gaussian dictionary for compressive sensing of the ecg signal

Introduction and Motivations Compressive Sensing Compressive Sensing of ECG signal Experimental Results Conclusion

Gaussian Dictionary for Compressive Sensing ofthe ECG Signal

Giulia Da Poian, Riccardo Bernardiniand Roberto RinaldoUniversity of Udine

2014 IEEE Workshop onBiometric Measurements and Systems for Security and Medical

ApplicationsRome, 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


Outline

1 Introduction and Motivations

2 Compressive Sensing

3 Compressive Sensing of ECG signal

4 Experimental Results

5 Conclusion

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Introduction

Wireless Body Sensor Nodes

(WBSNs)

Continuous monitoring ofbio-signals

Blood FlowRespirationECG

Three phases

AcquisitionProcessingWireless 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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WBANs Technical Challenges

Problem: Increase life time of sensors minimizing powerconsumption

Solution: Reduction of data to acquire and transmit

Old Paradigm

Conventional approaches to sampling signals require to sampledata at Nyquist rate and then compress

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Compressive Sensing Acquisition System

Compressive Sensing

When data is sparse/compressible, one can directly acquire acondensed representation with no/little information loss throughlinear dimensionality reduction

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CS: Acquisition

Give a k-sparse signal x of size N, than x can be recovered withoverwhelming 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. RandomGaussian Matrix)x is the input ECG vector of length N

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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

= a11

x1

+ a12

x2

+ a13

x3

x must be one of the threeintersections of the plane

add a measure, a2

y2

= a21

x1

+ a22

x2

+ a23

x3

x must belong to the lineresulting by the intersections of

the planes

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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

= a11

x1

+ a12

x2

+ a13

x3

x must be one of the threeintersections of the plane

add a measure, a2

y2

= a21

x1

+ a22

x2

+ a23

x3

x must belong to the lineresulting by the intersections of

the planes7 / 24


CS: Reconstruction

Goal: recover signal X from measurements YSolution: exploit the sparse/compressible geometry of acquiredsignal

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(Nk ))

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Sparsity

A signal x is k-sparse in the acquisition domain if it has at most knon-zero value:

ksk0

:= card(supp(s)) k

Sparsity - Compressibility

Bio-signals are highly sparse or compressible in a transformeddomain (Fourier, wavelets, ...)

The number of measurements required by CS depends on thesparsity level:

More sparse = Few measurements

M = O(k log(N

k))

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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 onsparsification domain

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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 complexPeriod 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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Proposed Method

Improve the Compressive Sensing technique exploiting the ECGsparsity in order to acquire a compressed version of the signalavoiding any pre-processing

Dictionary learning:

dictionary depends on training setneeds pre-processing stage (adding complexity to the encoder)

Proposed dictionary avoids the learning process

Composed using Gaussian-like functions

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Overcomplete Gaussian-Dictionary Design

ECG approximation

Approximation of ECG beats as a linear combinations of kGaussian functions:

x(t) =kX

i=1

sie

⇣t�piai

⌘2

Symmetric waves Q,R and Scan be approximated by 1Gaussian function

Asymmetric waves P, Trequire 2 or 3 Gaussianfunctions

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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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Experimental setup

Experimental database:

MIT-Arrhythmia ECG Database

First five minutes of each signal equally divided into segmentsof 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

Am

plit

ude

Sensing matrix with i.i.d. entries drown from a standardnormal distribution

Dictionary composed by 2816 atoms

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Experimental Setup

Recovery Algorithms

Convex optimization: Basis Pursuit Denoising (BPDN)Greed Algorithm: Orthogonal Matching Pursuit (OMP)

Performance Comparison

Daubechies Wavelet orthogonal transform (7 decompositionlevels) as sparsifying transformBSBL-BO algorithm (exploits the intra-block correlation)

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Performance metrics

CR: compression ratio:

CR(%) =N �m

N⇥ 100

PRD: Percent root mean square di↵erence

PRD(%) =

sPNn=1

(x(n)� x̂(n))2PN

n=1

x(n)2⇥ 100

x is the original zero-mean signal

PRD 2% for very good reconstructionPRD 9% for good reconstruction

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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

Am

plit

ude

(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

Am

plit

ude

(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]

Am

plit

ude

(c) Reconstructed signal using BP denoising and Wavelets

Proposed dictionary PRD=7.2%, Wavelet PRD=29.35%

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Performance Comparison (1/4)

Average PRD over all database records at di↵erent compressionratios

30 40 50 60 70 76 80 90 1000

VG

G10

15

20

25

30

35

Compression ratio (CR)

Ou

tpu

t P

RD

(av

erag

ed o

ver

all

rec

ord

s)

OMP using Gaussian DictionaryOMP using WaveletsBPDN using Gaussian DictionaryBPDN using Wavelets

Proposed method: PRD 9% for CR⇠ 76%Wavelet: PRD 9% for CR⇠ 50%

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Average PRD over all database records at di↵erent compressionratios

30 40 50 60 70 76 80 90 1000

VG

G10

15

20

25

30

35

Compression ratio (CR)

Ou

tpu

t P

RD

(av

erag

ed o

ver

all

rec

ord

s)

BSBL−BOOMP using Gaussian DictionaryBPDN using Gaussian Dictionary

Proposed method: PRD 9% for CR⇠ 76%BSBL-BO: PRD 9% for CR⇠ 69%

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Table : ECG compression results

Basis PRD% Time(s) PRD% Time(s)(BPDN) (BPDN) (OMP) (OMP)

Wavelet

CR=60% 13.61 1.116 14.16 0.012CR=70% 28.91 0.894 35.76 0.009CR=80% 56.86 0.556 88.69 0.006

Gaussian

CR=60% 3.43 2.823 6.94 0.054CR=70% 5.56 2.182 7.57 0.044CR=80% 11.92 1.410 11.02 0.033

Table : Average results over all records

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Proposedmethod

WaveletsBases

30 40 50 60 65 70 75 80 85 90

0

G

20

40

60

80

100

Compression Ratio (CR %)

PR

D

30 40 50 60 65 70 75 80 85 90

0

G

20

40

60

80

100

Compression Ratio (CR %)

PR

D

The proposed method shows a smaller variation of the PRDparameter for all the CR values

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Conclusion

CS is a viable solution for data reduction in ECG transmission

Reduction of the number of measurements necessary withouta↵ecting the accuracy of data recovery

The proposed overcomplete dictionary based on Gaussian-likefunctions

is independent from the training setdoes not require any pre-processingincreases the compression of:

25% respect to CS with Wavelets basis7% respect to BSBL-BO reconstruction algorithms

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Thank you!

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gaussian dictionary for compressive sensing of the ecg signal

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