3 Pulse Oximetry for the Measurement of Oxygen Saturation …
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3.11.1 SVD for Motion Artefact Reduction
In this method, K samples of a PPG signal corresponding to few cycles of heart rate
are arranged in matrix form, say matrix X with m rows and n columns. The matrix
thus obtained s subjected to singular value decomposition.
3.11.1.1 Singular Value Decomposition
The matrix manipulation tool of linear algebra called singular value decomposition
(SVD) was proposed in 1870. In SVD, a matrix X with m rows and n columns
containing real-valued data is decomposed into three sub-matrices, U, S and V. Here
it is assumed that m ≥ n and hence rank (p) of X ≤ n. Applying SVD to X we get
[76]:
X = U SV
T
(3.33)
In Eq. (3.33), U is an m x n matrix, S and V are n x n matrices. While S is a
diagonal matrix with (S = diag(s 1 ,…,s n )), U and V are unitary matrices. That is
U
T U = I and V
T V = I, where I is an identity matrix. Here, (s 1 , …, s n ) are called
the singular values (SV). SV represents the positive square roots of the eigen values
of the matrix X
T X. Also s q > 0 for 1 ≤ q ≤ p and S q = 0 for (p + 1) ≤ q ≤ n. The
columns of U are called left singular vectors of X, while the columns of V are called
right singular vectors of X. An important observation is that as the singular values
decay rapidly, with, S 1 ≥ S 2 ≥ … ≥ S n ≥ 0, we can expect that there will be a good
lower rank approximation ( X ) to X by setting the small singular values to zero. The
lower rank approximation X is given by.
X =
j
i=1
U i S i v
T
i
(3.34)
where S i assumed to be zero for i > j; u i and v i are the jth columns of U and V
respectively. The singular values of a given data matrix also contains information
about the noise level in the data, energy and rank of the matrix. This fact is exploited
for signal processing such as data compression, noise reduction and pattern matching
and extraction). This feature can also be exploited to remove motion artefacts from
corrupted PPG signals.
3.11.1.2 Principal Component Extraction Using SVD
Let X be an m by n matrix formed with data (samples) of a perfectly periodic waveform x(k), k = 1,2,…,mn having a period of n samples. If we place data such that
73
3.11.1 SVD for Motion Artefact Reduction
In this method, K samples of a PPG signal corresponding to few cycles of heart rate
are arranged in matrix form, say matrix X with m rows and n columns. The matrix
thus obtained s subjected to singular value decomposition.
3.11.1.1 Singular Value Decomposition
The matrix manipulation tool of linear algebra called singular value decomposition
(SVD) was proposed in 1870. In SVD, a matrix X with m rows and n columns
containing real-valued data is decomposed into three sub-matrices, U, S and V. Here
it is assumed that m ≥ n and hence rank (p) of X ≤ n. Applying SVD to X we get
[76]:
X = U SV
T
(3.33)
In Eq. (3.33), U is an m x n matrix, S and V are n x n matrices. While S is a
diagonal matrix with (S = diag(s 1 ,…,s n )), U and V are unitary matrices. That is
U
T U = I and V
T V = I, where I is an identity matrix. Here, (s 1 , …, s n ) are called
the singular values (SV). SV represents the positive square roots of the eigen values
of the matrix X
T X. Also s q > 0 for 1 ≤ q ≤ p and S q = 0 for (p + 1) ≤ q ≤ n. The
columns of U are called left singular vectors of X, while the columns of V are called
right singular vectors of X. An important observation is that as the singular values
decay rapidly, with, S 1 ≥ S 2 ≥ … ≥ S n ≥ 0, we can expect that there will be a good
lower rank approximation ( X ) to X by setting the small singular values to zero. The
lower rank approximation X is given by.
X =
j
i=1
U i S i v
T
i
(3.34)
where S i assumed to be zero for i > j; u i and v i are the jth columns of U and V
respectively. The singular values of a given data matrix also contains information
about the noise level in the data, energy and rank of the matrix. This fact is exploited
for signal processing such as data compression, noise reduction and pattern matching
and extraction). This feature can also be exploited to remove motion artefacts from
corrupted PPG signals.
3.11.1.2 Principal Component Extraction Using SVD
Let X be an m by n matrix formed with data (samples) of a perfectly periodic waveform x(k), k = 1,2,…,mn having a period of n samples. If we place data such that