Precise phase retrieval for propagation-based images using discrete mathematics

Propagation-based imaging is an X-ray phase-contrast method that does not require any additional optical elements in the imaging system. Instead, phase-contrast effects are observed as measurable intensity variations after an incident X-ray wave field has propagated some distance between the object and plane of detection. Subsequent phase-retrieval processing of the recorded images is a necessary step in order to obtain quantitative information and can greatly increase the image signal-to-noise ratio, in particular in computed tomography. Most commonly this is done by approximating the object as a single material, which cannot always be justified. An example of this is the presence of bones and soft-tissues in biomedical imaging, which motivated the work presented here. We developed the combination of spectral X-ray imaging and propagation-based X-ray phasecontrast imaging to overcome the limitations of a single-material approximation. The addition of spectral data enables phase-retrieval on multi-material objects, as well as the use of processing methods well-known from conventional dual-energy imaging, such as material decomposition, virtual monochromatic images and effective atomic-number mapping. Here, we present first computed tomography results of spectral propagation-based phase-contrast. We discuss how the phase retrieval process can be interpreted as low- and high-pass filters. Lastly, we take a look at current and future technological challenges, and the potential application areas of spectral propagation-based phase-contrast imaging.

A method of tomographic phase retrieval is developed for multi-material objects whose components each has a distinct complex refractive index. The phase-retrieval algorithm, based on the Transport-of-Intensity equation, utilizes propagation-based X-ray phase contrast images acquired at a single defocus distance for each tomographic projection. The method requires a priori knowledge of the complex refractive index for each material present in the sample, together with the total projected thickness of the object at each orientation. The requirement of only a single defocus distance per projection simplifies the experimental setup and imposes no additional dose compared to conventional tomography. The algorithm was implemented using phase contrast data acquired at the SPring-8 Synchrotron facility in Japan. The three-dimensional (3D) complex refractive index distribution of a multi-material test object was quantitatively reconstructed using a single X-ray phase-contrast image per projection. The technique is robust in the presence of noise, compared to conventional absorption based tomography.

The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.

In optical interferometric tomography (OIT), phase retrieval and alignment are crucial steps for achieving a precise measurement of the three-dimensional refractive index (3D RI) distribution within a sample. Traditionally, completing these two tasks requires multiple independent steps, each of which relies on extensive handcrafted prior knowledge, thereby constraining both the efficiency and precision of the measurements. To address this problem, we utilize deep learning techniques and propose a novel multi-task transformer network (MTTN) that can simultaneously retrieve the phase from single-frame interference patterns and align it with the scene center. We evaluate MTTN on both simulated and real-world data. The results demonstrate that MTTN achieves precise phase retrieval and alignment without additional prior knowledge. By integrating it with tomographic reconstruction techniques, we successfully reconstruct the 3D RI maps of micron-scale specialty optical fibers. We believe this framework provides a new technical solution for addressing challenges in OIT and related fields.

Accurate 3D Fluorophore Localization using Phase Retrieved Pupil Functions

The following article describes a method for 3D reconstruction of multi-material objects based on propagation-based X-ray phase-contrast tomography (PB-CT) with phase retrieval using the homogeneous form of the transport of intensity equation (TIE-Hom). Unlike conventional PB-CT algorithms that perform phase retrieval of individual projections, the described post-reconstruction phase-retrieval method is applied in 3D to a localized region of the CT-reconstructed volume. This work demonstrates, via numerical simulations, the accuracy and noise characteristics of the method under a variety of experimental conditions, comparing it with both conventional absorption tomography and 2D TIE-Hom phase retrieval applied to projection images. The results indicate that the 3D post-reconstruction method generally achieves a modest improvement in noise suppression over existing PB-CT methods. It is also shown that potentially large computational gains over projection-based phase retrieval for multi-material samples are possible. In particular, constraining phase retrieval to a localized 3D region of interest reduces the overall computational cost and eliminates the need for multiple CT reconstructions and global 2D phase retrieval operations for each material within the sample.

Phase retrieval is a promising method for optical system and surface metrology that makes use of intensity measurements of diffraction patterns. An iterative algorithm is used to solve the inverse problem to find the phase of the field producing the measured intensity distributions. For practical reasons, such as the reduction of coherent artifacts or to improve the signal-to-noise ratio of the measured data, it is often desirable to measure intensity distributions using broadband illumination. It is possible to perform phase retrieval with broadband data by incorporating a broadband model of the system into the phase retrieval algorithm. To do this, the system is modeled at several discrete wavelengths and the results from each are summed incoherently to produce a broadband result. This significantly increases the computational load. We show here that when aberrations are small, accurate estimates of the OPD distribution, on the level of λ/1000 RMS error, can be achieved using data with bandwidth up to about 10% as the input to a phase retrieval algorithm that assumes monochromatic data.

When the degradation is mainly caused by the camera linear motion, or the out of focus lens of the imaging systems, the model of the degradation functions or known as the point spread function (PSF) can be established with the corresponding parameters. In the situations, the identification of the PSF can be converted to the identification of the parameter of the degradation functions. In order to increase the identification accuracy of the degradation function and the computational efficiency in the identification process, the low pass finite impulse response (FIR) filter which is named as the inverse adaptive amplitude spectrum filter (IAASF) proposed in this paper is applied to identify the PSF. First, the amplitude spectra of the degraded images are calculated with discrete fast Fourier transform (DFFT). Second, filtering the amplitude spectra with IAASF in order to accurately measure the distance of the characteristic curves on the filtered amplitude spectra image frames, i.e. measuring the distance of the central two spectra lines on amplitude spectra image frames for the linear motion blur, and measuring the diameter of the inner circle of the concentric circles on the amplitude spectra image frames for the defocus blur, thus, the parameter of the PSF can be determined. Therefore, the PSF can be obtained from the parametric model of degradation function. The original images have been restored by the identified degradation function and the degraded image with Wiener filtering. The experimental results show that the algorithm proposed in this paper has higher computational efficiency and the higher estimation accuracy of the degradation function while the degraded images have higher signal to noise ratio (SNR). The algorithm has certain reference significance for the restoration and identification of the turbulence degraded images yielded by the aero-optical effects.

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

Including accurate modeling of the point spread function (PSF) in positron emission tomography (PET) reconstruction algorithms results in improvements in image spatial resolution and contrast. In this work, we sampled the PSF in our first-generation radio-frequency brain dedicated PET insert for simultaneous PET/MR imaging using a 100 µCi NEMA standard 250 µm diameter Na-22 point source at 13 different positions within a subsection of the system field of view (FoV). The acquired list mode data was converted into the canonical sinogram format from which the spatial positioning of the source and standard deviations were calculated. The subset was then used to extrapolate the PSF for the full system FoV. This model was then fed as an input parameter into a graphical processing unit based ordered subset expectation maximization (OSEM) reconstruction algorithm and used to generate reconstructed images with and without spatially varying PSF modeling for the Na-22 point source and a Hoffman brain phantom. Results indicate that for point source reconstruction, the FWHM of the horizontal profile of the point source is smaller with spatially variant PSF especially closer to the edges. Effect of spatially varying PSF modeling is also presented with Hoffman phantom reconstruction and CNR value has increased with spatially varying PSF.

Recent advances in photon counting detection technology have led to significant research interest in X-ray imaging. As a tutorial level review, this paper covers a wide range of aspects related to X-ray photon counting detector characterization. The tutorial begins with a detailed description of the working principle and operating modes of a pixelated X-ray photon counting detector with basic architecture and detection mechanism. Currently available methods and techniques for charactering major aspects including energy response, noise floor, energy resolution, count rate performance (detector efficiency), and charge sharing effect of photon counting detectors are comprehensively reviewed. Other characterization aspects such as point spread function (PSF), line spread function (LSF), contrast transfer function (CTF), modulation transfer function (MTF), noise power spectrum (NPS), detective quantum efficiency (DQE), bias voltage, radiation damage, and polarization effect are also remarked. A cadmium telluride (CdTe) pixelated photon counting detector is employed for part of the characterization demonstration and the results are presented. This review can serve as a tutorial for X-ray imaging researchers and investigators to understand, operate, characterize, and optimize photon counting detectors for a variety of applications.

Phase retrieval is an inverse problem which consists in recovering an unknown signal from a set of absolute squared projections. Recently, gradient descent algorithms have been developed to solve this problem. However, their optimization cost functions are non-convex and non-smooth. To address the non-smoothness of the cost function, some of these methods use truncation thresholds to calculate a truncated step gradient direction. But, the truncation requires designing parameters to obtain a desired performance in the phase recovery, which drastically modifies the search direction update, increasing the sampling complexity. Therefore, this paper develops the Phase Retrieval Smoothing Conjugate Gradient method (PR-SCG) which uses a smoothing function to retrieve the signal. PR-SCG is based on the smooth-ing projected gradient method which is useful for non-convex optimization problems. PR-SCG uses a nonlinear conjugate gradient of the smoothing function as the search direction to accelerate the convergence. Furthermore, the incremental Stochastic Smoothing Phase Retrieval algorithm (SSPR) is developed. SSPR involves a single equation per iteration which results in a simple, scalable, and fast approach useful when the size of the signal is large. Also, it is shown that SSPR converges linearly to the true signal, up to a global unimodular constant. Additionally, the proposed methods do not require truncation parameters. Simulation results are provided to validate the efficiency of PR-SCG and SSPR compared to existing phase retrieval algorithms. It is shown that PR-SCG and SSPR are able to reduce the number of measurements and iterations to recover the phase, compared with recently developed algorithms.

Phase retrieval for sub-Gaussian measurements

Transfer orthogonal sparsifying transform learning for phase retrieval

Phase-Contrast Tomography (PCT) is becoming an important technique for nondestructive, in-situ characterization of soft and hard condensed matter. This thesis sheds light on our progress in developing novel tomographic reconstruction algorithms in combination with in-situ experimental approaches that employ propagation-based PCT. These developments are aimed to improve the spatial and temporal resolutions of in-situ studies of the soft and hard condensed matter. The mechanical properties of materials such as steel are, to a large extent, defined by its microstructure. Propagation-based PCT (i.e. holotomography) can be used to visualize the microstructures associated with small variations in density or composition in the bulk of the specimen. This imaging technique has the advantage of being non-destructive and allows to carry out in-situ, time-resolved observations of miscrostructural changes during dynamic processes such as solid-state phase transformations. Sufficient spatial and temporal resolution can be achieved only when an extremely bright x-ray source is used. In order to test whether a novel table-top x-ray source, the MIRRORCLE-6X, can be used for implementation of a propagation-based PCT system, we have carried out a series of experiments. The characteristics of a propagation-based PCI system that employs the MIRRORCLE-6X were analyzed. In our further research we have focused on the experiments that could be performed at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. We reported on the investigation of the cementite microstructure in carbon steel. In this work we present the reconstruction of the three-dimensional morphology of cementite grains in the bulk of steel using a non-destructive imaging technique – x-ray Phase-Contrast Tomography (PCT). Complementary information about the crystalline structure of the ferrite grains surrounding the cementite is obtained using x-ray Diffraction-Contrast Tomography (DCT). The work was continued by a time-resolved x-ray PCT investigation of the austenite grain growth during the ferrite-to-austenite phase transformation in low-carbon steel. Experimental data allows to visualize the evolution of the three-dimensional morphology of the austenite-ferrite interfaces with a spatial resolution on the order of 1 ?m and a time resolution of less than 9 minutes. The in-situ measurements show that the theory for grain growth during diffusional solid-state phase transformations needs to be extended in order to describe the drastically lower mobility that we observed for flat interfaces compared to curved interfaces. We observe that some flat interfaces are completely immobile, which is not predicted by the state-of-the-art theories and showed - for the first time - that new computational model for grain growth are needed. Quantitative image reconstruction based on PCT data requires solving two inverse problems: phase retrieval and tomographic reconstruction. In most cases, the problem of phase retrieval is ill-posed and requires that some type of a prior knowledge about the reconstructed image (i.e. particular regularization method) is used. We present a novel algebraic approach suitable for phase retrieval using various linear models. In this approach Total Variation (TV) minimization is used for regularization of the linearized inverse problem by promoting the solution with a sparse gradient magnitude (i.e. piece-wise constant solution). In that case prior knowledge about the reconstructed image may allow to (partially) recover the unknown spatial frequencies that are undefined by the experimental data. The problem of tomographic reconstruction based on PCT data can be solved using a similar approach. We conclude our investigation of the image reconstruction algorithms by introducing several algebraic approaches suitable for tomographic reconstruction based on PCT data. In these approaches TV minimization is used to find a regularized solution to an underdetermined linear system based on a linearized representation of the PCT. When the density of the reconstructed object is piece-wise constant (or close to it) a virtually artifact-free solution can be computed for the tomographic problem. The proposed approach can also be used to radically improve the accuracy of the tomographic reconstruction based on incomplete data (e.g. small number of projections) or data with low signal to noise ratio.