Transport Of Intensity Equation Research Articles

Phase retrieval in lens-based Fresnel wave propagation model

The goal of phase retrieval is to extract the phase of an optical wave field from intensity measurements. The transport-of-intensity equation (TIE)-based method is a popular deterministic solution and has been applied in various fields such as optical microscopy, electron microscopy, and x-ray imaging. For macro-imaging, a camera is often used to capture the images, and thus the phase modulation of the lens should be considered. A new formulation is proposed to extend TIE for phase retrieval in a lens-based wave propagation model. To obtain the defocus step, a data-collection-reading device is designed by equipping a camera with a micrometer caliper. Simulation and real experiments are conducted to test the proposed method.

Iterative phase retrieval with partially coherent beams

This work studies the influence of finite (or partial) optical coherence on the transport of intensity for object images, and presents an iterative scheme to treat the phase retrieval. With the plane wave decomposition technique, the couplings between the fields of object images and that of optical (random) de-coherence are treated as a Fourier convolution, and the outcome fields follow the transport of intensity equation (TIE). An ensemble of realizations of the (outcome) fields thus determines the statistical properties that correspond to the actual measurements. Simulation results of the proposed method show a significant increase in the accuracy for phase retrievals that have finite optical coherence.

The transport of intensity equation for optical path length recovery using partially coherent illumination

We investigate the measurement of a thin sample's optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample's optical path length by the removal of illumination effects.

Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter - theory and applications

Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter--an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not only provides great insight into the behaviors, the shortcomings, and the performance of these existing intensity derivative estimation algorithms, but more important, it also suggests a new way of improving solution strategies by extending the applications of Savitzky-Golay differentiation filter in TIE. Two novel methods for phase retrieval based on TIE are presented--the first by introducing adaptive-degree strategy in spatial domain and the second by selecting optimal spatial frequencies in Fourier domain. Numerical simulations and experiments verify that the second method outperforms the existing methods significantly, showing reliable retrieved phase with both overall contrast and fine phase variations well preserved.

基于强度传输方程的多重网格算法

Multigrid algorithm based on the transport of intensity equation is researched to retrieval phase information from the intensity measurement.Given initial value and stared from the coarsest grid,a solution is obtained by calculating;regarded it as the finest initial value,an approximate solution is obtained by calculating;then its residual is calculated and it is restricted to coarser grid and solving,until to the coarsest grid,followed by corrected the solution of finer grid.Finally,different frequency error components are eliminated by cycle which on different sizes grids,and the accurate solution is obtained.The simulation and real experiments show that the multigrid algorithm can well retrieve the actual phase of the object from the intensity image.

Nonlinear diffusion regularization for transport of intensity phase imaging

We demonstrate a nonlinear diffusion (NLD) regularization method to solve the transport of intensity equation (TIE). A novel NLD regularization function is proposed to enforce piecewise-constant priors and to remove low-frequency artifacts in the TIE solution.

Strain mapping of LED devices by dark-field inline electron holography: Comparison between deterministic and iterative phase retrieval approaches

Dark-field inline electron holography has recently been established as a convenient method to map strain in semiconductor devices, combining high precision, low noise, sub-nm spatial resolution and fields-of-view larger than 1 μm. Here we compare two approaches to reconstruct the geometric phase from a transmission electron microscopy dark-field focal series and their effects on the strain measurement: the transport-of-intensity-equation (TIE) and a flux-preserving iterative approach. For this task, we used a GaN-based light emitting diode with a highly complex heterostructure as a model system. While the TIE relies on 3 images only but requires the optimization of two free parameters (defocus step and low-limit cut-off frequency), the iterative reconstruction algorithm involves no adjustable parameters and uses images recorded at 9 different planes of focus with quadratically increasing defocus values. Optimum parameters for the TIE-reconstruction could be identified. However, the iterative phase retrieval approach yields the strain values that agree best with the expected strain levels and provides also higher spatial resolution.

Enhanced deterministic phase retrieval using a partially developed speckle field

A technique for enhanced deterministic phase retrieval using a partially developed speckle field (PDSF) and a spatial light modulator (SLM) is demonstrated experimentally. A smooth test wavefront impinges on a phase diffuser, forming a PDSF that is directed to a 4f setup. Two defocused speckle intensity measurements are recorded at the output plane corresponding to axially-propagated representations of the PDSF in the input plane. The speckle intensity measurements are then used in a conventional transport of intensity equation (TIE) to reconstruct directly the test wavefront. The PDSF in our technique increases the dynamic range of the axial intensity derivative for smooth phase objects, resulting in a more robust solution to the TIE. The SLM setup enables a fast and accurate recording of speckle intensity. Experimental results are in good agreement with those obtained using the iterative phase retrieval and digital holographic methods of wavefront reconstruction.

Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression

It is an effective scheme to the phase retrieval for axial intensity derivative computing. In this paper, we demonstrate a method for estimating the axial intensity derivative and improving the calculation accuracy in the transport of intensity equation (TIE) from multiple intensity measurements. The method takes both the higher-order intensity derivatives and the noise into account, and minimizes the impact of detecting noise. The simulation results demonstrate that the proposed method can effectively reduce the error of intensity derivative computing.

Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes

The noise problem is generally inevitable for phase retrieval by solving the transport of intensity equation (TIE). The noise effect can be alleviated by using multiple intensities to estimate the axial intensity derivative in the TIE. In this study, a method is proposed for estimating the intensity derivative by using multiple unevenly-spaced noisy measurements. The noise-minimized intensity derivative is approximated by a linear combination of the intensity data, in which the coefficients are obtained by solving a constrained optimization problem. The performance of the method is investigated by both the error analysis and the numerical simulations, and the results show that the method can reduce the noise effect on the retrieved phase. In addition, guidelines for the choice of the number of the intensity planes are given.

Energy-filtered phase retrieval using the transport of intensity equation

Energy-filtered imaging was combined with phase retrieval to study plasmon scattering contributions to electron phase shifts surrounding gold nanoparticles. Spectral windows with sufficient energy resolution to exclude plasmon scattering were used to measure through-focal intensity derivatives and retrieve phase maps via the transport of intensity equation. Phase excursions in the phase maps that included plasmon scattering were found to disagree with theory by at least an order of magnitude. This discrepancy was attributed to subtle particle instabilities under the electron beam, which lead to systematic errors in the retrieved phase maps.

Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes

A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.

Robustness of phase retrieval methods in x‐ray phase contrast imaging: A comparison

The robustness of the phase retrieval methods is of critical importance for limiting and reducing radiation doses involved in x-ray phase contrast imaging. This work is to compare the robustness of two phase retrieval methods by analyzing the phase maps retrieved from the experimental images of a phantom. Two phase retrieval methods were compared. One method is based on the transport of intensity equation (TIE) for phase contrast projections, and the TIE-based method is the most commonly used method for phase retrieval in the literature. The other is the recently developed attenuation-partition based (AP-based) phase retrieval method. The authors applied these two methods to experimental projection images of an air-bubble wrap phantom for retrieving the phase map of the bubble wrap. The retrieved phase maps obtained by using the two methods are compared. In the wrap's phase map retrieved by using the TIE-based method, no bubble is recognizable, hence, this method failed completely for phase retrieval from these bubble wrap images. Even with the help of the Tikhonov regularization, the bubbles are still hardly visible and buried in the cluttered background in the retrieved phase map. The retrieved phase values with this method are grossly erroneous. In contrast, in the wrap's phase map retrieved by using the AP-based method, the bubbles are clearly recovered. The retrieved phase values with the AP-based method are reasonably close to the estimate based on the thickness-based measurement. The authors traced these stark performance differences of the two methods to their different techniques employed to deal with the singularity problem involved in the phase retrievals. This comparison shows that the conventional TIE-based phase retrieval method, regardless if Tikhonov regularization is used or not, is unstable against the noise in the wrap's projection images, while the AP-based phase retrieval method is shown in these experiments to be superior to the TIE-based method for the robustness in performing the phase retrieval.

On the Use of Low-Pass Filters for Image Processing with Inverse Laplacian Models

A novel signal processing-oriented approach to solving problems involving inverse Laplacians is introduced. The Monogenic Signal is a powerful method of computing the phase of discrete signals in image data, however it is typically used with band-pass filters in the capacity of a feature detector. Substituting low-pass filters allows the Monogenic Signal to produce approximate solutions to the inverse Laplacian, with the added benefit of tunability and the generation of three equivariant properties (namely local energy, local phase and local orientation), which allow the development of powerful numerical solutions for a new set of problems. These principles are applied here in the context of biological cell segmentation from brightfield microscopy image data. The Monogenic Signal approach is used to generate reduced noise solutions to the Transport of Intensity Equation for optical phase recovery, and the resulting local phase and local orientation terms are combined in an iterative level set approach to accurately segment cell boundaries. Potential applications of this approach are discussed with respect to other fields.

Phase retrieval using the transport of intensity equation solved by the FMG–CG method

The relationship between the object-plane phase and the intensity distribution in the Fresnel region can be described by the transport of intensity equation (TIE). The phase distribution can thus be uniquely determined by solving TIE. In this study, a full multigrid preconditioned conjugate gradient (FMG–CG) method is proposed to numerically solve the TIE for phase retrieval. The full multigrid method is a scalable algorithm, and can be parallelized readily and efficiently. By using this method as a preconditioner of the preconditioned conjugate gradient (PCG) method, fast convergence is obtained. The simulation experiments show that complicated phase distributions with fast convergence speed can be retrieved by this composite method.

Phase from chromatic aberrations

We show that phase objects may be computed accurately from a single color image in a brightfield microscope, with no hardware modification. Our technique uses the chromatic aberration that is inherent to every lens-based imaging system as a phase contrast mechanism. This leads to a simple and inexpensive way of achieving single-shot quantitative phase recovery by a modified Transport of Intensity Equation (TIE) solution, allowing real-time phase imaging in a traditional microscope.

Higher‐order phase shift reconstruction approach

Biological soft tissues encountered in clinical and preclinical imaging mainly consists of atoms of light elements with low atomic numbers and their elemental composition is nearly uniform with little density variation. Hence, x-ray attenuation contrast is relatively poor and cannot achieve satisfactory sensitivity and specificity. In contrast, x-ray phase-contrast provides a new mechanism for soft tissue imaging. The x-ray phase shift of soft tissues is about a thousand times greater than the x-ray absorption over the diagnostic x-ray energy range, yielding a higher signal-to-noise ratio than the attenuation contrast counterpart. Thus, phase-contrast imaging is a promising technique to reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues. Here the authors develop a novel phase retrieval method to reconstruct the phase image on the object plane from the intensity measurements. The reconstructed phase image is a projection of the phase shift induced by an object and serves as input to reconstruct the 3D refractive index distribution inside the object using a tomographic reconstruction algorithm. Such x-ray refractive index images can reveal structural features in soft tissues, with excellent resolution differentiating healthy and malignant tissues. A novel phase retrieval approach is proposed to reconstruct an x-ray phase image of an object based on the paraxial Fresnel-Kirchhoff diffraction theory. A primary advantage of the authors' approach is higher-order accuracy over that with the conventional linear approximation models, relaxing the current restriction of slow phase variation. The nonlinear terms in the autocorrelation equation of the Fresnel diffraction pattern are eliminated using intensity images measured at different distances in the Fresnel diffraction region, simplifying the phase reconstruction to a linear inverse problem. Numerical experiments are performed to demonstrate the accuracy and stability of the proposed approach. The proposed reconstruction formula is a generalization of the transport of intensity equation (TIE). It has the second-order accuracy compared to the linear model used in the conventional phase retrieval approach. The numerical experiments demonstrate that the accuracy and stability of the proposed phase reconstruction method outperforms the TIE-based reconstruction method. A novel approach has been proposed to retrieve an x-ray phase shift image induced by an object from intensity images measured at different distances in the Fresnel diffraction region. The authors' approach has the second-order accuracy and is able to retrieve the phase shift of an object stably, overcoming the restriction of slow phase variation assumed by the conventional phase retrieval techniques.

Transport of intensity phase imaging in a volume holographic microscope

We demonstrate a method for single-shot quantitative phase imaging based on the transport of intensity equation (TIE) in a volume holographic microscope (VHM). The VHM system uses a multiplexed volume hologram to laterally separate images from different focal planes. This axial intensity information is then used to solve the TIE and recover object phase quantitatively. Further, we show improved phase recovery by using five multiplexed gratings in one hologram.

Performance analysis of the attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging

The phase retrieval is an important task in x-ray phase contrast imaging. The robustness of phase retrieval is especially important for potential medical imaging applications such as phase contrast mammography. Recently the authors developed an iterative phase retrieval algorithm, the attenuation-partition based algorithm, for the phase retrieval in inline phase-contrast imaging [1]. Applied to experimental images, the algorithm was proven to be fast and robust. However, a quantitative analysis of the performance of this new algorithm is desirable. In this work, we systematically compared the performance of this algorithm with other two widely used phase retrieval algorithms, namely the Gerchberg-Saxton (GS) algorithm and the Transport of Intensity Equation (TIE) algorithm. The systematical comparison is conducted by analyzing phase retrieval performances with a digital breast specimen model. We show that the proposed algorithm converges faster than the GS algorithm in the Fresnel diffraction regime, and is more robust against image noise than the TIE algorithm. These results suggest the significance of the proposed algorithm for future medical applications with the x-ray phase contrast imaging technique.

MO‐E‐204C‐04: Improving Robustness of Phase Retrieval in X‐Ray Phase Contrast Imaging

Purpose: The phase retrieval, retrieving tissue phase maps from the x‐ray phase contrast images, is an important task in x‐ray phase‐sensitive imaging. The robustness of phase retrieval algorithms is of critical importance for reducing radiation doses in clinical applications. We show that the conventional phase retrieval method is actually unstable against the quantum noise. We present a more robust phase retrieval method that we developed. Method and Materials: We first studied the phase retrieval by means of the conventional Transport of Intensity Equation (TIE) method for phantom imaging (such as a nylon air bubble wrap). For improvement we developed a robust phase retrieval method based on our notion of the phase‐attenuation duality and attenuation partition, and derived a robust iterative phase retrieval algorithm. This algorithm had been applied to experimental phase contrast imaging with phantoms. In addition the phase retrieval accuracies had been analyzed with computer simulations of phase contrast imaging of a digital phantom of breast tissue characteristics. Results: We found that the TIE‐based phase‐retrieval suffers from an intrinsic singularity. The phantom imaging showed that the TIE‐based phase retrieval method is unstable against the noise and misalignment in phase contrast imaging, even if the Tikhonov regularization was employed. In contrast, our method based on the phase‐attenuation duality and attenuation partition is singularity‐free, and is robust in terms of the phase‐map image quality and phase‐map accuracies against the noise and misalignment, as is verified by means of experimental phantom imaging and computer simulation. Conclusion: The conventional TIE‐based phase retrieval method is unstable against noise and image misalignment. Our phase retrieval method method based on the phase‐attenuation duality and attenuation partition is robust against the noise and image misalignment. Research supported in part by DoD Breast Cancer Research Program.