The transport of intensity equation (TIE) is an elliptical, second order partial differential equation which relates the intensity variation along the optical axis to Laplacian-like expression involving of the unknown phase. Due to its simple mathematical formulation and the straight forward procedure to acquire the experimental data, the TIE has attracted enormous attentions from various research communities such as transmission electron microscopy, X-ray microscopy, neutron imaging, etc.. The FFT approach to the TIE due to its deterministic nature and computational speed is widely used. However, periodic boundary condition inherent to the FFT approach results in low frequency artifacts in case of non-periodic objects. Alternatively, one can impose an additional constraint on the solution in order to overcome the problem of only weakly encoded low spatial frequency phase information and unknown boundary conditions. For example, by total variation minimization approach employs gradient based optimization techniques to minimize the convex l₁-norm of the derivative of the phase is minimized, leading a solution which is preferentially flat. However, the TV-minimization approach should only be considered for piece-wise constant objects.
Here, we report on an iterative algorithm namely, the gradient flipping algorithm (GFA) [1,2], which imposes non-convex constraint on phase. The GFA assumes that the wavefront to be recovered is sparse in its gradient basis and therefore, combines the reciprocal space solution of the TIE with the principle of the charge flipping algorithm [3] by flipping the sign of phase gradients below a determined threshold. This leads to the sparsest solution in the gradient domain. In an iterative manner, the boundary conditions are updated in such a way that consistency of the recovered wavefront with the experimental data is assured and at the same time the solution is sparse. The algorithm iterates until the convergence criterion is fulfilled.
Figure 1(a) depicts the variation of the image intensity along the optical axis of images recorded of HeLa cells which was estimated from a focal series comprising 20 images with defocus step of 1 µm and laser illumination at a wavelength of λ = 520 nm. Figures 1(b), 1(c) and 1(d) show the phase reconstructed by a Tikhonov-regularized FFT-based solution of the TIE (q^-2 -> (q²+α)^-1, with q being the reciprocal space coordinate) for α =0.001 µm^-2,α=0.01 µm^-2 and α=0.1 µm^-2 where α is the regularization parameter. The phase retrieved by employing the TVAL3 package [4], (convex TV-minimization) is shown in Fig. 1(e). Finally, Fig. 1(f) presents the phase map reconstructed by the proposed approach. This GFA-reconstruction provides a physically very reasonable solution while that of TV-minimization suffers from missing low frequency information, and the Tikhonov regularized FFT-based reconstruction suffers from low frequency artifacts when ααα is small and is not capable of recovering low frequency information when the regularization parameter is increased. 

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EMC Abstracts - https://emc-proceedings.com/abstract/wave-front-reconstruction-vi-the-transport-of-intensity-equation-introduction-of-non-convex-constraints/