Phase retrieval in lens-based Fresnel wave propagation model

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.

Computational approaches in 3D quantitative phase Imaging: Current and future trends in cellular imaging.

When cells are observed with a microscope, the phase information for the sample is usually obtained from the acquired intensity images using the transport of intensity equation (TIE). However, to solve the TIE, it is necessary to obtain an in-focus image. The accuracy and speed of focus positioning affect the accuracy and time of phase retrieval. We propose a fast phase retrieval method based on the TIE and the threshold duty ratio. We first perform pixel reduction on a series of acquired intensity images, and these images are segmented using the optimal threshold to calculate the threshold duty ratios. The original image corresponding to the minimum threshold duty ratio is the optimal focus image. Finally, the optimal in-focus image and related defocused images are used to solve for the phase of the sample using the TIE under the boundary conditions. The correlation coefficient for the results of the simulation experiment reached 0.9317. In a microlens array experiment, the relative error between the results from our algorithm and the actual value was only 7.1%, thus, proving the correctness and effectiveness of the proposed method.

Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.

The transport of intensity equation (TIE) has been widely applied to phase imaging. A variety of methods for solving the TIE have been proposed. One of the most popular methods is an FFT-based method, which is simple and fast. However, this method has a strict restriction that the intensity of the object is assumed to be uniform. Otherwise, the accuracy of phase retrieval results may drop significantly. The transport of phase equation (TPE) is an equation coupled with the TIE, and both are derived from the Helmholtz equation. Few works have studied the role of the TPE in 3D imaging. In this work, a non-iterative FFT-based TIE with TPE correction is proposed. The phases at the object plane and four defocused planes are first calculated by the TIE. Ideally, the object intensity and computed phases are supposed to satisfy both the TIE and TPE. But the non-uniformity of object intensity, as well as the use of finite differences of intensities in the calculation of longitudinal derivatives for the FFT-based TIE method, introduce errors in phase retrieval result. We show that by using the TPE, the local refractive index during propagation can be updated and used as a correction in the TIE. The TIE is solved once again using the updated refractive index, and shows reduction of errors. This proposed technique can be extended to amplitude and phase imaging. It also offers the advantage of yielding the unwrapped phase with good accuracy performance and can be potentially applied to medical high-definition 3D imaging, for cells, micro bubbles, etc.

The problem of phase retrieval from intensity measurements plays an important role in many fields of physical research, e.g. optics, electron and x-ray microscopy, crystallography, diffraction tomography and others. in practice the recorded images contain information only on the intensity distribution I(x,y) = ψ*ψ*= |A|2 of the imaging wave function ψ = A*exp(-iϕ) and the phase information (ϕ(x,y) is usually lost. in general, the phase problem can be solved either by special holographic/interferometric methods, or by noninterferometric approaches based on intensity measurements in far Fraunhofer zone or in the Fresnel zone at two adjacent planes orthogonal to the optical axis. The latter approach uses the transport-of-intensity equation (TIE) formalism, introduced originally by Teague [1] and developed later in [2]. Applications of TIE to nonmagnetic materials and magnetic inductance mapping were successfully made in [3,4]. However, this approach still needs further improvement both in mathematics and in practical solutions, since the result is very sensitive to many experimental parameters.

Single beam phase retrieval techniques [1-8] allow the reconstruction of the object phase information of a general volume speckle field for single beam devices such as optical microscopes. These techniques are well-suited for simple optical setups and recover the phase using the intensity recorded at a single or multiple planes. Deterministic Phase Retrieval techniques, as e.g. approaches based on the Transport of Intensity Equation [4-6] or the Contrast Transfer Function [7,8] allow recovering directly the phase of a wavefront, but are limited either to the paraxial case or impose restrictions on the phase to be estimated. Available Iterative Phase Retrieval (IPR) techniques as the Single-Beam-Multiple-Intensity-Reconstruction (SBMIR) algorithm [1-3] overcome this problem and are applicable to the non-paraxial case without imposing restrictions on the object to be investigated. However, this family of IPR techniques are subject to varying levels of performance or stagnate after few numbers of iterations. The stagnation is a result of the slow elimination of the remaining low frequency error components usually in form of a phase tilt, a bell-shaped error, or other low frequency artefacts. This problem results in a poor reconstruction for the case of slow-varying objects or the measurement of aberrations.

Phase retrieval imaging is widely significant in the scientific, optical engineering, and medical fields. The transport of intensity equation (TIE) has been extensively applied to phase retrieval imaging. In this paper, we investigate a technique for phase retrieval based on the TIE and TPE equations employing electro-optic materials such as liquid crystals to create optical path length difference through application of a bias voltage across the cell.

A computational phase retrieval method is presented that utilizes the transport of intensity equation (TIE) and its solution as a physics-based prior in an iterative reconstruction process. TIE is a non-interferometric quantitative phase imaging (QPI) method that is experimentally simple to implement, needing a minimum of two through focus intensity measurements. However, the accuracy of TIE is severely affected by certain theoretical and practical limitations such as the requirement of uniform intensity, prior knowledge of boundary conditions. Moreover, the finite-difference approximation for intensity derivatives produces blurring for large defocus distances. We propose an iterative phase recovery algorithm that updates the required phase by minimizing the difference of axial intensity derivatives obtained using experiments and expected from angular spectrum propagation theory while integrating a TIE phase-compensation solution in the iterative steps for a fast and accurate phase recovery. We show the validity of the proposed algorithm with simulations for different cases of non-uniform intensity, noise, boundary conditions, and defocus distance. We experimentally verify the working of the proposed method for large defocus distances and compare the results with standard TIE solutions and algorithms to verify its robustness.

Phase retrieval from series of images obtained by defocus variation

Quantitative phase imaging (QPI) provides enhanced contrast for weakly absorbing specimens such as biological tissues under optical light and soft materials under X-ray. In this work, we develop a model-based phase retrieval framework by integrating the physics principles of phase imaging with the deep learning-based approach. Both measurements and the forward model are used as the inputs for a model-based neural network. The features of the object and the regularization weight of the established priors are learned by minimizing the difference between the network output to the ground truth during the training process. This method is tested on phase imaging of handwriting digital patterns and biological cells in a simulation of propagation-based TIE (transport of intensity equation) phase retrieval. We achieve enhanced accuracy for the phase retrieval compared to non-model based end-to-end neural networks and reduce the computation cost compared to traditional model-based iterative reconstruction algorithms.

The non-interferometric phase retrieval technique based on the transport-of-intensity equation (TIE) is employed to calculate the missing phase information from the intensity measurement. However, since the limiting effect of the lens aperture is existed in the actual imaging system, a certain effect on the resolution of the recovery phase is under consideration. A phase retrieval technology based on multi-angle tilt light modulation is proposed. Firstly, a spatial light modulator (SLM) is introduced in the original lens-based intensity acquisition system. Then the modulation function is displayed on the SLM to make the parallel illumination into tilted illumination with various incident angles. Lastly, the recorded intensity images are synthesized to compute the phase information. The experimental results show that the proposed method can improve the resolution of recovery phase without compromising the numerical error of the recovery results.

An atomic resolution phase map, which enables us to observe charge distribution or magnetic properties at an atomic scale, has been pointed out to be retrieved by transport of intensity equation (TIE) when taking two atomic-resolved transmission electron microscope (TEM) images of small defocus difference. In this work, we firstly obtained the atomic-resolved phase maps of an exfoliated molybdenum disulfide sheet using spherical aberration-corrected transmission electron microscope. We successfully observed 60° grain boundary of mechanically exfoliated monolayer molybdenum disulfide sheet. The relative phase shift of a single molybdenum atomic column to the column consisting of two sulfur atoms was obtained to be about 0.01 rad on average, which was about half lower than the simulated TIE phase map, indicating that the individual atomic sites can be distinguished qualitatively. The appropriate condition for retrieving atomic-resolved TIE phase maps was briefly discussed.

Practical procedure for retrieval of quantitative phase map for two-phase interface using the transport of intensity equation

Gaussian process (GP) regression is a nonparametric regression method that can be used to predict continuous quantities. Here, we show that the same technique can be applied to a class of phase imaging techniques based on measurements of intensity at multiple propagation distances, i.e. the transport of intensity equation (TIE). In this paper, we demonstrate how to apply GP regression to estimate the first intensity derivative along the direction of propagation and incorporate non-uniform propagation distance sampling. The low-frequency artifacts that often occur in phase recovery using traditional methods can be significantly suppressed by the proposed GP TIE method. The method is shown to be stable with moderate amounts of Gaussian noise. We validate the method experimentally by recovering the phase of human cheek cells in a bright field microscope and show better performance as compared to other TIE reconstruction methods.