Modified OMP Algorithm for Exponentially Decaying Signals

. 2014 Dec 24;15(1):234–247. doi: 10.3390/s150100234

Algorithm 2 Lorentzian peak matching pursuit.

Input:
- measurement matrix A ∈ M_K_×2_N₊₁(ℂ)
- measurement vector y ∈ ℂ^K
- accuracy ε > 0
- Lorentzian peaks $L_{l}^{i} \in C^{2 N + 1}$ where j ∈ [0, J], l ∈ [−N, N]
Output:
- f^_LPMP∈ ℂ²^N⁺¹
Initialization:
- f^_LPMP= 0, j = 0, b = 0, i = 0
The main loop:
while ‖y − Af^_LPMP‖₂ ≥ ε do
- i = i + 1
- $l_{i} ≔ \underset{l \in [- N, N]}{arg max} | A * {(y - A {\hat{f}}_{LPMP})}_{l} |$
- $(b_{l_{i}, j,} b_{l_{i - 1}, j_{j - 1}, \dots,} b_{l_{i}, j_{1}}) = \underset{z \in C^{i}}{argmin} {‖ y - A (z_{i} L_{l_{i}}^{j} + z_{i - 1} L_{l_{i - 1}}^{j_{i - 1}} + \dots z_{1} L_{l_{1}}^{j_{1}}) ‖}_{2}$
- $j_{i} = \underset{j \in [0, J]}{argmin} {‖ y - A (b_{l_{i}, j} L_{l_{i}}^{j} + b_{l_{i - 1}, j_{i - 1}} L_{l_{i} - 1}^{j_{i - 1}} + b_{l_{1}, j_{1}} L_{l_{1}}^{j_{1}}) ‖}_{2}$
- ${\hat{f}}_{LPMP} = b_{l_{i}, j_{i}} L_{l_{i}}^{j_{i}} + b_{l_{i - 1}, j_{i - 1}} L_{l_{i - 1}}^{j_{i - 1}} + b_{l_{1}, j_{1}} L_{l_{1}}^{j_{1}}$
end while