## 4 Numerical Experiments
### Reconstruction Results
Figures 5 and 6 illustrate the absolute phase reconstruction (3D/2D images) obtained in two different ways: by the proposed WM-APR algorithm and by the SPAR algorithm reconstructing the wrapped phases for the separate wavelength channels following by the phase unwrapping using the PUMA algorithm [42]. The conditions of these experiments are: \(SNR=6.4\) dB, \(N_{photon}=1.78\) for both test-objects. The multiwavelength WM-APR algorithm demonstrates a clear strong advantage in reconstruction of the absolute phases while the processing based on separate wavelength channels completely failed. The accuracy of the WM-APR reconstructions is very high despite of the very noisy observations. The wrapped phase for the true truncated Gaussian phase should have a form of circular fringes for the parts of the image corresponding to the Gaussian peak. Instead of it in Figure 2, we can see quite irregular structures which look like destroyed by random errors or moire patterns. This disturbance of the ideal circular fringes is an aliasing effect due to discretization (sampling) of the continuous Gaussian phase with a low sampling rate, which does not satisfy the Nyquist-Shannon conditions. It happens because the peak value of the truncated Gaussian phase is very high and the number of samples (100 \(\times\) 100) is comparatively small. This aliasing disturbance of the Gaussian phase is so strong that even advanced 2D unwrapping algorithms, in particular, the PUMA algorithm, failed to reconstruct the absolute truncated Gaussian phase from the precisely noiseless sampled data shown in Figure 2. The impressive high-accuracy performance of the proposed WM-APR algorithm in this extra difficult situation is amazing as demonstrating a strong robustness of this algorithm with respect to errors caused by aliasing effects. The reconstruction of the absolute phase for the binary logo TUT is also very difficult, in particular, due to huge differences of the phase values for adjacent pixels in the areas of the discontinuity of the phase. The sampling (pixelation) of this absolute phase also leads to the strong aliasing which can be noticed in 2D image in Figure 2. Again the unwrapping of the true noiseless wrapped phase by the 2D unwrapping algorithms is not possible, and then, the fail of the single wavelength based algorithms in Figure 6 is not surprising. Again, the amazing success of the WM-APR algorithm demonstrates that it is robust with respect to the sampling effects and is able to achieve the high-accuracy reconstruction in very difficult noisy scenarios. In conclusion, we wish to note that the pair-wise beat-frequency synthetic wavelengths, one of the conventional methods for phase retrieval from multiwavelength observations (e.g., [43; 46]), do not work in the considered scenario as the differences between the RGB wavelengths are not small. Figure 4: \(RRMSEs\) as functions of \(SNR\) and the number of experiments \(S\), 300 iterations, left and right 3D images are for Logo TUT and truncated Gaussian, respectively. Figure 5: Truncated Gaussian absolute phase reconstructions: \(RRMSE=0.056\) (\(SNR=6.3\) dB, \(N_{photon}=1.42\)), top row - 3D absolute phase surfaces, bottom row - the corresponding 2D absolute phase images. From left to right: WM-APR algorithm, and phase reconstructions obtained separately for \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\) wavelengths, respectively, followed by the 2D phase unwrapping. Figure 3: \(RRMSE\) and \(SNR\) as functions of the parameter \(\chi\) of the Poissonian distribution: solid triangles and dashed diamonds curves (blue in color images) show \(RRMSE\) for the truncated Gaussian and logo TUT objects, respectively (left y-axis); solid circle curve (orange) shows \(SNR\) of the observations (right y-axis). For simulations we use MATLAB R2016b on a computer with 32 Gb of RAM and CPU with 3.40 GHz Intel(r) CoreTM i7-3770 processor, 108.8 GFLOPS. The computation complexity of the algorithm is characterized by the time required for processing.