Ultrasound Radiation Force for the Assessment of Bone Fracture Healing in Children: An In Vivo Pilot Study

. 2019 Feb 24;19(4):955. doi: 10.3390/s19040955

Algorithm A1. Steps of optimization for variational mode decomposition.

\begin{array}{l} Initilize {{\hat{u}}_{p}^{1}}, {ω_{p}^{1}}, {\hat{λ}}^{1}, n \leftarrow 0 \\ repeat \end{array}

\begin{array}{l} n \leftarrow n + 1 \\ for p = 1 : P do \\ Update {\hat{u}}_{p} for all ω \geq 0 : \end{array}

{\hat{u}}_{p}^{n + 1} (ω) \leftarrow \frac{\hat{x} (ω) - \sum_{i < p} {\hat{u}}_{i}^{n + 1} (ω) - \sum_{i > p} {\hat{u}}_{i}^{n} (ω) + \frac{{\hat{λ}}^{n} (ω)}{2}}{1 + 2 α {(ω - ω_{p}^{n})}^{2}}

\begin{array}{l} Update ω_{p} : \\ ω_{p}^{n + 1} \leftarrow \frac{\int_{0}^{\infty} ω {| {\hat{u}}_{p}^{n + 1} (ω) |}^{2} d ω}{\int_{0}^{\infty} {| {\hat{u}}_{p}^{n + 1} (ω) |}^{2} d ω} \end{array}

\begin{array}{l} end for \\ Dual ascent for all ω \geq 0 : \\ {\hat{λ}}^{n + 1} (ω) \leftarrow {\hat{λ}}^{n} (ω) + τ (\hat{x} (ω) - \sum_{p} {\hat{u}}_{p}^{n + 1} (ω)) \end{array}

until convergence : \sum_{p = 1}^{P} {{‖ {\hat{u}}_{p}^{n + 1} - {\hat{u}}_{p}^{n} ‖}_{2}^{2} / ‖ {\hat{u}}_{p}^{} ‖}_{2}^{2} < ε