X-ray noninterferometric phase imaging: a unified picture

Investigating the robustness of a deep learning-based method for quantitative phase retrieval from propagation-based x-ray phase contrast measurements under laboratory conditions

Objective.Quantitative phase retrieval (QPR) in propagation-based x-ray phase contrast imaging of heterogeneous and structurally complicated objects is challenging under laboratory conditions due to partial spatial coherence and polychromaticity. A deep learning-based method (DLBM) provides a nonlinear approach to this problem while not being constrained by restrictive assumptions about object properties and beam coherence. The objective of this work is to assess a DLBM for its applicability under practical scenarios by evaluating its robustness and generalizability under typical experimental variations.Approach.Towards this end, an end-to-end DLBM was employed for QPR under laboratory conditions and its robustness was investigated across various system and object conditions. The robustness of the method was tested via varying propagation distances and its generalizability with respect to object structure and experimental data was also tested.Main results.Although the end-to-end DLBM was stable under the studied variations, its successful deployment was found to be affected by choices pertaining to data pre-processing, network training considerations and system modeling.Significance.To our knowledge, we demonstrated for the first time, the potential applicability of an end-to-end learning-based QPR method, trained on simulated data, to experimental propagation-based x-ray phase contrast measurements acquired under laboratory conditions with a commercial x-ray source and a conventional detector. We considered conditions of polychromaticity, partial spatial coherence, and high noise levels, typical to laboratory conditions. This work further explored the robustness of this method to practical variations in propagation distances and object structure with the goal of assessing its potential for experimental use. Such an exploration of any DLBM (irrespective of its network architecture) before practical deployment provides an understanding of its potential behavior under experimental settings.

Impact of a single distance phase retrieval algorithm on spatial resolution in X-ray inline phase sensitive imaging

A single-projection based phase retrieval method based on the phase attenuation duality principle (PAD) was used to compare the spatial resolution of the acquired phase sensitive and PAD processed phase retrieved images. An inline phase sensitive prototype was used to acquire the phase sensitive images. The prototype incorporates a micro-focus x-ray source and a flat panel detector with a 50 μm pixel pitch. A phantom composed of a 2 cm thick 50-50 adipose-glandular mimicking slab sandwiched with a 0.82 cm thick slanted PMMA sharp edge was used. Phase sensitive image of the phantom was acquired at 120 kV, 3.35 mAs with a 16 μm tube focal spot size under a geometric magnification (M) of 2.5. The PAD based method was applied to the acquired phase sensitive image for the retrieval of phase values. With necessary data processing, modulation transfer function (MTF) curves were determined for the estimation and comparison of the spatial resolution. The PAD processed phase retrieved values of the phantom were in good agreement with the theoretically calculated values. Phase sensitive images showed higher spatial resolution at all spatial frequencies compared to the phase retrieved images. It was noted that the high-frequency signal components in the retrieved image were suppressed that resulted in lower MTF values. When compared to the phase sensitive image, the cutoff resolution (10% MTF) for phase retrieved image dropped 32% from 15.6 lp/mm (32μm) to 10.6 lp/mm (47μm). The resolution offered by this phase sensitive prototype is radiographically enough to detect breast cancer.

3D Imaging Based on Depth Measurement Technologies

Three-dimensional (3D) imaging has attracted more and more interest because of its widespread applications, especially in information and life science. These techniques can be broadly divided into two types: ray-based and wavefront-based 3D imaging. Issues such as imaging quality and system complexity of these techniques limit the applications significantly, and therefore many investigations have focused on 3D imaging from depth measurements. This paper presents an overview of 3D imaging from depth measurements, and provides a summary of the connection between the ray-based and wavefront-based 3D imaging techniques.

Improved phase-attenuation duality method with space-frequency joint domain iterative regularization

PURPOSE

A common problem of in-line phase contrast imaging systems based on laboratory source and detector is the negative effects of finite source size, limited spatial resolution, and system noise. These negative effects swamp the fine phase contrast fringes and impede the precise retrieval of phase maps. This study aims to develop a novel phase retrieval method to restore phase information that is lost due to an imperfect system.

METHODS

An improved phase-attenuation duality (PAD) method based on space-frequency joint domain iterative regularization (JDIR) is proposed to overcome the problems of the analytical PAD method and the spatial-domain iterative regularization (SDIR) based PAD method. These problems include noise robustness and optical transfer function compensation. The proposed method was compared with the two former PAD methods through computer simulations and experiments for validation.

RESULTS

Results reveal that JDIR method outperforms the other two methods in terms of improving the visibility of structures in the retrieved phase maps. Among all the phase retrieval algorithms, the TV-norm-based JDIR method performed the best in considering the contrast and noise performance.

CONCLUSIONS

This paper provides a new method to investigate quantitative phase-contrast imaging when considering the negative effects of an imperfect system.

Coherent laser phase retrieval in the presence of measurement imperfections and incoherent light

Phase retrieval is a powerful numerical method that can be used to determine the wavefront of laser beams based only on intensity measurements, without the use of expensive, low-resolution specialized wavefront sensors such as Shack-Hartmann sensors. However, phase retrieval techniques generally suffer from poor convergence and fidelity when the input measurements contain electronic or optical noise and/or an incoherent intensity contribution overlapped with the otherwise spatially coherent laser beam. Here, we present an implementation of a modified version of the standard multiple-plane Gerchberg-Saxton algorithm and demonstrate that it is highly successful at extracting the intensity profile and wavefront of the spatially coherent part of the light from various lasers, including tapered laser diodes, at a very high fidelity despite the presence of incoherent light and noise.

Regularized Newton methods for x-ray phase contrast and general imaging problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95nm isotropic resolution.

Radiance and photon noise: imaging in geometrical optics, physical optics, quantum optics and radiology

The statistics of detector outputs produced by an imaging system are derived from basic radiometric concepts and definitions. We show that a fundamental way of describing a photon-limited imaging system is in terms of a Poisson random process in spatial, angular, and wavelength variables. We begin the paper by recalling the concept of radiance in geometrical optics, radiology, physical optics, and quantum optics. The propagation and conservation laws for radiance in each of these domains are reviewed. Building upon these concepts, we distinguish four categories of imaging detectors that all respond in some way to the incident radiance, including the new category of photon-processing detectors (capable of measuring radiance on a photon-by-photon basis). This allows us to rigorously show how the concept of radiance is related to the statistical properties of detector outputs and to the information content of a single detected photon. A Monte-Carlo technique, which is derived from the Boltzmann transport equation, is presented as a way to estimate probability density functions to be used in reconstruction from photon-processing data.

Phase retrieval method for in-line phase contrast x-ray imaging and denoising by regularization

Phase contrast X-ray imaging is increasingly popular in the past decade. In order to acquire phase contrast X-ray images, different types of imaging mechanisms have been proposed. Among them, in-line phase contrast X-ray imaging shows the highest potential because of its simplicity. In the study of in-line phase contrast imaging, based on different physical assumptions, many non-iterative phase retrieval methods, such as Bronnikov method, modified Bronnikov method, phase-attenuation duality (PAD) method, single-material method, and two-material method have been proposed. The main step of the non-iterative methods is a filtering process, thus different methods involve different filter design. In this paper we showed that every filter applied in the methods listed above is indeed the minimizer of a L(2)-norm regularization problem. In addition, two methods were proposed to overcome the over smoothing problem owing to the nature of L(2)-norm regularization.

Are human peripheral nerves sensitive to X-ray imaging?

Diagnostic imaging techniques play an important role in assessing the exact location, cause, and extent of a nerve lesion, thus allowing clinicians to diagnose and manage more effectively a variety of pathological conditions, such as entrapment syndromes, traumatic injuries, and space-occupying lesions. Ultrasound and nuclear magnetic resonance imaging are becoming useful methods for this purpose, but they still lack spatial resolution. In this regard, recent phase contrast x-ray imaging experiments of peripheral nerve allowed the visualization of each nerve fiber surrounded by its myelin sheath as clearly as optical microscopy. In the present study, we attempted to produce high-resolution x-ray phase contrast images of a human sciatic nerve by using synchrotron radiation propagation-based imaging. The images showed high contrast and high spatial resolution, allowing clear identification of each fascicle structure and surrounding connective tissue. The outstanding result is the detection of such structures by phase contrast x-ray tomography of a thick human sciatic nerve section. This may further enable the identification of diverse pathological patterns, such as Wallerian degeneration, hypertrophic neuropathy, inflammatory infiltration, leprosy neuropathy and amyloid deposits. To the best of our knowledge, this is the first successful phase contrast x-ray imaging experiment of a human peripheral nerve sample. Our long-term goal is to develop peripheral nerve imaging methods that could supersede biopsy procedures.

A simplified edge illumination set-up for quantitative phase contrast mammography with synchrotron radiation at clinical doses

This work presents the first study of x-ray phase contrast imaging based on a simple implementation of the edge illumination method (EIXPCi) in the field of mammography with synchrotron radiation. A simplified EIXPCi set-up was utilized to study a possible application in mammography at clinical doses. Moreover, through a novel algorithm capable of separating and quantifying absorption and phase perturbations of images acquired in EIXPCi modality, it is possible to extract quantitative information on breast images, allowing an accurate tissue identification. The study was carried out at the SYRMEP beamline of Elettra synchrotron radiation facility (Trieste, Italy), where a mastectomy specimen was investigated with the EIXPCi technique. The sample was exposed at three different energies suitable for mammography with synchrotron radiation in order to test the validity of the novel algorithm in extracting values of linear attenuation coefficients integrated over the sample thickness. It is demonstrated that the quantitative data are in good agreement with the theoretical values of linear attenuation coefficients calculated on the hypothesis of the breast with a given composition. The results are promising and encourage the current efforts to apply the method in mammography with synchrotron radiation.

High-speed multi-wavelength Fresnel diffraction imaging

We demonstrate a compact lensless microscope which can capture video-rate phase contrast images of moving objects and allows numerical scanning of the focal distance after recording. Using only an RGB-detector and illumination from a single mode fiber, diffraction patterns at three wavelengths are recorded simultaneously, enabling high-speed data collection and reconstruction of phase and amplitude. The technique is used for imaging of a moving test target, beads in a flow cell, and imaging of Caenorhabditis elegans moving in a droplet of liquid.

Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks

We report theoretical and experimental results for imaging of electromagnetic phase edge effects in lithography photomasks. Our method starts from the transport of intensity equation (TIE), which solves for phase from through-focus intensity images. Traditional TIE algorithms make an implicit assumption that the underlying in-plane power flow is curl-free. Motivated by our current study, we describe a practical situation in which this assumption breaks down. Strong absorption gradients in mask features interact with phase edges to contribute a curl to the in-plane Poynting vector, causing severe artifacts in the phase recovered. We derive how curl effects are coupled into intensity measurements and propose an iterative algorithm that not only corrects the artifacts, but also recovers missing curl components.

Simple solution to the Fresnel-Kirchoff diffraction integral for application to refraction-enhanced radiography

We present a simple solution to the Fresnel-Kirchoff diffraction integral that is appropriate for x-ray radiography of strongly absorbing and phase-shifting objects in the geometrical optics regime, where phase contrast enhancements can be considered to be caused by refraction by a semi-opaque object. We demonstrate its accuracy by comparison to brute-force numerical ray trace and diffraction calculations of a representative simulated object, and show excellent agreement for spatial scales corresponding to Fresnel numbers greater than unity. The result represents a significant improvement over approximate formulas typically used in analysis of refraction-enhanced radiographs, particularly for radiography of transient phenomena in objects that strongly refract and show significant absorption.

X-ray phase-contrast imaging: from pre-clinical applications towards clinics

Phase-contrast x-ray imaging (PCI) is an innovative method that is sensitive to the refraction of the x-rays in matter. PCI is particularly adapted to visualize weakly absorbing details like those often encountered in biology and medicine. In past years, PCI has become one of the most used imaging methods in laboratory and preclinical studies: its unique characteristics allow high contrast 3D visualization of thick and complex samples even at high spatial resolution. Applications have covered a wide range of pathologies and organs, and are more and more often performed in vivo. Several techniques are now available to exploit and visualize the phase-contrast: propagation- and analyzer-based, crystal and grating interferometry and non-interferometric methods like the coded aperture. In this review, covering the last five years, we will give an overview of the main theoretical and experimental developments and of the important steps performed towards the clinical implementation of PCI.

Phase retrieval in X-ray phase-contrast imaging suitable for tomography

In-line phase-contrast X-ray imaging provides images where both absorption and refraction contribute. For quantitative analysis of these images, the phase needs to be retrieved numerically. There are many phase-retrieval methods available. Those suitable for phase-contrast tomography, i.e., non-iterative phase-retrieval methods that use only one image at each projection angle, all follow the same pattern though derived in different ways. We outline this pattern and use it to compare the methods to each other, considering only phase-retrieval performance and not the additional effects of tomographic reconstruction. We also outline derivations, approximations and assumptions, and show which methods are similar or identical and how they relate to each other. A simple scheme for choosing reconstruction method is presented, and numerical phase-retrieval performed for all methods.

Ambiguity function and phase-space tomography for nonparaxial fields

A nonparaxial generalization of the ambiguity function that retains several properties of its paraxial counterpart is presented, in both two and three dimensions. This generalization is used to extend into the nonparaxial regime a scheme for the recovery of the coherence properties of scalar partially coherent fields in two-dimensional space.

Single-plane multiple speckle pattern phase retrieval using a deformable mirror

A design for a single-plane multiple speckle pattern phase retrieval technique using a deformable mirror (DM) is analyzed within the formalism of complex ABCD-matrices, facilitating its use in conjunction with dynamic wavefronts. The variable focal length DM positioned at a Fourier plane of a lens comprises the adaptive optical (AO) system that replaces the time-consuming axial displacements in the conventional free-space multiple plane setup. Compared with a spatial light modulator, a DM has a smooth continuous surface which avoids pixelation, pixel cross-talk and non-planarity issues. The calculated distances for the proposed AO-system are evaluated experimentally using the conventional free-space phase retrieval setup. Two distance ranges are investigated depending on whether the measurement planes satisfy the Nyquist detector sampling condition or not. It is shown numerically and experimentally that speckle patterns measured at the non-Nyquist range still yield good reconstructions. A DM with a surface height of 25 microns and an aperture diameter of 5.2 mm may be used to reconstruct spherical phase patterns with 50-micron fringe spacing.

Astigmatic phase retrieval: an experimental demonstration

We present the first experimental demonstration of the astigmatic phase retrieval technique, in which the diffracted wavefield is distorted by cylindrical curvature in two orthogonal directions. A charge-one vortex, a charge-two vortex, and a simple test image are all correctly reconstructed.

Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing

Wavefront reconstruction is carried out using sequentially recorded speckle patterns and an iterative phase retrieval method based on wave propagation. A novel fast-convergent algorithm that maintains the propagation distance in the iteration step equal to the distance between measurement planes is demonstrated. Employing the new algorithm, influences of the number of measurement planes, number of iterations, and uncertainties in the detector's transverse and axial positions on the rate of phase convergence are analyzed experimentally. A conceptual design for a dynamic wavefront sensor using arrays of beam splitters and detectors for parallel speckle recording is described.

Non-iterative solution of the phase retrieval problem using a single diffraction measurement

Coherent diffractive imaging is a method by which iterative methods are employed to recover image information about a finite object from its coherent diffraction pattern. We employ methods borrowed from density functional theory to show that an image can be recovered in a single non-iterative step for a finite sample subject to phase-curved illumination. The result also yields a new approach to quantitative x-ray phase-contrast imaging.

Noninterferometric phase retrieval using a fractional Fourier system

The signal extraction method based on intensity measurements in two close fractional Fourier domains is examined by using the phase space formalism. The fractional order separation has a lower bound and an upper bound that depend on the signal at hand and the noise in the optical system used for measurement. On the basis of a theoretical analysis, it is shown that for a given optical system a judicious choice of fractional order separation requires some a priori knowledge of the signal bandwidth. We also present some experimental results in support of the analysis.

Holography: an interpretation from the phase-space point of view

The formation of holograms is interpreted as the consequence of the bilinearity of the ambiguity function. Reconstruction can then be regarded as the manipulation of the ambiguity function. Specifically, we show that in the case of in-line holography, the reconstruction can be regarded as phase tomography. In this way we provide a unified picture for the formulation of both noninterferometric and interferometric phase-retrieval techniques.