In the CS context, instead of a full-length signal s(n), we are dealing with a random set of M measurements, where M < < N. The measurement process can be defined as:
$$ \mathbf{y}=\boldsymbol{\Phi} \mathbf{s}=\left[\begin{array}{c}{\boldsymbol{\Phi}}_1{\mathbf{s}}_1\\ {}{\boldsymbol{\Phi}}_2{\mathbf{s}}_2\\ {}\dots \\ {}{\boldsymbol{\Phi}}_P{\mathbf{s}}_P\end{array}\right], $$
(9)
where y denotes measurement vector, while s1, s2,…, sP are different parts of a signal vector. Having in mind that sp is sparse when represented in\( {\boldsymbol{\Psi}}^{\boxed{p}} \), for p=1,…, P, then we can write \( {\mathbf{s}}_p={\boldsymbol{\Psi}}^{\boxed{p}}{\mathbf{x}}^{\boxed{p}} \). Consequently, (9) becomes:
$$ \mathbf{y}=\boldsymbol{\Phi} \left[\begin{array}{c}{\boldsymbol{\Psi}}^{\boxed{1}}{\mathbf{x}}^{\boxed{1}}\\ {}{\boldsymbol{\Psi}}^{\boxed{2}}{\mathbf{x}}^{\boxed{2}}\\ {}\dots \\ {}{\boldsymbol{\Psi}}^{\boxed{P}}{\mathbf{x}}^{\boxed{P}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{A}}^{\boxed{1}}{\mathbf{x}}^{\boxed{1}}\\ {}{\mathbf{A}}^{\boxed{2}}{\mathbf{x}}^{\boxed{2}}\\ {}\dots \\ {}{\mathbf{A}}^{\boxed{P}}{\mathbf{x}}^{\boxed{P}}\end{array}\right], $$
(10)
with \( {\boldsymbol{\Psi}}^{\boxed{p}} \) being NP×NP transform matrix and \( {\mathbf{x}}^{\boxed{p}} \)is an NP × 1 vector of transform coefficients and \( {\mathbf{A}}^{\boxed{p}}={\boldsymbol{\Phi} \boldsymbol{\Psi}}^{\boxed{p}} \). Alternatively, we may write it in the matrix form as follows:
$$ {\mathbf{y}}_{M\times 1}={\left[\begin{array}{cccc}{{\mathbf{A}}^{\boxed{1}}}_{M_1\times {N}_1}& \mathbf{0}& \dots & \mathbf{0}\\ {}\mathbf{0}& {{\mathbf{A}}^{\boxed{2}}}_{M_2\times {N}_2}& \dots & \mathbf{0}\\ {}\dots & \dots & \dots & \dots \\ {}\mathbf{0}& \dots & \dots & {{\mathbf{A}}^{\boxed{P}}}_{M_P\times {N}_P}\end{array}\right]}_{M\times N}{\left[\begin{array}{c}{{\mathbf{x}}^{\boxed{1}}}_{N_1\times 1}\\ {}{{\mathbf{x}}^{\boxed{2}}}_{N_2\times 1}\\ {}\dots \\ {}{{\mathbf{x}}^{\boxed{P}}}_{N_P\times 1}\end{array}\right]}_{N\times 1} $$
i.e., y = AcsX. (11)
The combined transform domain representation is formed as:
$$ \mathbf{X}=\left[\;\begin{array}{c}{\mathbf{x}}^{\boxed{1}}\\ {}{\mathbf{x}}^{\boxed{2}}\\ {}\dots \\ {}{\mathbf{x}}^{\boxed{P}}\end{array}\;\right]. $$
(12)
Note that in the case of additive signal model the concatenated CS matrix would be \( {\mathbf{A}}^{CS}=\left[{\mathbf{A}}^{\boxed{1}}\kern0.24em {\mathbf{A}}^{\boxed{2}}\dots {\mathbf{A}}^{\boxed{P}}\right] \), where each of the sub-matrices is of size N×N. The reconstruction problem can now be defined as follows:
$$ {\displaystyle \begin{array}{c}\min {\left\Vert {\mathbf{x}}^{\boxed{1}}\right\Vert}_1\kern0.48em \wedge \kern0.36em \min {\left\Vert {\mathbf{x}}^{\boxed{2}}\right\Vert}_1\dots \kern0.36em \wedge \kern0.36em \min {\left\Vert {\mathbf{x}}^{\boxed{P}}\right\Vert}_1\kern0.36em \\ {}\kern0.24em \mathrm{subject}\ \mathrm{to}\kern0.49em \mathbf{y}={\mathbf{A}}^{CS}\mathbf{X}\end{array}}, $$
(13)
where the symbol ∧ is used to denote conjunction, i.e., logical and operation. This problem can be solved using the OMP algorithm as follows:
The core of the algorithm is a standard OMP algorithm which is used in the context of multi-base approach and reconstruction of different signal parts \( {\mathbf{x}}^{\boxed{1}} \),\( {\mathbf{x}}^{\boxed{2}} \),…,\( {\mathbf{x}}^{\boxed{P}} \) (extracted at line 11). Note that the problem defined by (13) can be solved by applying other existing reconstruction algorithms, such as convex optimization algorithms or thresholding methods. However, it will be shown that in the application with the ECG signal reconstruction, the OMP provides more accurate results comparing to other classical approaches. Furthermore, the OMP is more convenient due to its realization simplicity and the execution time when used in engineering applications [27].
In the application with ECG signals, the combination of two basis sets will be considered in the sequel, namely the set of discrete Hermite functions combined with the discrete Fourier basis. It has been proven in [9, 10] that the discrete Hermite functions provide the most compact representation of the most prominent ECG segments, i.e., QRS complexes having transient signal characteristics, because of the high similarity between QRS complexes and Hermite functions. Moreover, for this purpose, the discrete Hermite transform outperforms the DFT, DCT, but also the DWT [9]. The remaining parts of ECG signal are much smoother and locally quasi-periodic in nature as can be seen in Fig. 1, and consequently, the DFT basis appears as the most suitable choice for these signal parts.
Finally, we would like to remark that the proposed multi-base approach combined with the multi-structural signal recovery [13], could be considered as an interesting future topic and extension of this work. Namely, the reconstruction problem can be further extended as a linear combination of L1 and TV-norm minimization [13], to explore the advantages of additional structures (at the cost of higher algorithm complexity). It could be also of particular importance in the sense of generalization for different types of biomedical signals. However, the scalars in the linear combination of different constraints need to be determined optimally.