Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging

Physics-informed neural network for phase imaging based on transport of intensity equation

Non-interferometric quantitative phase imaging based on Transport of Intensity Equation (TIE) has been widely used in bio-medical imaging. However, analytic TIE phase retrieval is prone to low-spatial frequency noise amplification, which is caused by the illposedness of inversion at the origin of the spectrum. There are also retrieval ambiguities resulting from the lack of sensitivity to the curl component of the Poynting vector occurring with strong absorption. Here, we establish a physics-informed neural network (PINN) to address these issues, by integrating the forward and inverse physics models into a cascaded deep neural network. We demonstrate that the proposed PINN is efficiently trained using a small set of sample data, enabling the conversion of noise-corrupted 2-shot TIE phase retrievals to high quality phase images under partially coherent LED illumination. The efficacy of the proposed approach is demonstrated by both simulation using a standard image database and experiment using human buccal epitehlial cells. In particular, high image quality (SSIM = 0.919) is achieved experimentally using a reduced size of labeled data (140 image pairs). We discuss the robustness of the proposed approach against insufficient training data, and demonstrate that the parallel architecture of PINN is efficient for transfer learning.

3D particle tracking using transport of intensity equation (TIE)

This article presents a simple and high-speed approach for tracking colloidal spheres in three dimensions. The method uses the curvature of the wavefront as determined by the transport of intensity equation (TIE) technique. Due to the fact that the TIE is applicable under partially coherent light, our technique is fully compatible with standard bright field microscopes, requiring no demanding environmental stability requirements or restrictions on the noise produced by related laser speckles. The method was validated experimentally to determine the sedimentation and diffusion coefficients of two different sizes of microspheres, 20 and 3 microns. The 3D position of the microspheres was calculated with an accuracy greater than 350 nm. Moreover, we examined the calculated 3D positions to determine the parameters of the microsphere interaction with its surrounding media, such as the sedimentation and diffusion coefficients. The results show that the measured sedimentation and diffusion of the microspheres have a good agreement with predicted values of about 2% and 10%, respectively, demonstrating the robustness of our proposed method.

Principles of Different X-ray Phase-Contrast Imaging: A Review

Numerous advances have been made in X-ray technology in recent years. X-ray imaging plays an important role in the nondestructive exploration of the internal structures of objects. However, the contrast of X-ray absorption images remains low, especially for materials with low atomic numbers, such as biological samples. X-ray phase-contrast images have an intrinsically higher contrast than absorption images. In this review, the principles, milestones, and recent progress of X-ray phase-contrast imaging methods are demonstrated. In addition, prospective applications are presented.

Single-shot higher-order transport-of-intensity quantitative phase imaging based on computer-generated holography

The imaging quality of quantitative phase imaging (QPI) based on the transport of intensity equation (TIE) can be improved using a higher-order approximation for defocused intensity distributions. However, this requires mechanically scanning an image sensor or object along the optical axis, which in turn requires a precisely aligned optical setup. To overcome this problem, a computer-generated hologram (CGH) technique is introduced to TIE-based QPI. A CGH generating defocused point spread function is inserted in the Fourier plane of an object. The CGH acts as a lens and grating with various focal lengths and orientations, allowing multiple defocused intensity distributions to be simultaneously detected on an image sensor plane. The results of a numerical simulation and optical experiment demonstrated the feasibility of the proposed method.

A phase-retrieval toolbox for X-ray holography and tomography

Propagation-based phase-contrast X-ray imaging is by now a well established imaging technique, which - as a full-field technique - is particularly useful for tomography applications. Since it can be implemented with synchrotron radiation and at laboratory micro-focus sources, it covers a wide range of applications. A limiting factor in its development has been the phase-retrieval step, which was often performed using methods with a limited regime of applicability, typically based on linearization. In this work, a much larger set of algorithms, which covers a wide range of cases (experimental parameters, objects and constraints), is compiled into a single toolbox - the HoloTomoToolbox - which is made publicly available. Importantly, the unified structure of the implemented phase-retrieval functions facilitates their use and performance test on different experimental data.

Transport-of-intensity phase imaging with polarization directed flat lenses

A phase imaging technique based on the transport of intensity equation with polarization directed flat lenses is demonstrated. Transport-of-intensity phase imaging enables one to obtain a phase distribution from through-focus intensity distributions by solving the transport of intensity equation. In general, the through-focus intensity distributions are obtained by mechanical scanning of an image sensor or target object. Therefore, a precise alignment of an optical system is required. To solve this issue, the introduction of polarization directed flat lenses is presented. In the proposed method, two intensity distributions at different depth positions on the optical axis are obtained without mechanical scanning by changing polarization states of incident light. The feasibility of the proposed method is confirmed by an optical experiment.

Focus characterization of the NanoMAX Kirkpatrick-Baez mirror system

The focusing and coherence properties of the NanoMAX Kirkpatrick-Baez mirror system at the fourth-generation MAX IV synchrotron in Lund have been characterized. The direct measurement of nano-focused X-ray beams is possible by scanning of an X-ray waveguide, serving basically as an ultra-thin slit. In quasi-coherent operation, beam sizes of down to 56 nm (FWHM, horizontal direction) can be achieved. Comparing measured Airy-like fringe patterns with simulations, the degree of coherence |μ| has been quantified as a function of the secondary source aperture (SSA); the coherence is larger than 50% for SSA sizes below 11 µm at hard X-ray energies of 14 keV. For an SSA size of 5 µm, the degree of coherence has been determined to be 87%.

Fourier-based solving approach for the transport-of-intensity equation with reduced restrictions

The transport-of-intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, is based on a Fourier transform (FT). However, several assumptions have to be made to solve the TIE by these methods. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the parallelism of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.

Three-dimensional single-cell imaging with X-ray waveguides in the holographic regime

X-ray tomography at the level of single biological cells is possible in a low-dose regime, based on full-field holographic recordings, with phase contrast originating from free-space wave propagation. Building upon recent progress in cellular imaging based on the illumination by quasi-point sources provided by X-ray waveguides, here this approach is extended in several ways. First, the phase-retrieval algorithms are extended by an optimized deterministic inversion, based on a multi-distance recording. Second, different advanced forms of iterative phase retrieval are used, operational for single-distance and multi-distance recordings. Results are compared for several different preparations of macrophage cells, for different staining and labelling. As a result, it is shown that phase retrieval is no longer a bottleneck for holographic imaging of cells, and how advanced schemes can be implemented to cope also with high noise and inconsistencies in the data.

Probe reconstruction for holographic X-ray imaging

In X-ray holographic near-field imaging the resolution and image quality depend sensitively on the beam. Artifacts are often encountered due to the strong focusing required to reach high resolution. Here, two schemes for reconstructing the complex-valued and extended wavefront of X-ray nano-probes, primarily in the planes relevant for imaging (i.e. focus, sample and detection plane), are presented and compared. Firstly, near-field ptychography is used, based on scanning a test pattern laterally as well as longitudinally along the optical axis. Secondly, any test pattern is dispensed of and the wavefront reconstructed only from data recorded for different longitudinal translations of the detector. For this purpose, an optimized multi-plane projection algorithm is presented, which can cope with the numerically very challenging setting of a divergent wavefront emanating from a hard X-ray nanoprobe. The results of both schemes are in very good agreement. The probe retrieval can be used as a tool for optics alignment, in particular at X-ray nanoprobe beamlines. Combining probe retrieval and object reconstruction is also shown to improve the image quality of holographic near-field imaging.

Three-dimensional propagation in near-field tomographic X-ray phase retrieval

An extension of phase retrieval algorithms for near-field X-ray (propagation) imaging to three dimensions is presented, enhancing the quality of the reconstruction by exploiting previously unused three-dimensional consistency constraints.

This paper presents an extension of phase retrieval algorithms for near-field X-ray (propagation) imaging to three dimensions, enhancing the quality of the reconstruction by exploiting previously unused three-dimensional consistency constraints. The approach is based on a novel three-dimensional propagator and is derived for the case of optically weak objects. It can be easily implemented in current phase retrieval architectures, is computationally efficient and reduces the need for restrictive prior assumptions, resulting in superior reconstruction quality.

Fully automated, high speed, tomographic phase object reconstruction using the transport of intensity equation in transmission and reflection configurations

While traditional transport of intensity equation (TIE) based phase retrieval of a phase object is performed through axial translation of the CCD, in this work a tunable lens TIE is employed in both transmission and reflection configurations. These configurations are extended to a 360° tomographic 3D reconstruction through multiple illuminations from different angles by a custom fabricated rotating assembly of the phase object. Synchronization circuitry is developed to control the CCD camera and the Arduino board, which in its turn controls the tunable lens and the stepper motor to automate the tomographic reconstruction process. Finally, a MATLAB based user friendly graphical user interface is developed to control the whole system and perform tomographic reconstruction using both multiplicative and inverse radon based techniques.

Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction

A compound optical system for coherent focusing and imaging at the nanoscale is reported, realised by high-gain fixed-curvature elliptical mirrors in combination with X-ray waveguide optics or different cleaning apertures. The key optical concepts are illustrated, as implemented at the Göttingen Instrument for Nano-Imaging with X-rays (GINIX), installed at the P10 coherence beamline of the PETRA III storage ring at DESY, Hamburg, and examples for typical applications in biological imaging are given. Characteristic beam configurations with the recently achieved values are also described, meeting the different requirements of the applications, such as spot size, coherence or bandwidth. The emphasis of this work is on the different beam shaping, filtering and characterization methods.

Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs

We have performed x-ray phase-contrast tomography on mouse lung tissue. Using a divergent x-ray beam generated by nanoscale focusing, we used zoom tomography to produce three-dimensional reconstructions with selectable magnification, resolution, and field of view. Thus, macroscopic tissue samples extending over several mm can be studied in sub-cellular-level structural detail. The zoom capability and, in particular, the high dose efficiency are enabled by the near-perfect exit wavefront of an optimized x-ray waveguide channel. In combination with suitable phase-retrieval algorithms, challenging radiation-sensitive and low-contrast samples can be reconstructed with minimal artefacts. The dose efficiency of the method is demonstrated by the reconstruction of living macrophages both with and without phagocytized contrast agents. We also used zoom tomography to visualize barium-labelled macrophages in the context of morphological structures in asthmatic and healthy mouse lung tissue one day after intratracheal application. The three-dimensional reconstructions showed that the macrophages predominantly localized to the alveoli, but they were also found in bronchial walls, indicating that these cells might be able to migrate from the lumen of the bronchi through the epithelium.

Quantitative X-ray phase contrast waveguide imaging of bacterial endospores

Quantitative X-ray phase contrast imaging uniquely offers quantitative imaging information in terms of electron density maps allowing for mass and mass density determinations of soft biological samples (‘weighing with light’). Here, it was carried out using coherent X-ray waveguide illumination, yielding values of the mass and mass density of freeze-dried bacterial endospores (Bacillus spp.).

Quantitative waveguide-based X-ray phase contrast imaging has been carried out on the level of single, unstained, unsliced and freeze-dried bacterial cells of Bacillus thuringiensis and Bacillus subtilis using hard X-rays of 7.9 keV photon energy. The cells have been prepared in the metabolically dormant state of an endospore. The quantitative phase maps obtained by iterative phase retrieval using a modified hybrid input–output algorithm allow for mass and mass density determinations on the level of single individual endospores but include also large field of view investigations. Additionally, a direct reconstruction based on the contrast transfer function is investigated, and the two approaches are compared. Depending on the field of view and method, a resolution down to 65 nm was achieved at a maximum applied dose of below 5 × 10⁵ Gy. Masses in the range of about ∼110–190 (20) fg for isolated endospores have been obtained.

Phase retrieval through a one-dimensional ptychographic engine

Ptychographic techniques are currently the subject of increasing scientific interest due to their capability to retrieve the complex transmission function of an object at very high resolution. However, they impose a substantial burden in terms of acquisition time and dimension of the scanned area, which limits the range of samples that can be studied. We have developed a new method that combines the ptychographic approach in one direction with Fresnel propagation in the other by employing a strongly asymmetric probe. This enables scanning the sample in one direction only, substantially reducing exposure times while covering a large field of view. This approach sacrifices ptychographic-related resolution in one direction, but removes any limitation on the probe dimension in the direction orthogonal to the scanning, enabling the scan of relatively large objects without compromising exposure times.

Reconstruction of wave front and object for inline holography from a set of detection planes

We illustrate the errors inherent in the conventional empty beam correction of full field X-ray propagation imaging, i.e. the division of intensities in the detection plane measured with an object in the beam by the intensity pattern measured without the object, i.e. the empty beam intensity pattern. The error of this conventional approximation is controlled by the ratio of the source size to the smallest feature in the object, as is shown by numerical simulation. In a second step, we investigate how to overcome the flawed empty beam division by simultaneous reconstruction of the probing wavefront (probe) and of the object, based on measurements in several detection planes (multi-projection approach). The algorithmic scheme is demonstrated numerically and experimentally, using the defocus wavefront of the hard X-ray nanoprobe setup at the European Synchrotron Radiation Facility (ESRF).

Phase retrieval for object and probe using a series of defocus near-field images

Full field x-ray propagation imaging can be severely deteriorated by wave front aberrations. Here we present an extension of ptychographic phase retrieval with simultaneous probe and object reconstruction suitable for the near-field diffractive imaging setting. Update equations used to iteratively solve the phase problem from a set of near-field images in view of reconstruction both object and probe are derived. The algorithm is tested based on numerical simulations including photon shot noise. The results indicate that the approach provides an efficient way to overcome restrictive idealizations of the illumination wave in the near-field (propagation) imaging.

Versatility of a hard X-ray Kirkpatrick-Baez focus characterized by ptychography

In the past decade Kirkpatrick-Baez (KB) mirrors have been established as powerful focusing systems in hard X-ray microscopy applications. Here a ptychographic characterization of the KB focus in the dedicated nano-imaging setup GINIX (Göttingen Instrument for Nano-Imaging with X-rays) at the P10 coherence beamline of the PETRA III synchrotron at HASLYLAB/DESY, Germany, is reported. More specifically, it is shown how aberrations in the KB beam, caused by imperfections in the height profile of the focusing mirrors, can be eliminated using a pinhole as a spatial filter near the focal plane. A combination of different pinhole sizes and illumination conditions of the KB setup makes the prepared optical setup well suited not only for high-resolution ptychographic coherent X-ray diffractive imaging but also for moderate-resolution/large-field-of-view propagation imaging in the divergent KB beam.