Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography

Label-Free Three-Dimensional Morphological Characterization of Cell Death Using Holographic Tomography

This study presents a novel label-free approach for characterizing cell death states, eliminating the need for complex molecular labeling that may yield artificial or ambiguous results due to technical limitations in microscope resolution. The proposed holographic tomography technique offers a label-free avenue for capturing precise three-dimensional (3D) refractive index morphologies of cells and directly analyzing cellular parameters like area, height, volume, and nucleus/cytoplasm ratio within the 3D cellular model. We showcase holographic tomography results illustrating various cell death types and elucidate distinctive refractive index correlations with specific cell morphologies complemented by biochemical assays to verify cell death states. These findings hold promise for advancing in situ single cell state identification and diagnosis applications.

Dynamic tracking of onion-like carbon nanoparticles in cancer cells using limited-angle holographic tomography with self-supervised learning

This research presents a novel approach for the dynamic monitoring of onion-like carbon nanoparticles inside colorectal cancer cells. Onion-like carbon nanoparticles are widely used in photothermal cancer therapy, and precise 3D tracking of their distribution is crucial. We proposed a limited-angle digital holographic tomography technique with unsupervised learning to achieve rapid and accurate monitoring. A key innovation is our internal learning neural network. This network addresses the information limitations of limited-angle measurements by directly mapping coordinates to measured data and reconstructing phase information at unmeasured angles without external training data. We validated the network using standard SiO2 microspheres. Subsequently, we reconstructed the 3D refractive index of onion-like carbon nanoparticles within cancer cells at various time points. Morphological parameters of the nanoparticles were quantitatively analyzed to understand their temporal evolution, offering initial insights into the underlying mechanisms. This methodology provides a new perspective for efficiently tracking nanoparticles within cancer cells.

3D reconstruction of unstained weakly scattering cells from a single defocused hologram

We investigate the problem of 3D complex field reconstruction corresponding to unstained red blood cells (RBCs) with a single defocused off-axis digital hologram. The main challenge in this problem is the localization of cells to the correct axial range. While investigating the volume recovery problem for a continuous phase object like the RBC, we observe an interesting feature of the backpropagated field that it does not show a clear focusing effect. Therefore, sparsity enforcement within the iterative optimization framework using a single hologram data frame cannot effectively restrict the reconstruction to the true object volume. For phase objects, it is known that the amplitude contrast of the backpropagated object field at the focus plane is minimum. We use this information available in the recovered object field in the hologram plane to device depth-dependent weights that are proportional to the inverse of amplitude contrast. This weight function is employed in the iterative steps of the optimization algorithm to assist the object volume localization. The overall reconstruction process is performed using the mean gradient descent (MGD) framework. Experimental illustrations of 3D volume reconstruction of the healthy as well as malaria-infected RBCs are presented. A test sample of polystyrene microsphere bead is also used to validate the axial localization capability of the proposed iterative technique. The proposed methodology is simple to implement experimentally and provides an approximate tomographic solution, which is axially restricted and consistent with the object field data.

Near-infrared, wavelength, and illumination scanning holographic tomography

We present a holographic tomography technique in which the projections are acquired using both wavelength and illumination scanning in the near-infrared region. We show how to process the acquired data to obtain correct values of three-dimensional refractive index distributions in both single-wavelength and multi-wavelength data acquisition schemes and how to properly account for the dispersion of the sample. We perform numerical and experimental comparisons of different illumination scenarios to determine the most efficient measurement protocol. We show that the multi-wavelength protocol is advantageous in terms of signal-to-noise ratio and contrast-to-noise ratio over single-wavelength protocols, even for the same number of projections used for reconstructions. Finally, we show that this approach is suitable for providing high-quality refractive index distributions of relatively thick colon cancer samples.

Numerical refractive index correction for the stitching procedure in tomographic quantitative phase imaging

Tomographic quantitative phase imaging (QPI) lacks an absolute refractive index value baseline, which poses a problem when large dense objects extending over multiple fields of view are measured volume by volume and stitched together. Some of the measurements lack the natural baseline value that is provided by the mounting medium with a known refractive index. In this work, we discuss the problem of the refractive index (RI) baseline of individual reconstructed volumes that are deprived of access to mounting medium due to the extent of the object. The solution of this problem is provided by establishing the RI offsets based on the overlapping regions. We have proven that the process of finding the offset RI values may be justifiably reduced to the analogous procedure in the 2D baseline correction (2D-BC). Finally, we proposed the enhancement of the state-of-the-art 2D-BC procedure previously introduced in the context of 2D QPI. The processing is validated at the examples of a synthetic dataset and a liver organoid.

Developing a Reliable Holographic Flow Cyto-Tomography Apparatus by Optimizing the Experimental Layout and Computational Processing

Digital Holographic Tomography (DHT) has recently been established as a means of retrieving the 3D refractive index mapping of single cells. To make DHT a viable system, it is necessary to develop a reliable and robust holographic apparatus in order that such technology can be utilized outside of specialized optics laboratories and operated in the in-flow modality. In this paper, we propose a quasi-common-path lateral-shearing holographic optical set-up to be used, for the first time, for DHT in a flow-cytometer modality. The proposed solution is able to withstand environmental vibrations that can severely affect the interference process. Furthermore, we have scaled down the system while ensuring that a full 360° rotation of the cells occurs in the field-of-view, in order to retrieve 3D phase-contrast tomograms of single cells flowing along a microfluidic channel. This was achieved by setting the camera sensor at 45° with respect to the microfluidic direction. Additional optimizations were made to the computational elements to ensure the reliable retrieval of 3D refractive index distributions by demonstrating an effective method of tomographic reconstruction, based on high-order total variation. The results were first demonstrated using realistic 3D numerical phantom cells to assess the performance of the proposed high-order total variation method in comparison with the gold-standard algorithm for tomographic reconstructions: namely, filtered back projection. Then, the proposed DHT system and the processing pipeline were experimentally validated for monocytes and mouse embryonic fibroblast NIH-3T3 cells lines. Moreover, the repeatability of these tomographic measurements was also investigated by recording the same cell multiple times and quantifying the ability to provide reliable and comparable tomographic reconstructions, as confirmed by a correlation coefficient greater than 95%. The reported results represent various steps forward in several key aspects of in-flow DHT, thus paving the way for its use in real-world applications.

Numerical analysis of the effect of reduced temporal coherence in quantitative phase microscopy and tomography

We present the numerical analysis of the effect of the temporarily partially coherent illumination on the phase measurement accuracy in digital holography microscopy (DHM) and optical diffraction tomography (ODT), as reconstruction algorithms tend to assume purely monochromatic conditions. In the regime of reduced temporal coherence, we simulate the hologram formation in two different optical setups, representing classical off-axis two-beam and grating common-path configurations. We consider two ODT variants: with sample rotation and angle-scanning of illumination. Besides the coherence degree of illumination, our simulation considers the influence of the sample normal dispersion, shape of the light spectrum, and optical parameters of the imaging setup. As reconstruction algorithms we employ Fourier hologram method and first-order Rytov approximation with direct inversion and nonnegativity constraints. Quantitative evaluation of the measurement results deviations introduced by the mentioned error sources is comprehensively analyzed, for the first time to the best of our knowledge. Obtained outcomes indicate low final DHM/ODT reconstruction errors for the grating-assisted common-path configuration. Nevertheless, dispersion and asymmetric spectrum introduce non-negligible overestimated refractive index values and noise, and should be thus carefully considered within experimental frameworks.

Off-axis image plane hologram compression in holographic tomography - metrological assessment

In this paper, we present a novel study on the impact of lossy data compression on the metrological properties of holographic tomography reconstruction of the refractive index (RI). We use a spatial bandwidth-optimized compression procedure that leverages the properties of image plane off-axis holograms and standardized compression codecs, both widely applied in research and industry. The compression procedure is tested at multiple bitrates, for four different objects and against three reconstruction algorithms. The metrological evaluation is primarily done by comparison to the reconstruction from original data using the root-mean-squared error (RMSE). We show that due to differences between objects and different noise sensitivities of the reconstruction algorithms, the rate-distortion behaviour varies, but in most cases allows for the compression below 1 bit per pixel, while maintaining an RI RMSE less than 10^-4.

Optimizing sample illumination scanning for reflection and 4Pi tomographic diffractive microscopy

Tomographic diffractive microscopy (TDM) is increasingly gaining attention, owing to its high-resolution, label-free imaging capability. Fast acquisitions necessitate limiting the number of holograms to be recorded. Reconstructions then rely on optimal Fourier space filling to retain image quality and resolution, that is, they rely on optimal scanning of the tomographic illuminations. In this work, we theoretically study reflection TDM, and then the 4Pi TDM, a combination of transmission and reflection systems. Image simulations are conducted to determine optimal angular sweeping. We found that three-dimensional uniform scanning fills Fourier space the best for both reflection and 4Pi configurations, providing a better refractive index estimation for the observed sample.

Common-path intrinsically achromatic optical diffraction tomography

In this work we propose an open-top like common-path intrinsically achromatic optical diffraction tomography system. It operates as a total-shear interferometer and employs Ronchi-type amplitude diffraction grating, positioned in between the camera and the tube lens without an additional 4f system, generating three-beam interferograms with achromatic second harmonic. Such configuration makes the proposed system low cost, compact and immune to vibrations. We present the results of the measurements of 3D-printed cell phantom using laser diode (coherent) and superluminescent diode (partially coherent) light sources. Broadband light sources can be naturally employed without the need for any cumbersome compensation because of the intrinsic achromaticity of the interferometric recording (holograms generated by -1st and +1st conjugated diffraction orders are not affected by the illumination wavelength). The results show that the decreased coherence offers much reduced coherent noise and higher fidelity tomographic reconstruction especially when applied nonnegativity constraint regularization procedure.

DeepRegularizer: Rapid Resolution Enhancement of Tomographic Imaging Using Deep Learning

Optical diffraction tomography measures the three-dimensional refractive index map of a specimen and visualizes biochemical phenomena at the nanoscale in a non-destructive manner. One major drawback of optical diffraction tomography is poor axial resolution due to limited access to the three-dimensional optical transfer function. This missing cone problem has been addressed through regularization algorithms that use a priori information, such as non-negativity and sample smoothness. However, the iterative nature of these algorithms and their parameter dependency make real-time visualization impossible. In this article, we propose and experimentally demonstrate a deep neural network, which we term DeepRegularizer, that rapidly improves the resolution of a three-dimensional refractive index map. Trained with pairs of datasets (a raw refractive index tomogram and a resolution-enhanced refractive index tomogram via the iterative total variation algorithm), the three-dimensional U-net-based convolutional neural network learns a transformation between the two tomogram domains. The feasibility and generalizability of our network are demonstrated using bacterial cells and a human leukaemic cell line, and by validating the model across different samples. DeepRegularizer offers more than an order of magnitude faster regularization performance compared to the conventional iterative method. We envision that the proposed data-driven approach can bypass the high time complexity of various image reconstructions in other imaging modalities.

Limited-angle tomographic phase microscopy utilizing confocal scanning fluorescence microscopy

We present a multimodal imaging technique, combining tomographic phase microscopy with limited angular projection range and number, and two-channel spinning-disk confocal scanning fluorescence microscopy. This technique allows high-accuracy 3D refractive index (RI) profiling of live cells in spite of the missing projections. The cellular outer shape and its interior organelles measured by the confocal fluorescence imaging not only specify the cell in molecular levels, but also provide the 3D distributions of the whole cell as well as its organelles. We take these additional 3D morphological details as constraints in Gerchberg-Papoulis-based optical diffraction tomography algorithm. We then obtain an accurate 3D RI tomogram, even with a sparse angular range having a small number of perspective projections, otherwise providing low-accuracy RI reconstruction. Then, we obtain both cellular molecular specificity and inner RI values of the cell and its organelles. We compare the reconstructed 3D RI profiles of various samples, demonstrating the superiority of the proposed technique.

Refractive Index Changes of Cells and Cellular Compartments Upon Paraformaldehyde Fixation Acquired by Tomographic Phase Microscopy

Three-dimensional quantitative phase imaging is an emerging method, which provides the 3D distribution of the refractive index (RI) and the dry mass in live and fixed cells as well as in tissues. However, an insufficiently answered question is the influence of chemical cell fixation procedures on the results of RI reconstructions. Therefore, this work is devoted to systematic investigations on the RI in cellular organelles of live and fixed cells including nucleus, nucleolus, nucleoplasm, and cytoplasm. The research was carried out on four different cell lines using a common paraformaldehyde (PFA)-based fixation protocol. The selected cell types represent the diversity of mammalian cells and therefore the results presented provide a picture of fixation caused RI changes in a broader context. A commercial Tomocube HT-1S device was used for 3D RI acquisition. The changes in the RI values after the fixation process are detected in the reconstructed phase distributions and amount to the order of 10-3 . The RI values decrease and the observed RI changes are found to be different between various cell lines; however, all of them show the most significant loss in the nucleolus. In conclusion, our study demonstrates the evident need for standardized preparation procedures in phase tomographic measurements. © 2020 The Authors. Cytometry Part A published by Wiley Periodicals LLC. on behalf of International Society for Advancement of Cytometry.

Biological Applications of Short Wavelength Microscopy Based on Compact, Laser-Produced Gas-Puff Plasma Source

Over the last decades, remarkable efforts have been made to improve the resolution in photon-based microscopes. The employment of compact sources based on table-top laser-produced soft X-ray (SXR) in the “water window” spectral range (λ = 2.3–4.4 nm) and extreme ultraviolet (EUV) plasma allowed to overcome the limitations imposed by large facilities, such as synchrotrons and X-ray free electron lasers (XFEL), because of their high complexity, costs, and limited user access. A laser-plasma double stream gas-puff target source represents a powerful tool for microscopy operating in transmission mode, significantly improving the spatial resolution into the nanometric scale, comparing to the traditional visible light (optical) microscopes. Such an approach allows generating the plasma efficiently, without debris, providing a high flux of EUV and SXR photons. In this review, we present the development and optimization of desktop imaging systems: a EUV and an SXR full field microscope, allowing to achieve a sub-50 nm spatial resolution with short exposure time and an SXR contact microscope, capable to resolve internal structures in a thin layer of sensitive photoresist. Details about the source, as well as imaging results for biological applications, will be presented and discussed.

Variational Hilbert Quantitative Phase Imaging

Utilizing the refractive index as the endogenous contrast agent to noninvasively study transparent cells is a working principle of emerging quantitative phase imaging (QPI). In this contribution, we propose the Variational Hilbert Quantitative Phase Imaging (VHQPI)-end-to-end purely computational add-on module able to improve performance of a QPI-unit without hardware modifications. The VHQPI, deploying unique merger of tailored variational image decomposition and enhanced Hilbert spiral transform, adaptively provides high quality map of sample-induced phase delay, accepting particularly wide range of input single-shot interferograms (from off-axis to quasi on-axis configurations). It especially promotes high space-bandwidth-product QPI configurations alleviating the spectral overlapping problem. The VHQPI is tailored to deal with cumbersome interference patterns related to detailed locally varying biological objects with possibly high dynamic range of phase and relatively low carrier. In post-processing, the slowly varying phase-term associated with the instrumental optical aberrations is eliminated upon variational analysis to further boost the phase-imaging capabilities. The VHQPI is thoroughly studied employing numerical simulations and successfully validated using static and dynamic cells phase-analysis. It compares favorably with other single-shot phase reconstruction techniques based on the Fourier and Hilbert-Huang transforms, both in terms of visual inspection and quantitative evaluation, potentially opening up new possibilities in QPI.

Diffraction tomography with a deep image prior

We present a tomographic imaging technique, termed Deep Prior Diffraction Tomography (DP-DT), to reconstruct the 3D refractive index (RI) of thick biological samples at high resolution from a sequence of low-resolution images collected under angularly varying illumination. DP-DT processes the multi-angle data using a phase retrieval algorithm that is extended by a deep image prior (DIP), which reparameterizes the 3D sample reconstruction with an untrained, deep generative 3D convolutional neural network (CNN). We show that DP-DT effectively addresses the missing cone problem, which otherwise degrades the resolution and quality of standard 3D reconstruction algorithms. As DP-DT does not require pre-captured data or pre-training, it is not biased towards any particular dataset. Hence, it is a general technique that can be applied to a wide variety of 3D samples, including scenarios in which large datasets for supervised training would be infeasible or expensive. We applied DP-DT to obtain 3D RI maps of bead phantoms and complex biological specimens, both in simulation and experiment, and show that DP-DT produces higher-quality results than standard regularization techniques. We further demonstrate the generality of DP-DT, using two different scattering models, the first Born and multi-slice models. Our results point to the potential benefits of DP-DT for other 3D imaging modalities, including X-ray computed tomography, magnetic resonance imaging, and electron microscopy.

Optical diffraction tomography with finite object support for the minimization of missing cone artifacts

Limited-angle optical diffraction tomography suffers from strong artifacts in tomographic reconstructions. Numerous algorithms, mainly based on regularization methods, have been developed recently to overcome this limitation. However, the quality of results still needs further improvement. Here I present a simple yet extremely effective method of increasing the reconstruction quality in limited angle optical diffraction tomography that can be combined with known tomographic algorithms. In the method a finite object support is generated from the object data and utilized in the reconstruction procedure as an additional strong regularizer. Practical aspects of this method are given together with examples of application.

Fourier ptychography: current applications and future promises

Traditional imaging systems exhibit a well-known trade-off between the resolution and the field of view of their captured images. Typical cameras and microscopes can either "zoom in" and image at high-resolution, or they can "zoom out" to see a larger area at lower resolution, but can rarely achieve both effects simultaneously. In this review, we present details about a relatively new procedure termed Fourier ptychography (FP), which addresses the above trade-off to produce gigapixel-scale images without requiring any moving parts. To accomplish this, FP captures multiple low-resolution, large field-of-view images and computationally combines them in the Fourier domain into a high-resolution, large field-of-view result. Here, we present details about the various implementations of FP and highlight its demonstrated advantages to date, such as aberration recovery, phase imaging, and 3D tomographic reconstruction, to name a few. After providing some basics about FP, we list important details for successful experimental implementation, discuss its relationship with other computational imaging techniques, and point to the latest advances in the field while highlighting persisting challenges.

3D-printed biological cell phantom for testing 3D quantitative phase imaging systems

As the 3D quantitative phase imaging (QPI) methods mature, their further development calls for reliable tools and methods to characterize and compare their metrological parameters. We use refractive index engineering during two-photon laser photolithography to fabricate a life-scale phantom of a biological cell with internal structures that mimic optical and structural properties of mammalian cells. After verification with a number of reference techniques, the phantom is used to characterize the performance of a limited-angle holographic tomography microscope.

Projection extrapolation routine for tight-frame limited-angle optical diffraction tomography

We propose a data-replenishment-type expansion of the modified Gerchberg-Papoulis (GP) algorithm for limited-angle optical diffraction tomography (LAODT), which prevents artifact buildup in the GP reconstructions of confined bulk objects tightly fitting the active field of view (FoV) of the LAODT microscope. Objects crossing the FoV borders are not considered. The method relies on a Fourier-based forward projector complementary to the GP solver with no additional constraints. Fourier space regridding errors are minimized by means of one-dimensional oversampling in the axial direction, which is demonstrated to be more efficient than standard projection padding. Verification of both synthetic and experimental sinograms confirms the ability of the procedure to deduce missing projection parts necessary for the correct reconstruction.

A framework for directional and higher-order reconstruction in photoacoustic tomography

Photoacoustic tomography is a hybrid imaging technique that combines high optical tissue contrast with high ultrasound resolution. Direct reconstruction methods such as filtered back-projection, time reversal and least squares suffer from curved line artefacts and blurring, especially in the case of limited angles or strong noise. In recent years, there has been great interest in regularised iterative methods. These methods employ prior knowledge of the image to provide higher quality reconstructions. However, easy comparisons between regularisers and their properties are limited, since many tomography implementations heavily rely on the specific regulariser chosen. To overcome this bottleneck, we present a modular reconstruction framework for photoacoustic tomography, which enables easy comparisons between regularisers with different properties, e.g. nonlinear, higher-order or directional. We solve the underlying minimisation problem with an efficient first-order primal-dual algorithm. Convergence rates are optimised by choosing an operator-dependent preconditioning strategy. A variety of reconstruction methods are tested on challenging 2D synthetic and experimental data sets. They outperform direct reconstruction approaches for strong noise levels and limited angle measurements, offering immediate benefits in terms of acquisition time and quality. This work provides a basic platform for the investigation of future advanced regularisation methods in photoacoustic tomography.

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.

Illumination-related errors in limited-angle optical diffraction tomography

In the paper, the design and tolerances of optical systems and scanning components used in limited-angle optical diffraction tomography are analyzed in order to improve the performance of the measurement systems and to encourage the application of tomography as a standard method for quantitative analysis of 3D refractive index distribution in biological microstructures. The first part of the presented analysis consists of component selection for the scanning device and optical system in the illumination part of the setup and the influence of the illumination wavefront on reconstruction quality. In the second part, the sensitivity of the tomographic reconstruction quality to three representative measurement-related errors based on synthetic data is demonstrated. Finally, a configuration of the system, selected to minimize reconstruction errors, is proposed and alignment tolerances simulated using the Monte Carlo method are provided.

Morphometric analysis of erythrocytes from patients with thalassemia using tomographic diffractive microscopy

Complete blood count is the most common test to detect anemia, but it is unable to obtain the abnormal shape of erythrocytes, which highly correlates with the hematologic function. Tomographic diffractive microscopy (TDM) is an emerging technique capable of quantifying three-dimensional (3-D) refractive index (RI) distributions of erythrocytes without labeling. TDM was used to characterize optical and morphological properties of 172 erythrocytes from healthy volunteers and 419 erythrocytes from thalassemic patients. To efficiently extract and analyze the properties of erythrocytes, we developed an adaptive region-growing method for automatically delineating erythrocytes from 3-D RI maps. The thalassemic erythrocytes not only contained lower hemoglobin content but also showed doughnut shape and significantly lower volume, surface area, effective radius, and average thickness. A multi-indices prediction model achieved perfect accuracy of diagnosing thalassemia using four features, including the optical volume, surface-area-to-volume ratio, sphericity index, and surface area. The results demonstrate the ability of TDM to provide quantitative, hematologic measurements and to assess morphological features of erythrocytes to distinguish healthy and thalassemic erythrocytes.

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.

Optically driven full-angle sample rotation for tomographic imaging in digital holographic microscopy

This study presents a novel tomographic imaging technique for living biomedical samples using an optically driven full-angle rotation scheme based on digital holographic microscopy, in which the three-dimensional refractive index distribution inside the sample can be measured and analyzed. To accomplish the full-angle sample rotation, two optical traps are driven by highly focused spots on the top and bottom of the sample. The rim image of the sample outside the focal depth at the different rotation angles and propagation distances can be corrected and compensated, respectively, via numerical focusing; therefore, tomographic imaging of the sample can be conducted. The proposed approach shows that an entire symmetric spectrum can be acquired for tomographic reconstruction without the missing apple core problem as in traditional sample-rotation schemes. The three-dimensional refractive index of living yeast in a fluid medium is measured and verified.

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.

Angular spectrum method with compact space-bandwidth: generalization and full-field accuracy

A recent Letter [Opt. Lett.40, 3420 (2015)OPLEDP0146-959210.1364/OL.40.003420] reported a modified angular spectrum method that uses a sampling scheme based on a compact space-bandwidth product representation. That technique is useful for focusing and defocusing propagation cases and is generalized here for the case of propagation between two defocus planes. The proposed method employs paraxial spherical phase factors and modified propagation kernels to reduce the size of the numerical space-bandwidth product needed for wave field calculations. A Wigner distribution analysis is carried out in order to ensure high accuracy of the calculations in the entire computational domain. This is achieved by analyzing the evolution of the generalized space-bandwidth product when passing through the propagation algorithm for various space-frequency constraints. The results allow the derivations of sampling criteria, and, despite this, also show that a small amount of space/frequency zero padding significantly extends the capability of the recently reported modified angular spectrum method. Simulations validate the high accuracy of that method and verify a computational and memory gain of more than two orders of magnitude when comparing this technique with the conventional angular spectrum method.