s AT
(AAT
+ IM ) 1
(A˜s y). (7)
is equal to the l2 norm solution subject to k y As k2 ✏.
Solving the problem
2 2
. Let be the updated solution.
s, where has
n each column,
onal cost [8]. In
operations, with
s the problem in
ontinuous func-
ion to provide a
norm can be ap-
all values [6],
2
). (2)
s approximately
s2
i /2 2
). This
n with a convex
est ascent algo-
off between the
accuracy of the
f two nested it-
to gradually de-
ufficiently large,
min
ˆs
k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5)
rewritten as
min
ˆs
k Aˆs y k2
2 + k ˆs ˜s k2
2, (6)
ˆs = ˜s AT
(AAT
+ IM ) 1
(A˜s y). (7)
Solving the problem
• Equivalently, solve
atrices, where has
ries in each column,
utational cost [8]. In
dN) operations, with
solves the problem in
ith a continuous func-
function to provide a
e l0-norm can be ap-
or small values [6],
s2
i /2 2
). (2)
orm is approximately
exp( s2
i /2 2
). This
zation with a convex
steepest ascent algo-
rade-off between the
d the accuracy of the
ists of two nested it-
sible to gradually de-
is sufficiently large,
aximization of F (s)
min
ˆs
k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5)
rewritten as
min
ˆs
k Aˆs y k2
2 + k ˆs ˜s k2
2, (6)
ˆs = ˜s AT
(AAT
+ IM ) 1
(A˜s y). (7)
Solving the problem
2 2

equals the min-
x [3]. Therefore,
ss is usually set
n the feasible set
m, and updating
s µ , where:
(5)
he convex set to
:
). (6)
than BP, while
However, in the
thm needs to be
will propose a
orithms.
s0 = AT
(AAT
+ Im) 1
y; (12)
The proposed algorithm is summarized in 1.
Algorithm 1 -SL0
Input: µ step size, y, A, dec, min, , Kiter
Initialization: s0 AT
((AAT
) + I) 1
y,
1 = 2| max(s0)|
while k < min do
for k=1:Kiter do
s[e
s2
1
2 2
k , . . . , e
s2
K
2 2
k ]T
s s µ
Project s onto the feasible set: {s| k As y k2 ✏}
s s AT
((AAT
) + I) 1
(As y)
end for
k k dec
˜sk s
end while
Output: sOUT ˜sk

Results:simulatedfECG
Fig. 1. Reconstruction SNR versus input SNR obtained from
100 trials for simulated fECG signals, at CR=50%, using the
SL0, SL0 and BPBN (SPGL1) algorithms using the Wavelet
and the Gaussian Dictionary.
CR
0.3 0.4 0.5 0.6 0.7 0.8
AverageReconstructionSNR[dB]
0
10
20
30
40
50
λSL0 - Gaussian Dic.
SL0 - Gaussian Dic.
BPDN - Gaussian Dic.
λSL0 - Wavelet Basis
SL0 - Wavelet Basis
BPDN - Wavelet Basis
Fig. 2. Reconstruction SNR versus CR obtained from 100 tri-
als for simulated fECG signals.
4.1. fECG Reconstruction and Fetal Beats Detection
In [7] a framework for the compression of multichannel ab-
dominal fECG and joint detection of fetal beats has been pro-
posed. The compression of the signal is based on Compres-
sive Sensing and uses a binary sparse sensing matrix, con-
taining only d = 2 ones in random positions in each column,
in order to reduce the sensor complexity [8]. Before recon-
struction using SL0, Independent Component Analysis (ICA)
3. PERFORMANCE OF SL0
section, the effect of noise on the reconstruction per-
nce is experimentally analyzed. We compare the per-
nce of the proposed algorithm with the original SL0
e BPDN-SPGL1 algorithms. The signals used in these
ments are simulated fECG signals [10] with length
256. As sparsifying dictionaries we use a dictionary
ussian like functions [7], and the Wavelet basis with
chies’ length-4 ﬁlters. The sensing matrix elements
awn as independent Gaussian random variables [2]. We
the experiment 100 times with different source signals
erent noise levels, and using each time a different ran-
SNRin [dB]
10 20 30 40 50
AverageReconstructionSNR[dB]
-10
0
10
20
30
40
50
SL0 Gaussian Dic.
λSL0 Gaussian Dic.
SL0 Wavelet Basis
λSL0 Wavelet Basis
BPDN Wavelet Basis
BPDN Gaussian Dic.
Fig. 1. Reconstruction SNR versus input SNR obtained from
Reconstruction SNR versus input
SNR obtained from 100 trials for
simulated fECG signals*, at CR=50%,
using the SL0, λSL0 and BPDN
algorithms using the Wavelet and the
Gaussian Dictionary, N=256.
Reconstruction SNR
versus CR obtained from
100 trials for simulated
fECG signals, CR=50%.
*Behar et al., “An ECG simulator for generating maternal-
foetal activity mixtures on abdominal ecg recordings,”
Physiological measurement, vol. 35, no. 8, p. 1537, 2014.

Results:PhysionetChallengedatasetA*
Fig. 3. Sparse decomposition of the independent component
in (a) using the SL0 algorithm, for (b) CR=75% and (c)
CR=40% and (d) using the SL0 algorithm for CR=40%. In
the graphs, different intensities represent the weight of the ac-
tivated atoms.
Compression Ratio [%]
20 40 60 80 100
AveragesensitivityS[%]
0
20
40
60
80
100
SL0
λ-SL0
(a)
Compression Ratio [%]
20 40 60 80 100
AveragePRD[%]
0
10
20
30 SL0
λ-SL0
(b)
Fig. 4. (a) Detection performance for SL0 and SL0 algo-
rithm. The vertical coordinate gives the average Sensitivity
for dataset A at different CR values. (b) Comparison of aver-
age PRD when using the two algorithms at different CRs.
As an example, we show in Fig. 3 (a) a portion of the IC of
the expe
matrix (
badly an
value is
that the
almost i
at lower
In Fig. 4
outperfo
struction
4.2. Infl
In this s
tion pro
ent sens
variable
not theo
non-zero
imentall
(a) Detection performance for SL0 and λSL0 algorithm.The
vertical coordinate gives the average Sensitivity for dataset
A at different CR values. (b) Comparison of average PRD
when using the two algorithms at different CRs.
e, a second
pending on
recordings.
ks of length
, with only
position is
particularly
an efficient
we consider
M = 62
o CR=75%.
7 additions.
sed on the
hat it is the
quality of
, in Section
framework
ed with 16
signal ICs
ese can be
xing matrix
work we use
e Smoothed
performance figures usually applied for the assessment of QRS
detection algorithms, i.e., sensitivity (S) and positive predic-
tivity (P+). According to the American National Standard [36]
S and P+ are computed as
S =
TP
TP + FN
100, P+ =
TP
TP + FP
100, (15)
where TP is the number of true positives, FP of false positives
and FN of false negatives. A detected beat is considered to be
true positive if its time location differs less than 50 ms from
the reference markers (within a window of 100 ms centered
on the reference marker). The algorithm accuracy can be also
evaluated using the F1 measure, proposed in [37],
F1 = 2
S P+
S + P+
100 = 2
TP
2TP + FN + FP
100. (16)
Additionally, we apply the scoring methods proposed in
[4], using two metrics, i.e., fetal heart rate measurement and
RR interval measurement. The first one, denoted here as
HRmeas (bpm2
), is used to assess the ability of the algorithm
to provide valid fHR estimation. It is based on the squared
difference between matched reference (fHR) and detected
fHRd
measurements every 5 s (12 instances for 1 min long
signals)
HRmeas =
1
12
12X
(fHRi fHRd
i )2
. (17)
We repeat the experiment 20 times with different random
sparse binary matrix (d = 2), for all the signals in dataset A.
The reported values are the average of these simulations.
*“Physionet challenge 2013,” http: www.physionet.org/challenge/2013/.

Results:sensingmatrix
Table 1. Average performance of detection and reconstruction
for SL0 and SL0 for dataset A.
SL0 SL0
CR S PRD S PRD
% [%] [%] [%] [%]
40 Sparse 2 46 5.27 85 3.77
Gaussian 45 5.58 84 3.72
50 Sparse 2 77 5.93 85 4.49
Gaussian 75 5.71 85 4.27
75 Sparse 2 84 8.81 84 8.14
Gaussian 84 8.80 84 8.02
retical reconstruction performance for i.i.d. Gaussian sensing
matrices is well established, we can see experimentally that,
for the class of signals we are considering, sparse matrices
have similar performance. The use of a sparse sensing matrix
with d = 2 allows to achieve almost identical reconstruc-
tion results, besides the very low complexity implementation.
Finally, Table 1 summarizes the average reconstruction and
can efﬁcie
beats in th
ratios. Mo
ing matric
compares
permitting
[1] D. L
ory,
1306
[2] E. J
ceed
cian
[3] D. C
sign
IEEE
[4] G. D
Average performance of detection and
reconstruction for SL0 and λSL0 for dataset A.

Conclusions
The proposed method has been tested on public datasets (set A and set B of
the Physionet Challenge, Silesia dataset*), showing promising results for both
reconstruction quality and detection/classification performance
The use of the proposed λSL0 reconstruction algorithm is crucial for
consistent performance at all compression ratios
Experiments show an average reconstruction time for the λSL0 algorithm
ranging from 0.07 s, when CR=30%, to 0.01 s, when CR=80%. Thus, it
maintains approximately the same computational cost of the original SL0
algorithm (ranging from 0.03 s to 0.01 s), while being much faster than the
BPDN algorithm (1.6 s to 0.6 s). Programs are written in Matlab, running on an
Intel Core i7 processor, equipped with 16 GB memory.
The proposed framework has good performance and is suitable for real-time
implementation with low-power sensors and low complexity devices.
*A. L. Goldberger, L. A. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E.
Stanley, “Physiobank, physiotoolkit, and physionet components of a new research resource for complex physiologic signals,” Circulation, vol.
101, no. 23, pp. e215–e220, 2000.

ROBUST RECONSTRUCTION FOR CS-BASED FETAL BEATS DETECTION

Recomendados

Recomendados

Mais conteúdo relacionado

Semelhante a ROBUST RECONSTRUCTION FOR CS-BASED FETAL BEATS DETECTION

Semelhante a ROBUST RECONSTRUCTION FOR CS-BASED FETAL BEATS DETECTION (20)

Mais de Riccardo Bernardini

Mais de Riccardo Bernardini (11)

Último

Último (20)

ROBUST RECONSTRUCTION FOR CS-BASED FETAL BEATS DETECTION