Grating-free quantitative phase retrieval for x-ray phase-contrast imaging with conventional sources.

X-ray phase-contrast imaging can display subtle differences in low-density materials (e.g. soft tissues) more readily than conventional x-ray imaging. However, producing x-ray phase images requires significant spatial coherence of the beam which in turn requires highly specialized sources such as synchrotrons, small and low power microfocus sources, or complex procedures, such as multiple exposures with several carefully stepped precision gratings. To find appropriate approaches for producing x-ray phase-contrast imaging in a clinically meaningful way, we employed a grating-free method that utilized a low-cost, coarse wire mesh and simple processing. This method relaxes the spatial coherence constraint and allows quantitative phase retrieval for not only monochromatic but also polychromatic beams. We also combined the mesh-based system with polycapillary optics to significantly improve the accuracy of quantitative phase retrieval.

Phase and dark-field imaging with mesh-based structured illumination and polycapillary optics.

PURPOSE: X-ray phase and dark-field (DF) imaging have been shown to improve the diagnostic capabilities of X-ray systems. However, these methods have found limited clinical use due to the need for multiple precision gratings with limited field of view or requirements on X-ray coherence that may not be easily translated to clinical practice. This work aims to develop a practicable X-ray phase and DF imaging system that could be translated and practiced in the clinic. METHODS: This work employs a conventional source to create structured illumination with a simple wire mesh. A mesh-shifting algorithm is used to allow wider Fourier windowing to enhance resolution. Deconvolution of the source spot width and camera resolution improves accuracy. Polycapillary optics are employed to enhance coherence. The effects of incorporating optics with two different focal lengths are compared. Information apparent in enhanced absorption images, phase images, and DF images of fat embedded phantoms were compared and subjected to a limited receiver operator characteristic (ROC) study. The DF images of the moist and dry porous object (sponges) were compared. RESULTS: The mesh-based phase and DF imaging system constructs images with three different information types: scatter-free absorption images, differential phase images, and scatter magnitude/DF images, simultaneously from the same original image. The polycapillary optic enhances the coherence of the beam. The deblurring technique corrects the phase signal error due to geometrical blur and the limitation of the camera modulation transfer function (MTF) and removes image artifacts to improve the resolution in a single shot. The mesh-shifting method allows the use of a wider Fourier processing window, which gives even higher resolution, at the expense of an increased dose. The limited ROC study confirms the efficacy of the system over the conventional system. DF images of moist and dry porous object show the significance of the system in the imaging of lung infections. CONCLUSION: The mesh-based X-ray phase and DF imaging system is an inexpensive and easy setup in terms of alignment and data acquisition and can produce phase and DF images in a single shot with wide field of view. The system shows significant potential for use in diagnostic imaging in a clinical setting.

Comparative phase imaging of live cells by digital holographic microscopy and transport of intensity equation methods.

We describe a microscopic setup implementing phase imaging by digital holographic microscopy (DHM) and transport of intensity equation (TIE) methods, which allows the results of both measurements to be quantitatively compared for either live cell or static samples. Digital holographic microscopy is a well-established method that provides robust phase reconstructions, but requires a sophisticated interferometric imaging system. TIE, on the other hand, is directly compatible with bright-field microscopy, but is more susceptible to noise artifacts. We present results comparing DHM and TIE on a custom-built microscope system that allows both techniques to be used on the same cells in rapid succession, thus permitting the comparison of the accuracy of both methods.

Efficient propagation of the coherency matrix inspired by plenoptic field representations of nonparaxial Wigner functions.

The angle-impact Wigner function (AIWF) is a tool that assigns weights to rays in a way such that ray tracing can be used to exactly model the irradiance and polarization properties of nonparaxial, partially coherent, partially polarized electromagnetic fields. Although the AIWF offers computational advantages over more conventional wave propagation integrals, in practice, the large number of rays that must be traced has proved to be a significant bottleneck. We demonstrate an efficient method to implement AIWF propagation algorithms inspired by a plenoptic field representation. We illustrate the method by computing irradiance and polarization properties of several paraxial and nonparaxial fields.

Intracellular Macromolecules in Cell Volume Control and Methods of Their Quantification.

The traditional theories of cell volume regulation focus on monovalent ions and small organic osmolytes. The main subject of this review is macromolecular content of the cell and its role in cell volume. We start by reviewing general information about cellular macromolecules and present some quantitative relationships. Next, we review a wide range of methods for measuring intracellular macromolecular concentration and related parameters; in particular, a large section is devoted to the so-called quantitative phase imaging methods based on transmission light microscopy. In the last part, we discuss three specific biological examples where quantitative analysis of macromolecular concentrations is expected to generate valuable insights into biological processes: the biology of organelles, long-term cell volume maintenance and apoptotic volume decrease.

Optical convolution for quantitative phase retrieval using the transport of intensity equation.

Propagation-based phase imaging using the transport of intensity equation (TIE) allows rapid, deterministic phase retrieval from defocused images. However, computational solutions to the TIE suffer from significant low-frequency noise artifacts and are unique up to the application of boundary conditions on the phase. We demonstrate that quantitative phase can be imaged directly at the detector for a class of pure-phase samples by appropriately patterning the illumination to solve the TIE through an optical convolution with the source. This can reduce noise artifacts, obviates the need for user-supplied boundary conditions and is demonstrated via simulation and experiment.

Bright-Field Microscopy of Transparent Objects: A Ray Tracing Approach.

The formation of a bright-field microscopic image of a transparent phase object is described in terms of elementary geometrical optics. Our approach is based on the premise that the image replicates the intensity distribution (real or virtual) at the front focal plane of the objective. The task is therefore reduced to finding the change in intensity at the focal plane caused by the object. This can be done by ray tracing complemented with the requirement of energy conservation. Despite major simplifications involved in such an analysis, it reproduces some results from the paraxial wave theory. In addition, our analysis suggests two ways of extracting quantitative phase information from bright-field images: by vertically shifting the focal plane (the approach used in the transport-of-intensity analysis) or by varying the angle of illumination. In principle, information thus obtained should allow reconstruction of the object morphology.

Source diversity for transport of intensity phase imaging.

The transport of intensity equation (TIE) is a phase retrieval method that relies on measurements of the intensity of a paraxial field under propagation between two or more closely spaced planes. A limitation of TIE is its susceptibility to low frequency noise artifacts in the reconstructed phase. Under Köhler illumination, when both illumination power and exposure time are limited, the use of larger sources can improve low-frequency performance although it introduces blurring. Appropriately combining intensity measurements taken with a diversity of source sizes can improve both low- and high-frequency performance in phase reconstruction.

Phase Imaging Using Focused Polycapillary Optics.

Conventional radiographic techniques depend on attenuation, which provides low contrast between soft tissues. However, X rays can accumulate large differential phase delays even in weakly absorbing materials. This can produce significantly higher contrast. One technique for taking advantage of phase effects, propagation-based phase imaging, can yield marked edge enhancement but requires spatially coherent intense sources. Microfocus sources have sizes on the order of tens of microns but necessarily are low power and hence require long exposures. In this project, X-ray optical and computational techniques were explored to develop both edge-enhancement and phase imaging using a large spot conventional source. A polycapillary optic was employed to create a small secondary source from a large spot rotating anode X-ray generator. The secondary spot created by the focusing polycapillary optic was 114 µm ± 50 µm. Images of a 1.6 mm polyethylene rod were taken at varying distances from the optic. Edge enhancement was observed with a maximum edge-enhancement-to-noise ratio of 6.5. Insect images were also acquired and analyzed. Phase reconstructions were computed using two different approaches, weak attenuation and phase attenuation duality. Pure phase images were successfully reconstructed from the phase contrast images by employing the weak attenuation model.

Compressive x-ray phase tomography based on the transport of intensity equation.

We develop and implement a compressive reconstruction method for tomographic recovery of refractive index distribution for weakly attenuating objects in a microfocus x-ray system. This is achieved through the development of a discretized operator modeling both the transport of intensity equation and the x-ray transform that is suitable for iterative reconstruction techniques.

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.

Wigner function measurement using a lenslet array.

Geometrical-optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave-optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam's WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross-talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell-model beam.

Wigner functions for evanescent waves.

We propose phase space distributions, based on an extension of the Wigner distribution function, to describe fields of any state of coherence that contain evanescent components emitted into a half-space. The evanescent components of the field are described in an optical phase space of spatial position and complex-valued angle. Behavior of these distributions upon propagation is also considered, where the rapid decay of the evanescent components is associated with the exponential decay of the associated phase space distributions. To demonstrate the structure and behavior of these distributions, we consider the fields generated from total internal reflection of a Gaussian Schell-model beam at a planar interface.

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.

Wigner functions defined with Laplace transform kernels.

We propose a new Wigner-type phase-space function using Laplace transform kernels--Laplace kernel Wigner function. Whereas momentum variables are real in the traditional Wigner function, the Laplace kernel Wigner function may have complex momentum variables. Due to the property of the Laplace transform, a broader range of signals can be represented in complex phase-space. We show that the Laplace kernel Wigner function exhibits similar properties in the marginals as the traditional Wigner function. As an example, we use the Laplace kernel Wigner function to analyze evanescent waves supported by surface plasmon polariton.

Generalized radiometry model for the propagation of light within anisotropic and chiral media.

An exact radiometric model is proposed for the nonparaxial propagation of fully or partially coherent electromagnetic radiation within anisotropic and/or chiral media in terms of rectilinear ray propagation. The definition of the generalized radiance is motivated by a geometric interpretation, based on the plane-wave decomposition of the field in the medium. The new representations are illustrated through two examples, corresponding to the propagation of two-dimensional fields within birefringent and chiral media.

Phase space distributions tailored for dispersive media.

New phase space distributions are proposed for describing pulse propagation in dispersive media for one spatial dimension. These distributions depend on time, position, and velocity, so that the pulse's spatial propagation or temporal evolution is described by a free-particle-like transformation followed by integration over velocity. Examples are considered for approximate Lorentz-model dielectrics and metallic waveguides.

Propagation of nonparaxial partially coherent fields across interfaces using generalized radiometry.

A radiometric framework is described for modeling the propagation of nonparaxial scalar fields of any degree of coherence past planar boundaries (or composite interfaces) between homogeneous, isotropic nonabsorptive media in three dimensions. The transformation is shown to be, to lowest order, that predicted by classical radiometry but potentially including a Goos-Hänchen shift. Higher-order corrections take the form of coefficients multiplied by derivatives of the basic estimate. The accuracy of the radiometric term, along with second-order derivative corrections, are examined for Gaussian Schell-model fields of varying width and states of coherence. This technique is found to work well for most such fields but to fail in reflection for fields with significant total-internally-reflected components.

Ray-based propagation of the cross-spectral density.

We explore the propagation of the cross-spectral density for scalar and electromagnetic fields based on generalized radiances that are exactly conserved along rays. Two formulas are derived: The first uses all rays to calculate the cross-spectral density exactly, while the second uses only the subset of those rays that pass through a single spatial point to construct an infinite series expression for the cross-spectral density. The evaluation of the truncated series is examined numerically for a variety of fields of varying angular width and coherence and is found to exhibit better convergence to the cross-spectral density when the rays through the centroid between the two observation points are used, when the fields are less coherent, and when the fields are more paraxial. In generalizing the series formula, two new cross-spectral correlations associated with the flux and energy density are examined.

Propagation of partially coherent fields through planar dielectric boundaries using angle-impact Wigner functions I. Two dimensions.

We examine the angle-impact Wigner function (AIW) as a computational tool for the propagation of nonparaxial quasi-monochromatic light of any degree of coherence past a planar boundary between two homogeneous media. The AIWs of the reflected and transmitted fields in two dimensions are shown to be given by a simple ray-optical transformation of the incident AIW plus a series of corrections in the form of differential operators. The radiometric and leading six correction terms are studied for Gaussian Schell-model fields of varying transverse width, transverse coherence, and angle of incidence.