## Appendix A Absolute Phase Reconstruction (APR) Algorithm  
The minimization of \(\mathcal{L}_{1}\) (Equation (21)) on \(\varphi_{o}\) and \(\delta_{\lambda}\) is an absolute phase estimation (unwrapping) problem where the joint use of the \(\lambda\)-channels is required. The algorithm is composed from three successive stages \(A\), \(B\), \(C\). We name this algorithm _Absolute phase reconstruction (APR) algorithm_. 1. _Phase synchronization._ Let \(\lambda^{\prime}\in\Lambda\) be a reference channel. Define the estimates for \(\delta_{\lambda}\) in (Equation 21) as follows \[\delta_{\lambda}=\mathcal{W}(\delta_{\lambda^{\prime}}\cdot\mu_{\lambda}/\mu_ {\lambda^{\prime}}+\delta_{\lambda,\lambda^{\prime}}\mu_{\lambda})  
In general case, when \(\delta_{\lambda^{\prime}}\
eq\hat{\delta}_{\lambda^{\prime}}\), the in-phase situation is not achieved. Going back to the criterion (Equation (A3)) we note that with this compensation it takes the form  
\[\mathcal{L}_{1}(\varphi_{o}\,,\,\delta_{\lambda})=\sum_{\lambda}|||u_{o, \lambda}|[\exp(j(\mu_{\lambda}\cdot\varphi_{o}))-\exp(j(\psi_{o,\lambda}- \delta_{\lambda}))]|||_{2}^{2}\] (A5)  
and using (Equation (A4)) it can be rewritten as  
\[\mathcal{L}_{1}(\varphi_{o}\,,\,\delta_{\lambda})=\sum_{\lambda}|||u_{o, \lambda}|[\exp(j(\mu_{\lambda}(\varphi_{o}+\Delta\varphi_{o}))-|u_{o,\lambda}| \exp(j(\psi_{o,\lambda}-\delta_{\lambda}))]|||_{2}^{2},\,\Delta\varphi_{o}=( \delta_{\lambda^{\prime}}-\delta_{\lambda^{\prime}})/\mu_{\lambda^{\prime}}.\]  
Thus, the accurate compensation of the phase shifts \(\delta_{\lambda}\) in the wrapped phases \(\psi_{o,\lambda}\) is achieved while the absolute phase \(\varphi_{o}\) can be estimated within an unknown but invariant phase shift \(\Delta\varphi_{o}\). This is not essential error as in the phase retrieval setup, \(\varphi_{o}\) can be estimated only within an invariant shift. The equivalence of the calculations (Equations (A1) and (A2)) to the phase manipulations shown in Equation (A4) cannot be proved analytically but careful numerical experiments confirm that they are quite precise even for noisy data. 2. _Minimization of_ Equation (A5) on \(\varphi_{o}\) is a solution of the problem \[\phi_{o}(x,\,\delta_{\lambda^{\prime}}) = \arg\min_{\varphi_{o}(x)}\mathcal{L}_{1}(\varphi_{o}\,,\,\hat{ \delta}_{\lambda}),\] (A6) \[\hat{\delta}_{\lambda} = \delta_{\lambda^{\prime}}\cdot\mu_{\lambda}/\mu_{\lambda^{\prime }}+\delta_{\lambda^{\prime}}\mu_{\lambda},\] (A7) where \[\mathcal{L}_{1}(\varphi_{o},\,\hat{\delta}_{\lambda})=2\sum_{\lambda}\sum_{x }|u_{o,\lambda}(x)|^{2}[1-\cos(\mu_{\lambda}\cdot\varphi_{o}(x)-(\psi_{o, \lambda}(x)-\hat{\delta}_{\lambda}))].\] (A8) _Minimization of_ Equation (A8) on \(\varphi_{o}\) gives the estimate of the absolute phase provided given \(\hat{\delta}_{\lambda}\). It has no an analytical solution and is obtained by numerical calculations. The calculations in Equation (A8) are produced pixel-wise for each pixel \(x\) independently. Note that \(\hat{\delta}_{\lambda}\) as well as the solution \(\hat{\varphi}_{o}\) both depend on the unknown invariant \(\hat{\delta}_{\lambda^{\prime}}\). 3. _Minimization of_ Equation (A8) on \(\delta_{\lambda^{\prime}}\) leads to the following scalar numerical optimization: \[\hat{\delta}_{\lambda^{\prime}}=\min_{\delta_{\lambda^{\prime}}}\sum_{\lambda }\sum_{x}|u_{o,\lambda}|^{2}[1-\cos(\mu_{\lambda}\cdot\hat{\varphi}_{o}(x, \delta_{\lambda^{\prime}})-(\psi_{o,\lambda}(x)-(\delta_{\lambda^{\prime}} \cdot\mu_{\lambda}/\mu_{\lambda^{\prime}}+\delta_{\lambda,\lambda^{\prime}}\mu _{\lambda}))].\] (A9) Finalization of estimation. When \(\hat{\delta}_{\lambda^{\prime}}\) is found the optimal values for \(\hat{\varphi}_{o}\) are calculated as \[\hat{\varphi}_{o}(x)=\hat{\varphi}_{o}(x,\hat{* _[PERSON] et al. (2014)_ [PERSON]; [PERSON]; [PERSON] [PERSON] Digital off-axis holographic interferometry with simulated wavefront. _Opt. Express_**2014**, _22_, 28363-28376. [CrossRef] [PubMed]
* _[PERSON] et al.