Abstract
The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green's function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.
Highlights
Phase is an important component of an optical wavefield providing information of the refractive index, optical thickness, and the topology of the specimen
The commonly known transport of intensity equation (TIE) proposed by Teague [1] is a non-interferometric method for the optical phase retrieval
It is found that the Green's function under Neumann boundary conditions with a rectangular region can be represented as an infinite-series expansion in terms of Fourier cosine harmonics, leading to a fast and efficient numerical implementation of the TIE solution with use of fast discrete cosine transform (DCT)
Summary
Phase is an important component of an optical wavefield providing information of the refractive index, optical thickness, and the topology of the specimen. An alternative FFT-based method proposed by Volkov et al [15] extends the measured intensity by a mirror padding scheme, which provides homogenous Neumann boundary conditions rather than the periodic ones It still does not fundamentally solve the boundary artifacts problem since the boundary conditions are formed purely by a mathematical trick which is generally not physically grounded. We will solve the TIE with experimentally measurable inhomogeneous Neumann boundary conditions, through an efficient numerical algorithm using the fast discrete cosine transform (DCT) This approach addresses the above mentioned difficulties such as obtaining boundary conditions under non-uniform intensity distribution, distinguishing the boundary signal from the interior intensity derivative, and defining the correct domain for the solution, and retains the major advantages of the FFT-based methods – simple and fast for the rectangular domain. It is seen that the TIE links the longitudinal intensity derivative with the slope and curvature of the wavefront that produces the changes in intensity as the wavefront propagates
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