Slideshow of the presentation given at the IEEE Workshop on Biometric Measurements and Systems for Security and Medical applications (2014) Compressive Sensing (CS) is a newly introduced signal processing technique that enables to recover sparse signals from fewer samples than the Shannon sampling theorem would typically require. It is based on the assumption that, for a sparse signal, a small collection of linear measurements contains enough information to allow its reconstruction. Combining the acquisition and compression stages, CS is a very promising technique to develop ultra low power wireless bio-signal monitoring systems. In this paper we present a Compressive Sensing framework for ECG signals based on a universal Gaussian over-complete dictionary that permits to successfully increase the reconstruction quality performance. The purpose of the proposed dictionary is to improve ECG signal sparsity in order to achieve a higher compression ratio. Numerical experiments demonstrate that our method achieves improved performance with respect to state-of-the-art CS schemes.

1. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Gaussian Dictionary for Compressive Sensing of
the ECG Signal
Giulia Da Poian, Riccardo Bernardini
and Roberto Rinaldo
University of Udine
2014 IEEE Workshop on
Biometric Measurements and Systems for Security and Medical
Applications
Rome, October 17, 2014
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See also:
http://ieeexplore.ieee.org/document/7305770/
http://www.mdpi.com/1424-8220/17/1/9/htm

2. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Outline
1 Introduction and Motivations
2 Compressive Sensing
3 Compressive Sensing of ECG signal
4 Experimental Results
5 Conclusion
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3. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Introduction
Wireless Body Sensor Nodes
(WBSNs)
Continuous monitoring of
bio-signals
Blood Flow
Respiration
ECG
Three phases
Acquisition
Processing
Wireless Transmission
Challenge
Increase Sensors Lifetime
Ultra long for implants (Up to 5 years for implants)
Long for wearable (Up to 1 week for wearable)
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4. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
WBANs Technical Challenges
Problem: Increase life time of sensors minimizing power
consumption
Solution: Reduction of data to acquire and transmit
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5. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
WBANs Technical Challenges
Problem: Increase life time of sensors minimizing power
consumption
Solution: Reduction of data to acquire and transmit
Old Paradigm
Conventional approaches to sampling signals require to sample
data at Nyquist rate and then compress
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6. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Compressive Sensing Acquisition System
Compressive Sensing
When data is sparse/compressible, one can directly acquire a
condensed representation with no/little information loss through
linear dimensionality reduction
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7. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS: Acquisition
Give a k-sparse signal x of size N, than x can be recovered with
overwhelming probability by sensing it M times, with M << N.
y is the measurements vector of length M
is the (M ⇥ N) measurements matrix (i.e. Random
Gaussian Matrix)
x is the input ECG vector of length N
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8. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS: Recovery of Sparse Vector
Example: sparse vector x 2 R3 with one non-zero coe cient,
x belongs to one of the coordinate axes
measurement vector a1
y1 = a11x1 + a12x2 + a13x3
x must be one of the three
intersections of the plane
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9. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS: Recovery of Sparse Vector
Example: sparse vector x 2 R3 with one non-zero coe cient,
x belongs to one of the coordinate axes
measurement vector a1
y1 = a11x1 + a12x2 + a13x3
x must be one of the three
intersections of the plane
add a measure, a2
y2 = a21x1 + a22x2 + a23x3
x must belong to the line
resulting by the intersections of
the planes
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10. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS: Reconstruction
Goal: recover signal X from measurements Y
Solution: exploit the sparse/compressible geometry of acquired
signal
P0,✏
Find the sparsest solution:
minkxk0 subject to ky xk2  ✏
only M = 2K , NP-hard
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11. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS: Reconstruction
Goal: recover signal X from measurements Y
Solution: exploit the sparse/compressible geometry of acquired
signal
P0,✏
Find the sparsest solution:
minkxk0 subject to ky xk2  ✏
only M = 2K , NP-hard
Convex optimization: BP,
BPDN
Greedy Algorithms: MP,
OMP, CoSaMP ...
P1,✏
Use the convex relaxation l1
minkxk1 subject to ky xk2  ✏
M = O(k log(N
k ))
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12. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Sparsity
A signal x is k-sparse in the acquisition domain if it has at most k
non-zero value:
ksk0 := card(supp(s))  k
Sparsity - Compressibility
Bio-signals are highly sparse or compressible in a transformed
domain (Fourier, wavelets, ...)
The number of measurements required by CS depends on the
sparsity level:
More sparse = Few measurements
M = O(k log(
N
k
))
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13. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
CS and compressible signals
When x has a sparse representations in
Given the measurements vector y and a dictionary solve:
minkxk1 subject to ky xk2  ✏
Compressive Sensing acquisition process does not depend on
sparsification domain
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14. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Prior Works in Compressive Sensing of the ECG signal
Analytical sparsifying transform:
DCT Transform
Wavelet Transform
Use of Compressed Sansing as a compression technique
Dictionary Learning
Pre-processing stage to find the QRS complex
Period normalization (each beat cycle of the same length)
Exploit correlation among leads (require to acquire more data)
Exploit correlation among beats
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15. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Proposed Method
Improve the Compressive Sensing technique exploiting the ECG
sparsity in order to acquire a compressed version of the signal
avoiding any pre-processing
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16. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Proposed Method
Improve the Compressive Sensing technique exploiting the ECG
sparsity in order to acquire a compressed version of the signal
avoiding any pre-processing
Dictionary learning:
dictionary depends on training set
needs pre-processing stage (adding complexity to the encoder)
Proposed dictionary avoids the learning process
Composed using Gaussian-like functions
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17. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Overcomplete Gaussian-Dictionary Design
ECG approximation
Approximation of ECG beats as a linear combinations of k
Gaussian functions:
x(t) =
kX
i=1
si e
⇣
t pi
ai
⌘2
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18. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Overcomplete Gaussian-Dictionary Design
ECG approximation
Approximation of ECG beats as a linear combinations of k
Gaussian functions:
x(t) =
kX
i=1
si e
⇣
t pi
ai
⌘2
Symmetric waves Q,R and S
can be approximated by 1
Gaussian function
Asymmetric waves P, T
require 2 or 3 Gaussian
functions
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19. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Overcomplete Gaussian-Dictionary Design
Dictionary is designed for ECG segments of length 256
Scale parameters used ai 2 {1, 2, 3, 4, 5, 6, 7, 8, 50, 52}
All shift parameters pi within the vector length
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20. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Experimental setup
Experimental database:
MIT-Arrhythmia ECG Database
First five minutes of each signal equally divided into segments
of 256 samples
0 256 512 768 1024
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Samples
Amplitude
Sensing matrix with i.i.d. entries drown from a standard
normal distribution
Dictionary composed by 2816 atoms
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21. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Experimental Setup
Recovery Algorithms
Convex optimization: Basis Pursuit Denoising (BPDN)
Greed Algorithm: Orthogonal Matching Pursuit (OMP)
Performance Comparison
Daubechies Wavelet orthogonal transform (7 decomposition
levels) as sparsifying transform
BSBL-BO algorithm (exploits the intra-block correlation)
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22. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Performance metrics
CR: compression ratio:
CR(%) =
N m
N
⇥ 100
PRD: Percent root mean square di↵erence
PRD(%) =
sPN
n=1(x(n) ˆx(n))2
PN
n=1 x(n)2
⇥ 100
x is the original zero-mean signal
PRD  2% for very good reconstruction
PRD  9% for good reconstruction
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23. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Visual evaluation of reconstructed ECG
ECG database record 221 has been ”acquired” using M=63
measurements, with a compression ratio CR=76%
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−1
−0.5
0
0.5
1
1.5
Amplitude
(a) Original MIT−BIH record 221
Time [s]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−1
−0.5
0
0.5
1
1.5
Amplitude
(b) Reconstructed signal using BP denoising and Gaussain Dictioanry
Time [s]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−1
−0.5
0
0.5
1
1.5
Time [s]
Amplitude
(c) Reconstructed signal using BP denoising and Wavelets
Proposed dictionary PRD=7.2%, Wavelet PRD=29.35%
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24. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Performance Comparison (1/4)
Average PRD over all database records at di↵erent compression
ratios
30 40 50 60 70 76 80 90 100
0
VG
G
10
15
20
25
30
35
Compression ratio (CR)
OutputPRD(averagedoverallrecords)
OMP using Gaussian Dictionary
OMP using Wavelets
BPDN using Gaussian Dictionary
BPDN using Wavelets
Proposed method: PRD 9% for CR⇠ 76%
Wavelet: PRD 9% for CR⇠ 50%
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25. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Performance Comparison (2/4)
Average PRD over all database records at di↵erent compression
ratios
30 40 50 60 70 76 80 90 100
0
VG
G
10
15
20
25
30
35
Compression ratio (CR)
OutputPRD(averagedoverallrecords)
BSBL−BO
OMP using Gaussian Dictionary
BPDN using Gaussian Dictionary
Proposed method: PRD 9% for CR⇠ 76%
BSBL-BO: PRD 9% for CR⇠ 69%
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26. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Performance Comparison (3/4)
Table : ECG compression results
Basis PRD% Time(s) PRD% Time(s)
(BPDN) (BPDN) (OMP) (OMP)
Wavelet
CR=60% 13.61 1.116 14.16 0.012
CR=70% 28.91 0.894 35.76 0.009
CR=80% 56.86 0.556 88.69 0.006
Gaussian
CR=60% 3.43 2.823 6.94 0.054
CR=70% 5.56 2.182 7.57 0.044
CR=80% 11.92 1.410 11.02 0.033
Table : Average results over all records
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27. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Performance Comparison (4/4)
Proposed
method
Wavelets
Bases
30 40 50 60 65 70 75 80 85 90
0
G
20
40
60
80
100
Compression Ratio (CR %)
PRD
30 40 50 60 65 70 75 80 85 90
0
G
20
40
60
80
100
Compression Ratio (CR %)
PRD
The proposed method shows a smaller variation of the PRD
parameter for all the CR values
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28. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Conclusion
CS is a viable solution for data reduction in ECG transmission
Reduction of the number of measurements necessary without
a↵ecting the accuracy of data recovery
The proposed overcomplete dictionary based on Gaussian-like
functions
is independent from the training set
does not require any pre-processing
increases the compression of:
25% respect to CS with Wavelets basis
7% respect to BSBL-BO reconstruction algorithms
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29. Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion
Thank you!
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