![]() Zuegel, Optical testing using the transport-of-intensity equation. Retsch, Noninterferometric quantitative phase imaging with soft X-rays. Nishchal, “Single-shot phase imaging based on transport of intensity equation,” Opt. Nomura, Phase imaging based on modified transport of intensity equation using liquid crystal variable waveplate and partial coherent illumination. Nugent, Non-interferometric phase imaging with partially-coherent light. De Graef, A new symmetrized solution for phase retrieval using the transport of intensity equation. Chen, “Transport of intensity equation: a tutorial,” Opt. Asundi, Transport of intensity equation: a new approach to phase and light field. #Intensity equation series#Oxley, Phase retrieval from series of images obtained by defocus variation. Nugent, Rapid quantitative phase imaging using the transport of intensity equation. Roddier, Wavefront sensing and the irradiance transport equation. Takeda, Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments. Goodman, Introduction to Fourier Optics, McGraw-Hill (1996). Cowley, Diffraction Physics, Elsevier (1995). Streibl, Phase imaging by the transport equation of intensity. Teague, Deterministic phase retrieval: a Green’s function solution. ![]() As a proof-of-concept, experiments have been carried out with two different samples a micro-lens and USAF chart. A combined image with the high- and low-pass filtered phase images has been reconstructed which provide both higher and lower spatial frequency information. To overcome the difficulty of high and low spatial frequency content in a single phase image, a composite phase image can be reconstructed. On the other hand, a phase image recovered from small defocus intensity data provides higher spatial frequency content at the cost of lower spatial frequencies. A phase image recovered from intensity data captured at large defocus distance provides well-resolved low spatial frequencies at the expense of higher spatial frequencies. Similarly, to recover accurately the lower spatial frequencies, larger separation is required. Therefore, we need smaller image separation to recover accurately the higher spatial frequencies. In TIE-based phase imaging, with the increase in distance between imaging planes, the range of well-resolved spatial frequencies in the retrieved phase reduces. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.The paper demonstrates a composite method of transport-of-intensity equation (TIE)-based phase imaging for broad range of spatial frequency recovery. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. 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. ![]() 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. ![]() 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. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, the boundary conditions are difficult to obtain in practice. The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. ![]()
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