This document proposes a Bayesian framework to recover physiological signals from ambulatory devices using sparse regularization. It suggests minimizing l2-l2 cost functions on devices for initial processing and l2-TV cost functions on servers for improved accuracy. The approach models signal gradients as sparse and estimates signals iteratively using majorization-minimization. It applies these techniques to recover electrocardiogram signals from ambulatory recordings.

1. Recovering vital
physiological signals from
ambulatory devices
Praveen Pankajakshan and Rangavittal
Narayanan
Samsung Advanced Institute of Technology, India
13 February 2013
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3. Context Methodology Tools
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4. 3
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5. Motivation and challenges
• Ambulatory monitoring: Record
vital signal continuously
‣ Mostly non-invasive or
minimally invasive
‣ Patients asymptotic at hospital
and monitor disease
progression Image Courtesy: Cambridge Consultants
• Challenges:
‣ SNR is low [1]
‣ Available storage, processing
power and battery is low
[1] G. Garner et al., EP2327360A1, Nov. 2010. 4
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6. Bayesian framework
• Data from sensor: y[n], n=0, 1, 2, … N-1 ∈ N0.
• From Bayesian theorem [2], estimate x[n] of the signal y[n] can be
realized from
‣ p(y|x) is the likelihood
‣ p(x) is the knowledge on x[n].
• Likelihood is given by the normal distribution
‣ Assumption: Residual noise is asymptotically Gaussian,
variance σ2.
2
‣ ||•|| 2 is the l2 norm.
[2] J. Idier, 2008.
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7. Sparsity of gradient
[2] J. Idier, 2008.
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8. Sparsity of gradient
[2] J. Idier, 2008.
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9. Sparsity of gradient
l1norm of
the gradient
[2] J. Idier, 2008.
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10. Sparsity of gradient
l1norm of
the gradient
Many coefficients are
[2] J. Idier, 2008.
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small!
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11. Sparsity of gradient
• The estimated signal x[n]
must respect:
‣ Bounded signal and
gradient: x[n]>- ∞ and
x[n]<∞
‣ Distribution: Positive
skewed, long tail with
small values.
• These are satisfied by: l1norm of
‣ λ: trade-off parameter, the gradient
E(x) is: Many coefficients are
[2] J. Idier, 2008.
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small!
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12. Sparsity of gradient
• The estimated signal x[n]
must respect:
‣ Bounded signal and
gradient: x[n]>- ∞ and
x[n]<∞
‣ Distribution: Positive
skewed, long tail with
small values.
• These are satisfied by: l1norm of
‣ λ: trade-off parameter, the gradient
E(x) is: Many coefficients are
[2] J. Idier, 2008.
6
small!
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13. A holistic solution
• x[n] can be realized from y[n] by
• Equivalent convex primal problem:
(1)
‣ R is a NxN Toeplitz matrix, p lies between [1, 2].
• x[n] can be estimated directly or piece-wise from y[n] by
minimizing (1) using convex optimization ([2])
[2] J. Idier, 2008.
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14. Solution conceptualization
In-Phone Server-level
processing processing
l2-l2 minimization l2-TV minimization
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15. Majorization-minimization
• Find a surrogate function
J(x, xk) to E(x) such that
‣ J(x, xk) must be convex
‣ J(x, xk)≥E(x)
‣ At xk, E(xk)=J(xk, xk)
• We choose J(x, xk) [3] as:
• The iterative solution is [3]:
[3] M. Figueiredo et al. 2006
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16. Case: Content selection
ECG signal from ML-II lead [4,5]). 48 hour ambulatory ECG with fs=360Hz, 200mV 11 bit
resolution over 10mV amplitude range.
[4] G. B. Moody and R. G. Mark, 2001. 10
[5] A. L. Goldberger, et al. 2000.
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17. Progress and convergence
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18. Progress and convergence
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19. Progress and convergence
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20. Progress and convergence
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21. Progress and convergence
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22. Progress and convergence
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23. Progress and convergence
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24. Baseline correction
Acquired Data
Estimated Baseline
Corrected signal
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25. Baseline correction
Acquired Data
Estimated Baseline
Corrected signal
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26. Baseline correction
Acquired Data
Estimated Baseline
Corrected signal
40 minutes of data processed in 0.7 seconds with 3.2GHz and 4GB memory!
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27. Peak detection on
recovered signal
A 3 second recording of a z-normalized ECG and peak detection [6]
on the restored signals
[6] J. Pan and W. J. Tompkins, March 1985.
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28. Summary Highlight
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29. Summary
• Scope: Restore vital physiological
signals from ambulatory conditions.
• Processing:
‣ For handheld-devices by minimizing a
l2-l2 cost function.
‣ For accuracy at servers by minimizing
a l2-TV cost function.
•Performance: Outperforms classical
approaches.
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30. Thats all folks!
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