Numerical Reconstruction Algorithm Research Articles

Optical property discrepancies found between healthy and unhealthy skin cells using digital holographic microscopy with three wavelengths

Cancer and other health disorders can be differentiated by changes in cell optical properties such as their refractive index, thickness, and topology (height and width). Here, we employ three wavelengths simultaneously in digital holographic microscopy (3λ-DHM) to visualize the whole cell topology as 3D images through a numerical reconstruction algorithm applied to a hologram. By identifying the cell state and the changes in its optical properties, it is possible to discern between healthy and unhealthy cells. The simultaneous use of three wavelengths provides a rapid and straightforward quantitative reconstruction of the whole cell without the need for an unwrapping algorithm. This is a benefit over traditional methods, which often require complicated procedures. The performance of the approach was first validated in a known sample, a silicon dioxide thin film, where we were able to corroborate its refractive index with the values reported in the literature. Then the method was applied to fixed skin cells finding a refractive index of 1.3443 for healthy cells and 1.3246 for cells found in tumor tissue. We discuss and highlight differences based on the refractive index to demonstrate that the employed process can provide reliable information to distinguish characteristics between healthy and unhealthy cells.

Lensless Mueller holographic microscopy with robust noise reduction for multiplane polarization imaging

Lensless holographic microscopy has emerged as a powerful and cost-effective tool for computational imaging, offering high resolution over a large field of view, beneficial for various biological applications. However, conventional approaches can struggle with contrast and accurate visualization of diverse components over the samples, which can directly affect the diagnostic precision of the techniques. Mueller imaging, while offering detailed, stain-free observations of polarized light responses in samples, often has a limited field of view and single plane information. This is due to the use of high NA microscope objectives and generally complex hardware setups, thus narrowing its practical effectiveness. This work introduces a Lensless Mueller Holographic Microscopy (LMHM) system that overcomes these limitations, enabling large field of view, volumetric multi-layer imaging, and Mueller matrix computation using in-line lens-free holography setup. The proposed system provides precision visualization of polarization information in samples, offering high-quality features due to the incorporation of a numerical multi-height Gerchberg-Saxton reconstruction algorithm with additional complex field filtering and a physical rotating diffuser. The proposed LMHM framework is validated with a calibrated USAF 1951 birefringent test target. A multiplane sample containing cloth fiber is utilized to study the LMHM capabilities of imaging volumetric samples. Finally, the LMHM is used to analyze two mice’s brain slices, effectively showcasing this organ’s anatomy. Among other structures in the brain, the proposed method easily allows the visualization of, e.g., the corpus callosum. These results constitute a proof-of-concept evaluation for bioimaging applications.

Wide-angle digital holography with aliasing-free recording

High-quality wide-angle holographic content is at the heart of the success of near-eye display technology. This work proposes the first digital holographic (DH) system enabling recording wide-angle scenes assembled from objects larger than the setup field of view (FOV), which can be directly replayed without 3D deformation in the near-eye display. The hologram formation in the DH system comprises free space propagation and Fourier transform (FT), which are connected by a rectangular aperture. First, the object wave propagates in free space to the rectangular aperture. Then, the band-limited wavefield is propagated through the single lens toward the camera plane. The rectangular aperture can take two sizes, depending on which DH operates in off-axis or phase-shifting recording mode. An integral part of the DH solution is a numerical reconstruction algorithm consisting of two elements: fringe processing for object wave recovery and wide-angle propagation to the object plane. The second element simulates propagation through both parts of the experimental system. The free space part is a space-limited angular spectrum compact space algorithm, while for propagation through the lens, the piecewise FT algorithm with Petzval curvature compensation is proposed. In the experimental part of the paper, we present the wide-angle DH system with FOV 25°×19°, which allows high-quality recording and reconstruction of large complex scenes.

NoQSM-net: Combining Convolutional Neural Network With Numerical Optimization Algorithm for Quantitative Susceptibility Mapping Reconstruction

NoQSM-net: Combining Convolutional Neural Network With Numerical Optimization Algorithm for Quantitative Susceptibility Mapping Reconstruction

Pore-scale modeling and investigation on the effect of calendering on lithium-ion battery cathodes

The cathode of lithium-ion batteries (LIBs) is a porous electrode that has a crucial influence on cell performance and durability. In making the electrode, a metal foil is first coated with a cathode slurry and dried. The electrode then undergoes a hard-pressing by rollers, i.e., calendering process. In the present study, the effects of calendering on the cathode's effective transport properties and its impact on the battery's charging performance are investigated using a multiscale approach. Microscopic images of LiNi0.8Co0.1Mn0.1O2 samples are obtained using X-ray computed tomography and SEM. This material's three-dimensional microstructure is generated using a numerical reconstruction algorithm based on the distribution characteristics of active material, additives, and pores observed from the microscopic images. Pore-scale simulations are performed on models of various calendering ratios (CRs) to evaluate the pore distribution, effective lithium-ion diffusivity, and electrical conductivity of the cathode material at different CRs. The computed effective transport properties of the cathode are applied to a macroscopic model to study the charging performance of LIBs at various CRs. It is found that the battery performance increases with CR and reaches optimal electrochemical performance at CR = 20 %. When the cathode is calendered to CR = 35 %, the battery's charging performance deteriorates, and this effect is more pronounced at high charging rates. The present study demonstrates a multiscale approach combining experiment characterization and pore-scale simulation for investigating the effects of calendering on LIB cathode. This approach can be employed for the future optimization of LIB fabrication processes.

Two-dimensional reconstruction of filament temperatures and densities with the thermal helium beam at ASDEX Upgrade

In the scrape-off layer (SOL) of fusion plasmas, radial particle and energy transport is mainly carried by plasma filaments. The plasma parameters of such filaments can be measured by the thermal helium beam diagnostic (THB). By means of an extended collisional radiative model (CRM) used for the evaluation of the diagnostic data, the light intensity response of neutral helium transitions from filaments is studied, which depends on the parameters of the filament and the background. The electron density of the filament is found to be the dominant parameter impacting the line intensities. By applying a numerical reconstruction algorithm based on the CRM, electron temperatures and densities as well as the sizes of experimentally measured filaments are obtained. This method allows for the first time measurements of the temporal decay rates of filament density and temperature in the co-moving frame of the filament and its radial propagation velocity.

Radon transform with Gaussian beam: Theoretical and numerical reconstruction scheme

The Radon transform and its various types have been studied since its introduction by Johann Radon in 1917. Since the Radon transform is an integral transform that maps a given function to its line integral, it has been studied in the field of computerized tomography, which deals with electromagnetic waves that primarily travel along straight lines, such as X-rays. However, in many laser optics applications, it is assumed that the laser beam is shaped like a Gaussian bell rather than a straight line. Therefore, in tomographic modalities using optical beams, such as optical projection tomography, images reconstructed with the inversion algorithms for the standard Radon transform are subject to a loss of quality. To address this issue, one needs to consider theoretical inversion methods for Radon transforms with Gaussian beam kernels and associated numerical reconstruction methods. In this study, we consider a Radon transform with a Gaussian beam kernel (also known as the point spread function) and show the uniqueness of the inversion of this transform. Furthermore, we provide an accurate and stable numerical reconstruction algorithm using the point spread function-sequential quadratic Hamiltonian scheme. Numerical experiments with disk and Shepp–Logan phantoms demonstrate that the proposed framework provides superior reconstructions compared to the traditional filtered back-projection algorithm.

Combined macroscopic and pore scale modeling of direct contact membrane distillation with micro-porous hydrophobic membranes

The transport processes of a direct contact membrane distillation (DCMD) module are investigated with a multiscale approach in this study. A stochastic, numerical reconstruction algorithm is first developed to generate 3D virtual membranes based on the membrane's pore size and fiber orientation distributions. Pore-scale simulations are then performed on the reconstructed membrane model. The membrane's effective thermal conductivity and mass diffusivity are computed using AVIZO software and OpenFOAM solver, respectively. A 2D macroscopic, multiphysics model considering the conservation of mass, species, momentum, and energy is developed, where the effective transport properties obtained from pore-scale simulations are utilized, to study the heat and mass transfer through the membrane. Two commercially available polytetrafluoroethylene (PTFE) membranes (GE-Osmonics/0.22 μm and MS-3010/0.45 μm) are chosen as examples for the present multiscale approach. The computed transport properties of the membranes are found to yield good distillate flux predictions with data reported in the literature. The MS-3010/0.45 μm membrane is found to produce 24% higher freshwater on average than GE-Osmonics/0.22 μm membrane. Finally, the performance of DCMD of three hypothetical membranes with different specifications is compared to the reconstructed MS-3010/0.45 μm membrane to study the factors that need to be taken into account for optimal design of the membrane for DCMD applications.

Dual-channel illumination surface plasmon resonance holographic microscopy for resolution improvement.

Surface plasmon resonance holographic microscopy (SPRHM), combining digital holographic microscopy with surface plasmon resonance (SPR), can simultaneously obtain the amplitude and phase distributions of the reflected beam carrying specimen information in SPR. Due to the decaying length of the surface plasmon wave as large as tens of micrometers, the spatial resolution of SPRHM is lower than that of ordinary optical microscopes. In this work, we propose a scheme to improve the spatial resolution of SPRHM by applying dual-channel SPR excitations. Through the polarization multiplexing technique, two holograms carrying the information of SPR excited in orthogonal directions are simultaneously acquired. Via a numerical reconstruction and filtering algorithm for holograms, the lateral spatial resolution of SPRHM can be effectively enhanced to reach nearly 1 µm at a wavelength of 632.8 nm. This is comparable to the resolution of traditional optical microscopes, while possessing the advantages of wide-field imaging and high measurement sensitivity of SPR.

High-throughput label-free cell detection and counting from diffraction patterns with deep fully convolutional neural networks.

.Significance: Digital holographic microscopy (DHM) is a promising technique for the study of semitransparent biological specimen such as red blood cells (RBCs). It is important and meaningful to detect and count biological cells at the single cell level in biomedical images for biomarker discovery and disease diagnostics. However, the biological cell analysis based on phase information of images is inefficient due to the complexity of numerical phase reconstruction algorithm applied to raw hologram images. New cell study methods based on diffraction pattern directly are desirable.Aim: Deep fully convolutional networks (FCNs) were developed on raw hologram images directly for high-throughput label-free cell detection and counting to assist the biological cell analysis in the future.Approach: The raw diffraction patterns of RBCs were recorded by use of DHM. Ground-truth mask images were labeled based on phase images reconstructed from RBC holograms using numerical reconstruction algorithm. A deep FCN, which is UNet, was trained on the diffraction pattern images to achieve the label-free cell detection and counting.Results: The implemented deep FCNs provide a promising way to high-throughput and label-free counting of RBCs with a counting accuracy of 99% at a throughput rate of greater than 288 cells per second and field of view at the single cell level. Compared to convolutional neural networks, the FCNs can get much better results in terms of accuracy and throughput rate.Conclusions: High-throughput label-free cell detection and counting were successfully achieved from diffraction patterns with deep FCNs. It is a promising approach for biological specimen analysis based on raw hologram directly.

Boundary Control and Observation to Inverse Coefficient Problem for Heat Equation With Unknown Source and Initial Value

This article investigates an inverse problem of determining the spatially variable diffusion coefficient of a one-dimensional heat equation with an unknown spatial varying source term and initial value. The big challenge of the problem comes from the multiple unknowns and very limited available data that are only boundary control and boundary observation at one end, in addition to the ill-posed nature of the inverse problem. We first design a switch <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on/off</small> boundary control and show that the diffusion coefficient can be uniquely determined by the boundary observation. Next, a stable numerical algorithm for reconstruction of the diffusion coefficient is proposed by means of the matrix pencil method and optimal perturbation regularization technique. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed identification algorithm.

Double moiré localized plasmon structured illumination microscopy

Abstract Structured illumination microscopy (SIM) is a well-established fluorescence imaging technique, which can increase spatial resolution by up to a factor of two. This article reports on a new way to extend the capabilities of structured illumination microscopy, by combining ideas from the fields of illumination engineering and nanophotonics. In this technique, plasmonic arrays of hexagonal symmetry are illuminated by two obliquely incident beams originating from a single laser. The resulting interference between the light grating and plasmonic grating creates a wide range of spatial frequencies above the microscope passband, while still preserving the spatial frequencies of regular SIM. To systematically investigate this technique and to contrast it with regular SIM and localized plasmon SIM, we implement a rigorous simulation procedure, which simulates the near-field illumination of the plasmonic grating and uses it in the subsequent forward imaging model. The inverse problem, of obtaining a super-resolution (SR) image from multiple low-resolution images, is solved using a numerical reconstruction algorithm while the obtained resolution is quantitatively assessed. The results point at the possibility of resolution enhancements beyond regular SIM, which rapidly vanishes with the height above the grating. In an initial experimental realization, the existence of the expected spatial frequencies is shown and the performance of compatible reconstruction approaches is compared. Finally, we discuss the obstacles of experimental implementations that would need to be overcome for artifact-free SR imaging.

Automated plankton classification from holographic imagery with deep convolutional neural networks

AbstractIn situ digital inline holography is a technique which can be used to acquire high‐resolution imagery of plankton and examine their spatial and temporal distributions within the water column in a nonintrusive manner. However, for effective expert identification of an organism from digital holographic imagery, it is necessary to apply a computationally expensive numerical reconstruction algorithm. This lengthy process inhibits real‐time monitoring of plankton distributions. Deep learning methods, such as convolutional neural networks, applied to interference patterns of different organisms from minimally processed holograms can eliminate the need for reconstruction and accomplish real‐time computation. In this article, we integrate deep learning methods with digital inline holography to create a rapid and accurate plankton classification network for 10 classes of organisms that are commonly seen in our data sets. We describe the procedure from preprocessing to classification. Our network achieves 93.8% accuracy when applied to a manually classified testing data set. Upon further application of a probability filter to eliminate false classification, the average precision and recall are 96.8% and 95.0%, respectively. Furthermore, the network was applied to 7500 in situ holograms collected at East Sound in Washington during a vertical profile to characterize depth distribution of the local diatoms. The results are in agreement with simultaneously recorded independent chlorophyll concentration depth profiles. This lightweight network exemplifies its capability for real‐time, high‐accuracy plankton classification and it has the potential to be deployed on imaging instruments for long‐term in situ plankton monitoring.

Partial data inverse problem with 𝐿^{𝑛/2} potentials

We construct an explicit Green’s function for the conjugated Laplacian e − ω ⋅ x / h Δ e − ω ⋅ x / h e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h} , which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ L n / 2 q \in L^{n/2} . Separately, we also use this Green’s function to derive L p L^p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

Kohn-Vogelius formulation and high-order topological asymptotic formula for identifying small obstacles in a fluid medium

Abstract This paper concerns the identification of a small obstacle immersed in a Stokes flow from boundary measurements. The proposed approach is based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. We derive a high order asymptotic formula describing the variation of a Kohn-Vogelius type functional with respect to the insertion of a small obstacle inside the fluid flow domain. The obtained asymptotic formula will serve as very useful tools for developing accurate and robust numerical reconstruction algorithms.

On implied volatility recovery of a time-fractional Black-Scholes equation for double barrier options

ABSTRACT This paper investigates a time-fractional Black-Scholes equation with an implied volatility which is assumed to be associated with underlying price. Two aspects are considered. One is for the forward problem, i.e. a robust -CDIA (-central difference implicit approximation) scheme is utilized to solve the initial boundary value problem effectively. Remarkably, convergence analysis for the forward problem solver is also given. The other is for the inverse problem, i.e. recovery of the implied volatility via additional data. By linearization which is similar as [Bouchouev et al. Quant. Finance, 2 (2002) 257–263.], a Fredholm integral equation of the first kind for the unknown volatility is obtained. Based upon the integral equation, a uniqueness of the recovery is shown under some assumptions and an efficient numerical reconstruction algorithm is proposed to reconstruct the coefficient. Numerical results for both direct problem and inverse problem are presented to illustrate the validity and effectiveness of the proposed schemes.

High-precision phase extraction of biological tissues based on lateral shear with zig-zag scanning in off-axis digital holography

We propose a method combining off-axis digital holography (DH) with one-dimensional (1D) lateral shear based on Zig-zag scanning to extract high-precision phase distribution of biological tissues. Firstly, object wavefront is reconstructed using the numerical reconstruction algorithm from digital hologram. Secondly, Zig-zag scanning is introduced to reorder the object wavefront from matrix to 1D vector, allowing the retrieved phase value to change smoothly without value saltation. Then, 1D lateral shear process is performed to acquire the shear vector, from which the phase difference vector is extracted by arctangent calculation. Given the small amount of shear, the phase difference vector is extracted without unwrapping process. Afterwards, the phase vector is obtained through numerical accumulation. Finally, the real phase distribution matrix is retrieved by inverse Zig-zag scanning. The proposed method has high precision in phase extraction of biological tissues. The simulated and experimental results demonstrate the feasibility and precision superiority of the proposed method.

Direct three-dimensional tomographic reconstruction and phase retrieval of far-field coherent diffraction patterns

We present an alternative numerical reconstruction algorithm for direct tomographic reconstruction of a sample refractive indices from the measured intensities of its far-field coherent diffraction patterns. We formulate the well-known phase-retrieval problem in ptychography in a tomographic framework which allows for simultaneous reconstruction of the illumination function and the sample refractive indices in three dimensions. Our iterative reconstruction algorithm is based on the Levenberg-Marquardt algorithm. We demonstrate the performance of our proposed method with simulation studies.

Reconstruction of digital rock considering micro-fracture based on multi-point statistics

The macroscopic transport properties of shale are determined by microscopic pore structure, which displays typical multi- scale characteristic with pore size ranging from nanometers to microns and well development of natural micro cracks. The physical experimental method is limited by the contradiction between the selected resolution and imaging sample size. Therefore it is significant to obtain the representative elementary volume to simultaneously incorporate nano-micro pore information and micro-fracture characteristics. To deal with this, numerical reconstruction algorithms are commonly applied to reconstruct pore structure based on easily obtained 2D images and in recent years are applied to reconstruct multi-scale shale porous media. The reconstruction algorithms mainly include the truncated Gaussian random field, simulated annealing, Markov chain Monte Carlo sequential indicator simulation, multiple-point statistics, phase-recovery algorithm and process-based method. In this study, a pattern based on multipoint statistics (MPS) method is proposed to account for multi-scale pore structure characteristics in shale. MPS comes from a statistical method to reconstruct large-scale patterns using pixel-based representations and can characterize the correlation between multiple points compared with traditional two-point statistics. First, different pore types and fractures are identified based on the gray scale shale image. Second, multi-scale shale images are constructed by containing formation of cracks with different properties as well as randomly pore-fracture distribution pattern and then treated as the basic training images. Next, the search tree is used to store and classify occurrences of all pore-fracture distribution patterns that are obtained after the multiple data template scanning the training image. Data template capturing pattern library and conditional probability functions are both stored in the search tree to reflect the long-range connectivity. Accordingly, a random path is defined to visit all unsampled voxels. For every new voxel, the data event is obtained by using the previous template and the conditional probability distribution function (CPDF) of this voxel is established from the search tree. The state value is taken stochastically and put into conditional events for the future voxel simulation. The following voxels are simulated on the random path until a new 2D image is produced. Finally, the type-method interpolation approach is employed to construct the potential simulation space. In order to evaluate the consequence of reconstruction, the constructed images are compared with the training image in terms of autocorrelation function, pore morphology, topology structure and permeability value. Different fracture distribution patterns including orthogonal fractures, dipping fractures and fractal rough fractures are generated. The structure and transport properties of generated pore-fracture models are compared with those of the training images to test the robustness of the proposed reconstruction algorithm. The results show that the constructed images display similar characteristics as training images and their property values are identical with each other under the conditions of adding different types of cracks, which verifies the rationality of proposed method.

Single-shot incoherent digital holography based on spatial light modulator

Fresnel incoherent correlation holography (FINCH) is a relatively innovative technology, which can achieve incoherent holograms by using the correlation between the object information and a Fresnel zone plate. In this method, the optical wave front scattered from an object propagates and is incident on a spatial light modulator which a phase mask is mounted on, and then the optical beam is split and phase shifted. The biggest advantage of the FINCH is that it can be matched with any standard optical imaging technology, which can realize microscopic imaging, telescopic imaging, spectroscopic imaging, etc. based on incoherent digital holography, and has important application prospect in remote sensing, astronomy, microscopy, and material analysis. In this paper, based on phase modulation characteristic of spatial light modulator, two types of masks are used. The first mask has an optical axis. And the results show that when the distribution intervals of the three phases on the spatial light modulator (SLM) are larger, the reconstruction image is clearer. On this basis, a new method of mode mounting on the SLM is put forward. The second mask has dual-lens array mode with three phases of 0°, 120°, and 240°, and the three phases respectively correspond to their corresponding optical axis, which means that the mask has three optical axes. Both of the two masks can achieve the single-shot of FINCH. By comparing the two mask forms, we find that the field-of-view of the first mask is larger, which can image the entire resolution board; however, because the sub-phase shift holograms are mixed together and cannot be extracted, the quality of the reconstructed image is worse. The second one can extract three sub-holograms, and the reconstructed image has better quality; but because of smaller imaging field of view, it is suitable for the real-time imaging of micro-organisms and objects. Experiments show that a compound digital hologram including three phase-shifting elements is recorded in charge-coupled device in this way. Three sub-holograms with different phase shift angles are extracted from the compound hologram, and there is no overlapping among the three phase shift holograms. Therefore, the three-phase-shifting technique is usually employed. The sample is reconstructed by numerical reconstruction algorithm. The proposed method may be useful in dynamic process real-time measurement and three-dimensional analysis of the object, and thus providing a new way to promote the development of incoherent digital holography.