Page 122 - AIH-2-3

Artificial Intelligence in Health Opportunities for AI-based arrhythmia screening

and the respective duration between sequential R peaks. is introduced, and samples are taken with a time constant
The heart rate is subsequently calculated by counting the θ , as defined in Equation V. This time constant is
χi
number of R peaks over a duration of 60 s. determined by searching for local maxima using the
Under the compressive sensing method described matched filter approach. 35,38,39
above, the one-pulse ECG signal, U , is transformed The QRS complex can now be expressed in the
r
into a J-dimensional vector: compressed domain using Equation X. The solution to the
signal data stream measurements, as defined by:
U r U (VIII)  
w v (X)
While incorporating several random measurements
acquired during the recording, represented by: Can be derived using direct estimators. The associated
   correlation R xψ(n) can be assessed by using the direct
˘
y U (IX) estimator (  Uϕ ) through matrix multiplication, where
   Ωφ is decomposed in terms of its rows, using:
n
Where y falls in the domain ∈ y R , while τ identifies

the compressed noise and random signal influences based w, n (XI)
n
on the measurement and skin preparation techniques. The
solution involves calculating the position θ of the R-wave Hence, the correlation can be obtained:
χi
peak in the acquired signal using the information ˘


embedded in y for the short compressive sensing of the  U 1 ,, ,, n 1 , (XII)
3
2
n
signal. The scalar products, or dot products, in Equations XI
Based on the conditions of operating with only and XII, involve vectors of length n, illustrating the degree
compressive measurements, the data do not allow for pre- of alignment between the respective vectors. This does not
processing, such as removing other signal components directly require the reconstruction of the measured signal.
(e.g., the P and T waves in the ECG) or artifacts like baseline Alternatively, the use of the orthogonalized estimator
˘
drift. Furthermore, it is not a common practice to perform (  xnϕ ) is often more relevant, as it is calculated by the
pre-processing within the compressive sensing sensor. averaged matrix product, denoted by the chevrons in the
Mathematical operations are designed to account for all following:
deviations, allowing them to be processed appropriately.
1
˘
The matched filter approach will effectively determine xn J . w T n (XIII)
the relevant depolarization peak. With an appropriate I
choice of signal template, the signal-to-noise ratio can be The baseline drift can be compensated for by subtracting
significantly enhanced without pre-processing. However, the signal mean over time from each signal block before
the variability in biological data poses a significant risk of applying the direct estimator in Equation XIII. The signal
false positives or false negatives when applying a healthy mean can be estimated using an appropriately chosen
or pathological template to compressive sensing signal symmetric multiplication vector, defined by the transposed
analysis. unified vector array:
In the QRS complex, the matched filter approach is t
used in compressive sensing to locate and determine its L 1111 1 (XIV)
,,,,

magnitude, based on an appropriately defined filter. This m JJJJ J
is done by performing compressive matched filtering on
a relatively small number of random frequency-domain Subsequently yielding the vector average resulting from
samples. In this approach, the data stream measurements the vector product:
are considered projections under the application of
a random sensing matrix. The complete signal can U r U , m (XV)
r
subsequently be reconstructed using the results obtained ˘
from the compressive sensing approach. This orthogonal estimator (ζ U r ) can be calculated by
multiplying the data acquisition sensing matrix (Ω) with
To determine the correlation [R (τ)] between the its transposed (Ω ), as expressed by:
T
xφ
compressed template and the compressed cardiac muscle 
1
˘
depolarization pattern [expressed as U ], white noise U r . w T  m (XVI)
r
Volume 2 Issue 3 (2025) 116 doi: 10.36922/aih.8468