## 3 Algorithm Development
### Algorithm's Implementation  
Combining the above solutions, the iterative algorithm is developed of the structure shown in Table 1. We use an abbreviation WM-APR for this algorithm as corresponding to Wavelength Multiplexing Absolute Phase Retrieval (WM-APR). The initialization by the complex-valued \(u_{o,\lambda}^{1}\) is obtained from the observations \(\{z_{s,c}\}\) separately for each wavelength by the SPAR algorithm [32] modified for the data provided by CFA. This initialization procedure is identical to the algorithm in Table 1, where Step 4 is omitted. The main iterations start from the forward propagation (Step 1) and follows by the amplitude update and the interpolation for the wavefront pixels where there are no observations (Step 2). The operator \(\Phi_{1}\) for this update is defined by Equations (15)-(18) for Poissonian and Gaussian noise models. The back propagation in Step 3, the operator \(\Phi_{2}\), is defined by Equation (20). The absolute phase reconstruction from the wrapped phases of \(u_{o,\lambda}^{t+1}\) is produced in Step 4 by the \(APR\) algorithm presented in Appendix. It is based on the grid optimization with respect to \(\varphi_{o}\) and the phase shift \(\delta_{\lambda^{\prime}}\) for the reference channel-\(\lambda^{\prime}\). The obtained amplitude and phase update \(u_{o,\lambda}^{t+1}\) at Step 5. The number of iteration is fixed in this implementation of the algorithm. The steps 3 and 4 are complemented by the Block-Matching 3 Dimensions (BM3D) filtering. In Step 3, it is the filtering of complex-valued \(u_{o,\lambda}^{t+1/2}\) produced separately for the wrapped phase and amplitude of \(u_{o,\lambda}^{t+1/2}\). In Step 4, this filtering is applied to the absolute phase \(\phi_{o}^{t+1/2}\). These BM3D filters are derived from the group-wise sparsity priors for the filtered variables. The derivation is based on the Nash equilibrium formulation for the phase retrieval instead of the more conventional constrained optimization with a single criterion function as it is in Equation (14). We do not show here this derivation as it is quite similar to developed in [32]. \begin{table}
\begin{tabular}{l l} \hline \hline  & **Input:**\(\{z_{s,c}\}\), \(s=1,...,S,c\in(R,G,B)\), \\  & **Initialization:**\(u_{o,\lambda}^{1},\lambda\in\Lambda\) \\  & **Main iterations:**\(t=1,2,...,T\) \\ \hline
**1.** & **Forward propagation and mixing** \\  & \(u_{o,\lambda}^{t+1/2}=\mathcal{P}_{s}u_{o,\lambda}^{t},s=1,...,S,\lambda\in\Lambda\); \\  & \(u_{o,\lambda}^{t+1/2}\to u_{o,\lambda}^{t+1/2}\); \\ \hline
**2.** & **Noise suppression**: \\  & \(u_{o,\lambda}^{t}=\Phi_{1}(u_{o,\lambda}^{t+1/2},z_{s,c})\); \\  & \(u_{o,\lambda}^{t}\to u_{o,\lambda}^{t}\); \\ \hline
**3.** & **Backward propagation and filtering**: \\  & \(u_{o,\lambda}^{t+1/2}=\Phi_{2}(u_{o,\lambda}^{t}),u_{o,\lambda}^{t+1}=BM3D(u_{o, \lambda}^{t+1/2})\); \\ \hline
**4.** & **Absolute phase retrieval and filtering**: \\  & \([\varphi_{o}^{t+1/2},b_{o,\lambda}^{t+1/2})=\mathcal{APR}(u_{o,\lambda}^{t+1})\); \\  & \(\varphi_{o}^{t+1}=BM3D(\varphi_{o}^{t+1/2})\); \\ \hline
**5.** & **Object wavefront update**: \\  & \(u_{o,\lambda}^{t+1}=b_{o,\lambda}^{t+1}\exp(iq_{o}^{t+1}\mu_{\lambda}),\lambda\in\Lambda\); \\ \hline
**Output:**\(\varphi_{o}^{t+1},u_{o,\lambda}^{t+1}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Wavelength Multiplexing Absolute Phase Retrieval (WM-APR) Algorithm. Sparsity and Block-Matching 3 Dimensions Filtering  
The sparsity rationale assumes that there is a transform of image or signal such that it can be represented with a small number of transform coefficients or in a bit different terms with a small number of basis functions [58]. This idea is confirmed and supported by a great success of many sparsity based techniques developed for image or signal processing problems. Overall, the efficiency of the sparsity depends highly on selection of the transforms, i.e., basic functions relevant to the problem at hand.