IEEE Circuits and Systems Magazine - Q1 2020 - 42

ECG
Monitor

HQ
HC

NVM

Figure 1. A dual mode ECG monitor that locally stores compressed data for medical-grade analysis wile a subset of the
compressed data are transmitted to a gateway for wellness
oriented applications.

on the whole vector x and thus the number of measurements passed to the decoder is a simple but effective
way of administering the rate-distortion trade-off.
As reported in Figure 1, the monitoring device is
equipped with a Non-Volatile-Memory (NVM) that stores
the m HQ measurements needed to reconstruct an HighQuality medical-grade ECG (HQ) and with a transmitter
sending only m LQ 1 m HQ of those measurements to a personal device, such as a smartwatch or a smartphone, whose
functionalities depend on the reconstruction of a Low-Quality ECG (LQ), e.g., sufficient to reliably estimate heart rate.
Actually, the proposed device can switch from LQ to HQ
due to an external trigger either from the patient or from
the heart rate monitoring device in case of critical events
whose scrutiny is needed at medical-grade precision.
Clearly, both storage and transmission take advantage of compression in terms of hardware cost, memory
footprint, computation time and, most important, energy consumption. Adaptation plays a fundamental role in
this. In fact, once target qualities are fixed, the adoption
of classical CS like in [11] compresses with a factor 2.2:1
in the HQ setting and 3.7:1 in the LQ setting. If methods
among these presented in this paper are employed, the
above figures become a compression ratio of 6.0:1 for
the HQ setting and 14.2:1 in the LQ setting.
The illustrated application briefly highlights how the
adaptation can be directly employable in the design of
market-ready systems. This is also an example of the
way in which we expect adaptive CS to play a significant
role e in the incoming future.
As an additional example, it is also worth mentioning
that the U.S. Food and Drug Administration has recently
approved a magnetic resonance imaging scan1 that uses
CS to speed up the images acquisition.
1

online available on: https://www.healthimaging.com/topics/cardiovascular
-imaging/fda-clears-compressed-sensing-mri-acceleration-technology
-siemens

IEEE CIRCUITS AND SYSTEMS MAGAZINE

The rest of the paper is organized as follows. Section II reports the basics of CS and three of the most
useful explanation of its working principle. Section III
is devoted to the review of methods that adapt the encoder to the class of signals to acquire, distinguishing
between methods inspired by mutual coherence arguments and methods driven by average energy considerations. Improvements with respect to non-adaptive
CS provided by encoder adaption are assessed in Section IV. Section V deals with adaptation at the decoder.
Section VI develops the ECG monitoring application
sketched above by applying the best performing methods both at the encoder and at the decoder. Section VIII
is finally devoted to describe adaptation policies that
were devised in CS hardware implementations.
II. Basics of CS
The most general model for CS signal encoding is
y = A (x + o x ) + o y, where o x ! R n and o y ! R m are vectors of disturbances corresponding to the errors implicitly attached to the signal (o x ) and those due to the processing producing the measurements (o y ).
For simplicity's sake, most of the contributions that
tune A neglect o x as it makes the overall disturbance
contribution Ao x + o y dependent on A. We adhere to
such a simplification, neglect such a dependency, and
assume y = Ax + o where o is a vector of white and
Gaussian disturbances o + N (0, v 2 I m), with I m the
m # m identity matrix and v 2 the power of the equivalent noise affecting each measurement. Paired with that,
the prototype decoding algorithm is the so called basis
pursuit denoising (BPDN) that estimates xt = Dpt with
pt = argmin p ! R d p

s.t.

y - Bp

(1)

where i$ ip indicates the p-norm, B = AD and h = max o 2
takes into account the maximum deviation from the
original signal due to disturbances. Assuming that
x = Dpr for a certain pr one aims at pt = pr.
To grasp the core of CS theory, assume first that h = 0,
the noiseless case in which BPDN reduces to Basis Pursuit
(BP). Since m 1 n, then A and B are slanted matrices and,
assuming that they are full rank, finding pr means picking
the right solution among the infinite number of candidates
that satisfy the ill-conditioned system of equations y = Bp.
A. Explaining CS With Restricted Isometries
First, let us try to explain how the previous problem
may admit a solution by assuming that matrix B satisfies the well-known Restricted Isometry Property (RIP)
[13]. Start by noting that an obvious requirement is that
no two l-sparse vectors pl ! pm solve y = Bp j . In fact, if
y = Bpl = Bp m , i.e., if B (pl - pm ) = 0, it is impossible to
tell from y which of the two solutions is the true signal.
FIRST QUARTER 2020

https://www.healthimaging.com/topics/cardiovascular-imaging/fda-clears-compressed-sensing-mri-acceleration-technology-siemens https://www.healthimaging.com/topics/cardiovascular-imaging/fda-clears-compressed-sensing-mri-acceleration-technology-siemens https://www.healthimaging.com/topics/cardiovascular-imaging/fda-clears-compressed-sensing-mri-acceleration-technology-siemens

IEEE Circuits and Systems Magazine - Q1 2020

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