Phase retrieval for objects in rain based on a combination of variational image decomposition and variational mode decomposition in FPP

As far as we know, there is no paper reported to retrieve the phase of an object in rain by the fringe projection profilometry (FPP) method. The fringe projection pattern taken in rain contains much rain noise, which makes it difficult to accurately retrieve the phase of the object. In this paper, we focus on the phase retrieval of the object in rain by the FPP method. We first decompose the original fringe projection pattern into a series of band-limited intrinsic mode functions by the two-dimensional variational mode decomposition (2D-VMD) method. Then we screen out fringe-associated modes adaptively based on mutual information and reconstruct the fringe projection pattern. Next, we decompose the reconstructed fringe projection pattern by the TGV-Hilbert-BM3D variational model to obtain the de-rained fringe component. Finally, we use the Fourier transform method, phase unwrapping method, and carrier-removal method to obtain the unwrapped phase. We test the proposed method on three fringe projection patterns taken in simulated rain weather, and we compare our proposed method with the phase-shifting method, windowed Fourier method, morphological operation-based bidimensional empirical mode decomposition method, 2D-VMD method, and the TGV-Hilbert-BM3D method. The experimental results demonstrate that, for the first time to our knowledge, our method can effectively retrieve the phase of an object in rain from a single fringe projection pattern.