Phase retrieval methods in inline-holography

Abstract

Quantitative phase imaging, i.e. the detection of the phase delay imposed by a biological cell or thin transparent sample on the incoming electromagnetic wave, allows not only to visualize the otherwise hidden structure of a sample under inves- tigation but also, observe and study the dynamics of refractive index and thickness fluctuation. One well-known approach toward phase reconstruction based on images acquired at different planes of focus is to solve the transport of intensity equation (TIE), which, owing to its simple mathematical formulation and straight forward procedure of acquiring the corresponding experimental data, has gained attention at different research communities over the past decades. The TIE is a second order, elliptical, non-separable partial differential equation which relates the intensity variation along the optical axis to a Laplace-like function of the phase and yields a unique solution for the phase, provided the measured intensity at the principle plane is strictly positive and the boundary conditions are well defined. However, boundary conditions are not accessible in general and therefore, the TIE is ill-conditioned and the solution is not unique. In order to get around the problem of non-uniqueness and ill-conditionedness of the TIE, different algorithm are presented throughout my thesis. The first algorithm presented in this research work to solve the TIE is based on the prior knowledge of free area in the image plane. This is realized by applying Dirichlet boundary conditions on the perimeter of a polygon where the phase is constant. The Neumann boundary condition is imposed to the boundary of the padded area. The TIE is solved by the finite element method in which a multigrid solver is employed in order to minimize the computation time. The second approach towards phase retrieval from intensity measurements is called the Gradient Flipping (GF) algorithm, in which the l0 norm of the gradient of the phase. To accomplish this task, an algorithm devised to combine the well-known fast Fourier transform (FFT) solution of the TIE with the constraint projection principle adapted from the charge flipping algorithm. In an iterative manner, the boundary condition is updated in such way that consistency between experimental data and reconstructed phase is assured. The algorithm iterates until convergence is reached. Experimental demonstration shows the superiority of the GF algorithm over the conventional FFT solution to the TIE and a l1 minimization approach. Finally, unlike the TIE which relies on defocused intensity measurements, a new approach based on the astigmatic intensity equation (AIE) which relies on intensity measurements at different angle of a rotating cylindrical lens is presented. The AIE offers an over-determined system of equations . An iterative algorithm based on a FFT-approach to solve the AIE has been developed. Numerical experiments demonstrate its capability to reconstruct the phase of weakly scattering object.