2.3 Modified Variational Mode Decomposition Based Method
2.3.1 Proposed AO peak detection method
2.3.1.1 Proposed modified VMD (MVMD) based SCG filtering
The VMD is an adaptive and non-recursive data decomposition model, which computes the relevant spectral bands and the associated modes [122]. It can decompose a signal into a set of modes in such a way that modes collectively reconstruct the input signal in a least-square sense. Extracted modes are band-limited about central frequencies, which are updated in each iteration. The main constituents of VMD are Wiener filter for noise reduction, Hilbert transform for unilateral band analytic signal, and harmonic mixing for frequency translation into baseband. A real valued multicomponent signal x(t) may be decomposed using VMD into a finite number of modesmk(t), which are mostly centered around a central frequency ωk. The bandwidth of each mode is assessed as follows: (i) construction of an analytic signal for each mode mk, (ii) mixing of each mode with an exponential (e−jωkt) tuned to the corresponding estimated ωk, (iii) performing squaredL2-norm operation on the gradient. The resultant constrained variational problem is following [122]:
mmink,ωk
K
X
k=1
∂
∂t
mk(t) +j 1
πt∗mk(t)
e−jωkt
2 2
s.t.
K
X
k=1
mk(t) =x(t) (2.22)
where ‘∗’ is convolution operator. {mk}:={m1, m2, ..., mK} and{ωk}:={ω1, ω2, ..., ωK}denote the set of modes of the signal x(t) and the corresponding central frequencies, respectively. This constraint optimization problem (2.22) is solved by using Lagrange multipliers and a quadratic penalty term.
The quadratic penalty term is used to stimulate reconstruction fidelity, if additive white Gaussian noise is present. Moreover, Lagrange multipliers are generally used to enforce constraints strictly. The resultant unconstrained problem is as follows:
L(mk, ωk, λ) =α
K
X
k=1
∂
∂t [mk(t) +jmbk(t)]e−jωkt
2 2
+
x(t)−
K
X
k=1
mk(t)
2 2
+
*
λ(t), x(t)−
K
X
k=1
mk(t) +
(2.23)
where, α denotes the balancing parameter of the data fidelity constraint and (·) is Hilbert transformc operator. Then, the alternate direction method of multipliers (ADMM) is used to solve Eq. (2.23).
The algorithm for complete optimization of VMD is summarized in Algorithm 1 (steps 1 to 9), where
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2.3 Modified Variational Mode Decomposition Based Method
Algorithm 1 Pseudocode for proposed MVMD optimization
Initialization: K,α= 10, FH
Input: x(n)
1: Initialize: me1k,ω1k,eλ1,n←0 2: repeat
3: n←n+ 1 .Number of iterations
4: fork= 1 toKdo
5: Updatemek for all frequenciesω≥0:
men+1k ← ex(ω)−P
i<kmen+1i (ω)−P
i>kmeni(ω) +eλn2(ω)
1 + 2α(ω−ωnk)2 (2.24)
6: Updateωk:
ωn+1k ← R∞
0 ω|men+1k (ω)|2dω R∞
0 |men+1k (ω)|2dω (2.25)
7: end for
8: Updateλefor all frequenciesω≥0:
λen+1(ω)←λen(ω) +τ ex(ω)−X
k
men+1k (ω)
!
(2.26) 9: untilconvergence criterion: PK
k=1
kmen+1k −menkk22 kmenkk22 <
10: if anyωk>FH,∀k= 1 :Kthen 11: repeat
12: ∆k=ωk−FH (2.27)
13: η=max(∆k)
P
k|∆k| (2.28)
14: α←α(1 +η) (2.29)
15: Repeat steps 1 to 9 with updatedα 16: untilallωk≤FH,∀k= 1 :K
17: end if
18: if anyDk:= (ωk+1−ωk)<0,∀k= 1 :K−1then 19: repeat
20: η= min(Dk)
P
k|Dk| (2.30)
21: α←α(1− |η|/2) (2.31)
22: Repeat steps 1 to 9 with updatedα 23: untilallωk≤FH,∀k= 1 :K
24: end if
Output: αoptimum←α
f(·) is Fourier transform operator. Minimization of the argument of unconstrained problem (2.23) with respect tomk andωk shows Wiener filter behaviour (Eq. (2.24)) and expression for center of gravity of the respective mode’s power spectral density (Eq. (2.25)), respectively. The mode in the temporal domain is estimated as the real part of the inverse Fourier transform of the respective filtered analytic signal. The VMD process depends on numerous input quantities: the number of the modes (K), the bandwidth constraint (α), the initial central frequency (ω), the tolerance for convergence condition (), and the time step of the dual ascent (τ).
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2. SCG Waveform Delineation - Standalone Approaches
The variational mode decomposition mainly depends on two major parameters: the bandwidth constraint (α) and the number of modes (K). For a specific event detection, the optimal selection of both the parameters is essential. The parameterα controls the bandwidth of a mode, andK controls the distribution of energy. A large value ofα allows small frequency bands in the decomposed modes and provides less spectral separability among the modes. Similarly, a small value ofαyields the modes with large bandwidths, which are highly separable in the spectral domain [122].
The modified VMD (MVMD) automatically update α for a given number of modes. The model is initialized with a small α (say, α=10) and a fixed K value. The model employs VMD recursively until the following two nested criteria are satisfied:
• Criterion 1: Each of the decomposed modes must have central frequency less than a fixed cut off frequency FH. In other cases,αis incremented by (η/100)%, whereηis maximum normalized central frequency difference from FH. Thus, α is updated adaptively with FH. The updation of αis given by Eqs. (2.27)–(2.29) in Algorithm 1. After updatingα, VMD is applied to the signal, and criterion 1 verifies the estimated ωk’s of decomposed modes repeatedly until this criterion is satisfied.
• Criterion 2: The central frequencies of consecutive modes must be in the increasing order.
Otherwise,α is decremented by a small value of (η/200)%, where η is the minimum normalized difference of consecutive central frequencies. Thus, α is being updated with central frequency variations. The updation of α is given by Eqs. (2.30)–(2.31) in Algorithm 1, where Dk = ωk+1 −ωk. Subsequently, the signal is decomposed with an updated version of α, and the estimated ωk’s are cross-checked by criterion 2 recursively. Optimum α is obtained when this criterion is satisfied.
The SCG signal, denoted by x[n] (n = 1,2, ..., N), is decomposed using two stage VMD. The decompositions are done to remove low-frequency (LF) artifacts and to segregate all the essential SCG informative components. The LF artifacts mainly reside in the frequency range of 0 to 1.5 Hz, and the major systole profile information of the SCG lies under 45 Hz frequency [3]. Thus, the parameter FHis set at 1.5 Hz in stage-I for capturing LF artifacts and 45 Hz in stage-II for estimating systole profile components. In the first stage of VMD, x[n] is decomposed in two modes (K = 2) with α = 8 ×106, which is determined from the proposed MVMD model. This process is shown in
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2.3 Modified Variational Mode Decomposition Based Method
0 50 100 150 200 250
I = 100
0 50 100 150 200 250
Frequency (Hz) I = 200
0 50 100 150 200 250
σ-1 = 10
0 50 100 150 200 250
Frequency (Hz) σ
-1 = 50
(a) (b)
I = 100 σ
-1 = 50
Figure 2.18: Effects of GDF parameter variation (I,σ) on magnitude spectrum. This experiment is performed on multi-sinusoid signal having all harmonic components of 2 Hz frequency in the range of 0–200 Hz.
Figure 2.22. It captures baseline drift in the first mode and other low frequency motion artifacts in the second mode as shown in Figure 2.22(b)–(c), which combinedly show the trend of the input signal.
The detrended SCG signal is estimated as:
xd[n] =x[n]−
2
X
φ=1
mIφ[n] (2.32)
where φdenotes the mode index, mIφ[n] represents decomposed modes of the first stage of the VMD process, andxd[n] is the resulting detrended signal.
In the second stage, all information (systolic profile, diastolic profile, and motion artifacts) present in the detrended SCG are resolved into different modes, which are denoted by mIIγ[n] (γ corresponds mode index for second stage VMD) individually. For this purpose, an average value of α is obtained by the proposed MVMD model for four number of modes. To represent the relevant systolic profile information, a minimum of four modes are found optimal. From the decomposition results, as shown in Figure 2.23(a) forα= 800, it is observed that modes containing diagnostic information are spectrally separated. It is desired to select the modes having significant systolic profile information. Before this, the information of the systolic profile is enhanced from each of the modes by a Gaussian derivative filtering technique.