Image formation in holographic tomography: high-aperture imaging conditions

Recent Advances and Current Trends in Transmission Tomographic Diffraction Microscopy

Optical microscopy techniques are among the most used methods in biomedical sample characterization. In their more advanced realization, optical microscopes demonstrate resolution down to the nanometric scale. These methods rely on the use of fluorescent sample labeling in order to break the diffraction limit. However, fluorescent molecules' phototoxicity or photobleaching is not always compatible with the investigated samples. To overcome this limitation, quantitative phase imaging techniques have been proposed. Among these, holographic imaging has demonstrated its ability to image living microscopic samples without staining. However, for a 3D assessment of samples, tomographic acquisitions are needed. Tomographic Diffraction Microscopy (TDM) combines holographic acquisitions with tomographic reconstructions. Relying on a 3D synthetic aperture process, TDM allows for 3D quantitative measurements of the complex refractive index of the investigated sample. Since its initial proposition by Emil Wolf in 1969, the concept of TDM has found a lot of applications and has become one of the hot topics in biomedical imaging. This review focuses on recent achievements in TDM development. Current trends and perspectives of the technique are also discussed.

Multi-slice ptychographic imaging with multistage coarse-to-fine reconstruction

The ability to image 3D samples with optical sectioning is essential for the study of tomographic morphology in material and biological sciences. However, it is often hampered by limitations of acquisition speed and equipment complexity when performing 3D volumetric imaging. Here, we propose, to the best of our knowledge, a new method for 3D reconstruction from a minimum of four intensity-only measurements. The complementary structured patterns provided by the digital micromirror device (DMD) irradiate the outermost layer of the sample to generate the corresponding diffraction intensities for recording, which enables rapid scanning of loaded patterns for fast acquisition. Our multistage reconstruction algorithm first extracts the overall coarse-grained information, and then iteratively optimizes the information of different layers to obtain fine features, thereby achieving high-resolution 3D tomography. The high-fidelity reconstruction in experiments on two-slice resolution targets, unstained Polyrhachis vicina Roger and freely moving C. elegans proves the robustness of the method. Compared with traditional 3D reconstruction methods such as interferometry-based methods or Fourier ptychographic tomography (FPT), our method increases the reconstruction speed by at least 10 times and is suitable for label-free dynamic imaging in multiple-scattering samples. Such 3D reconstruction suggests potential applications in a wide range of fields.

Optimizing sample illumination scanning in transmission tomographic diffractive microscopy

Due to the sequential nature of data acquisition, it is preferable to limit the number of illuminations to be used in tomographic diffractive microscopy experiments, especially if fast imaging is foreseen. On the other hand, for high-quality, high-resolution imaging, the Fourier space has to be optimally filled. Up to now, the problem of optimal Fourier space filling has not been investigated in itself. In this paper, we perform a comparative study to analyze the effect of sample scanning patterns on Fourier space filling for a transmission setup. Optical transfer functions for several illumination patterns are studied. Simulation as well as experiments are conducted to compare associated image reconstructions. We found that 3D uniform angular sweeping best fills the Fourier space, leading to better quality images.

Iterative optical diffraction tomography for illumination scanning configuration

Optical diffraction tomography (ODT) is used to reconstruct refractive-index distributions from multiple measurements in the object rotating configuration (ORC) or the illumination scanning configuration (ISC). Because of its fast data acquisition and stability, ISC-based ODT has been widely used for biological imaging. ODT typically fails to reconstruct multiply-scattering samples. The previously developed iterative ODT (iODT) was for the multiply-scattering objects in ORC, and could not be directly applied to ISC. To resolve this mismatch, we developed an ISC update and numerically demonstrated its accuracy. With the same prior knowledge, iODT-ISC outperforms conventional ODT in resolving the missing-angle problem.

Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem

Diffraction calculations are widely used in applications that require numerical simulation of optical wave propagation. Different numerical diffraction calculation methods have their own transform and sampling properties. In this study, we provide a unified analysis where five popular fast diffraction calculation methods are analyzed from the perspective of phase space optics and the sampling theorem: single fast Fourier transform-based Fresnel transform, Fresnel transfer function approach, Fresnel impulse response approach, angular spectrum method, and Rayleigh-Sommerfeld convolution. The evolutions of an input signal's space-bandwidth product (SBP) during wave propagation are illustrated with the help of a phase space diagram (PSD) and an ABCD matrix. It is demonstrated that all of the above methods cannot make full use of the SBP of the input signal after diffraction; and some transform properties have been ignored. Each method has its own restrictions and applicable range. The reason why different methods have different applicable ranges is explained with physical models. After comprehensively studying and comparing the effect on the SBP and sampling properties of these methods, suggestions are given for choosing the proper method for different applications and overcoming the restrictions of corresponding methods. The PSD and ABCD matrix are used to illustrate the properties of these methods intuitively. Numerical results are presented to verify the analysis, and potential ways to develop new diffraction calculation methods are also discussed.

High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images

Optical diffraction tomography (ODT) reconstructs a sample's volumetric refractive index (RI) to create high-contrast, quantitative 3D visualizations of biological samples. However, standard implementations of ODT use interferometric systems, and so are sensitive to phase instabilities, complex mechanical design, and coherent noise. Furthermore, their reconstruction framework is typically limited to weakly scattering samples, and thus excludes a whole class of multiple-scattering samples. Here, we implement a new 3D RI microscopy technique that utilizes a computational multi-slice beam propagation method to invert the optical scattering process and reconstruct high-resolution (NA > 1.0) 3D RI distributions of multiple-scattering samples. The method acquires intensity-only measurements from different illumination angles and then solves a nonlinear optimization problem to recover the sample's 3D RI distribution. We experimentally demonstrate the reconstruction of samples with varying amounts of multiple-scattering: a 3T3 fibroblast cell, a cluster of C. elegans embryos, and a whole C. elegans worm, with lateral and axial resolutions of ≤ 240 nm and ≤ 900 nm, respectively. The results of this work lays groundwork for future studies into using optical wavelengths to probe 3D RI distributions of highly scattering biological organisms.

Tomographic phase microscopy: principles and applications in bioimaging [Invited]

Tomographic phase microscopy (TPM) is an emerging optical microscopic technique for bioimaging. TPM uses digital holographic measurements of complex scattered fields to reconstruct three-dimensional refractive index (RI) maps of cells with diffraction-limited resolution by solving inverse scattering problems. In this paper, we review the developments of TPM from the fundamental physics to its applications in bioimaging. We first provide a comprehensive description of the tomographic reconstruction physical models used in TPM. The RI map reconstruction algorithms and various regularization methods are discussed. Selected TPM applications for cellular imaging, particularly in hematology, are reviewed. Finally, we examine the limitations of current TPM systems, propose future solutions, and envision promising directions in biomedical research.

Beyond Born-Rytov limit for super-resolution optical diffraction tomography

Optical diffraction tomography (ODT) using Born or Rytov approximation suffers from severe distortions in reconstructed refractive index (RI) tomograms when multiple scattering occurs or the scattering signals are strong. These effects are usually seen as a significant impediment to the application of ODT because multiple scattering is directly linked to an unknown object itself rather than a surrounding medium, and a strong scatter invalidates the underlying assumptions of the Born and Rytov approximations. The focus of this article is to demonstrate for the first time that multiple scattering and high material contrast, if handled aptly, can significantly improve the image quality of the ODT thanks to multiple scattering inside a sample. Experimental verification using various phantom and biological cells substantiates that we not only revealed the structures that were not observable using the conventional approaches but also resolved the long-standing problem of missing cones in the ODT.

Holographic tomography with object rotation and two-directional off-axis illumination

A hybrid system of holographic tomography, which utilizes rotation of a sample and two-directional, off-axis illumination is proposed. The applied type of illumination brings two major benefits. First, it offers theoretical potential for the resolution improvement with respect to conventional tomography. Second, it enables effective, numerical compensation of the defocus error, which is achieved with an accurate, noise-immune autofocusing. Hence, the main practical obstacle of hybrid tomography is removed and its high-resolution potential is put into practice. The utility of the proposed concept is experimentally demonstrated with the tomographic measurement of a photonic crystal fiber.

Modeling the depth-sectioning effect in reflection-mode dynamic speckle-field interferometric microscopy

Unlike most optical coherence microscopy (OCM) systems, dynamic speckle-field interferometric microscopy (DSIM) achieves depth sectioning through the spatial-coherence gating effect. Under high numerical aperture (NA) speckle-field illumination, our previous experiments have demonstrated less than 1 μm depth resolution in reflection-mode DSIM, while doubling the diffraction limited resolution as under structured illumination. However, there has not been a physical model to rigorously describe the speckle imaging process, in particular explaining the sectioning effect under high illumination and imaging NA settings in DSIM. In this paper, we develop such a model based on the diffraction tomography theory and the speckle statistics. Using this model, we calculate the system response function, which is used to further obtain the depth resolution limit in reflection-mode DSIM. Theoretically calculated depth resolution limit is in an excellent agreement with experiment results. We envision that our physical model will not only help in understanding the imaging process in DSIM, but also enable better designing such systems for depth-resolved measurements in biological cells and tissues.

Holographic tomography with scanning of illumination: space-domain reconstruction for spatially invariant accuracy

The paper presents two novel, space-domain reconstruction algorithms for holographic tomography utilizing scanning of illumination and a fixed detector that is highly suitable for imaging of living biomedical specimens. The first proposed algorithm is an adaptation of the filtered backpropagation to the scanning illumination tomography. Its space-domain implementation enables avoiding the error-prone interpolation in the Fourier domain, which is a significant problem of the state-of-the-art tomographic algorithm. The second proposed algorithm is a modified version of the former, which ensures the spatially invariant reconstruction accuracy. The utility of the proposed algorithms is demonstrated with numerical simulations and experimental measurement of a cancer cell.

Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors

This paper explores the concept of image-plane holographic tomography applied to the measurements of laser-induced thermal gradients in an aqueous solution of a photosensitizer with respect to the reconstruction accuracy of three-dimensional variations of the refractive index. It uses the least-squares estimation algorithm to reconstruct refractive index variations in each holographic projection. Along with the bitelecentric optical system, transferring focused projection to the sensor plane, it facilitates the elimination of diffraction artifacts and noise suppression. This work estimates the influence of typical random and systematic errors in experiments and concludes that random errors such as accidental measurement errors or noise presence can be significantly suppressed by increasing the number of recorded digital holograms. On the contrary, even comparatively small systematic errors such as a displacement of the rotation axis projection in the course of a reconstruction procedure can significantly distort the results.

Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography

In optical tomography, there exist certain spatial frequency components that cannot be measured due to the limited projection angles imposed by the numerical aperture of objective lenses. This limitation, often called as the missing cone problem, causes the under-estimation of refractive index (RI) values in tomograms and results in severe elongations of RI distributions along the optical axis. To address this missing cone problem, several iterative reconstruction algorithms have been introduced exploiting prior knowledge such as positivity in RI differences or edges of samples. In this paper, various existing iterative reconstruction algorithms are systematically compared for mitigating the missing cone problem in optical diffraction tomography. In particular, three representative regularization schemes, edge preserving, total variation regularization, and the Gerchberg-Papoulis algorithm, were numerically and experimentally evaluated using spherical beads as well as real biological samples; human red blood cells and hepatocyte cells. Our work will provide important guidelines for choosing the appropriate regularization in ODT.

Enhanced 3D spatial resolution in quantitative phase microscopy using spatially incoherent illumination

We describe the use of spatially incoherent illumination to make quantitative phase imaging of a semi-transparent sample, even out of the paraxial approximation. The image volume electromagnetic field is collected by scanning the image planes with a quadriwave lateral shearing interferometer, while the sample is spatially incoherently illuminated. In comparison to coherent quantitative phase measurements, incoherent illumination enriches the 3D collected spatial frequencies leading to 3D resolution increase (up to a factor 2). The image contrast loss introduced by the incoherent illumination is simulated and used to compensate the measurements. This restores the quantitative value of phase and intensity. Experimental contrast loss compensation and 3D resolution increase is presented using polystyrene and TiO(2) micro-beads. Our approach will be useful to make diffraction tomography reconstruction with a simplified setup.

Modeling quantitative phase image formation under tilted illuminations

A generalized product-of-convolution model for simulation of quantitative phase microscopy of thick heterogeneous specimen under tilted plane-wave illumination is presented. Actual simulations are checked against a much more time-consuming commercial finite-difference time-domain method. Then modeled data are compared with experimental measurements that were made with a quadriwave lateral shearing interferometer.

Partially coherent lensfree tomographic microscopy [Invited]

Optical sectioning of biological specimens provides detailed volumetric information regarding their internal structure. To provide a complementary approach to existing three-dimensional (3D) microscopy modalities, we have recently demonstrated lensfree optical tomography that offers high-throughput imaging within a compact and simple platform. In this approach, in-line holograms of objects at different angles of partially coherent illumination are recorded using a digital sensor-array, which enables computing pixel super-resolved tomographic images of the specimen. This imaging modality, which forms the focus of this review, offers micrometer-scale 3D resolution over large imaging volumes of, for example, 10-15 mm(3), and can be assembled in light weight and compact architectures. Therefore, lensfree optical tomography might be particularly useful for lab-on-a-chip applications as well as for microscopy needs in resource-limited settings.

Realistic 3D coherent transfer function inverse filtering of complex fields

We present a novel technique for three-dimensional (3D) image processing of complex fields. It consists in inverting the coherent image formation by filtering the complex spectrum with a realistic 3D coherent transfer function (CTF) of a high-NA digital holographic microscope. By combining scattering theory and signal processing, the method is demonstrated to yield the reconstruction of a scattering object field. Experimental reconstructions in phase and amplitude are presented under non-design imaging conditions. The suggested technique is best suited for an implementation in high-resolution diffraction tomography based on sample or illumination rotation.

Quantitative phase restoration by direct inversion using the optical transfer function

Quantitative phase recovery of phase objects is achieved by a direct inversion using the defocused weak object transfer function. The presented method is noniterative and is based on partially coherent principles. It also takes into account the optical properties of the system and gives the phase of the object directly. The proposed method is especially suitable for application to weak phase objects, such as live and unstained biological samples but, surprisingly, has also been shown to work with comparatively strong phase objects.