# Publications

JOURNAL PUBLICATIONS

**Deep speckle correlation: a deep learning approach towards scalable imaging through scattering media
**Yunzhe Li, Yujia Xue, Lei Tian

arXiv:1806.04139, 2018

Imaging through scattering is an important, yet challenging problem. Tremendous progress has been made by exploiting the deterministic input-output relation for a static medium. However, this approach is highly susceptible to speckle decorrelations – small perturbations to the scattering medium lead to model errors and severe degradation of the imaging performance. In addition, this is complicated by the large number of phase-sensitive measurements required for characterizing the input-output ‘transmission matrix’. Our goal here is to develop a new framework that is highly scalable to both medium perturbations and measurement requirement. To do so, we abandon the traditional deterministic approach, instead propose a statistical framework that permits higher representation power to encapsulate a wide range of statistical variations needed for model generalization. Specifically, we develop a convolutional neural network (CNN) that takes intensity-only speckle patterns as input and predicts unscattered object as output. Importantly, instead of characterizing a single input-output relation of a fixed medium, we train our CNN to learn statistical information contained in several scattering media of the same class. We then show that the CNN is able to generalize over a completely different set of scattering media from the same class, demonstrating its superior adaptability to medium perturbations. In our proof of concept experiment, we first train our CNN using speckle patterns captured on diffusers having the same macroscopic parameter (e.g. grits); the trained CNN is then able to make high-quality reconstruction from speckle patterns that were captured from an entirely different set of diffusers of the same grits. To investigate the physical underpinnings of our CNN, we conduct correlation analysis and show that the captured speckle patterns, although are decorrelated (e.g. < e^{−1} ) using the classical Pearson correlation coefficient metric, still contain statistically invariant information. These invariance is hard to invert using deterministic models, but can be effectively utilized using our statistical CNN model. Our work paves the way to a highly scalable deep learning approach for imaging through scattering media.

**Convolutional neural network for Fourier ptychography video reconstruction: learning temporal dynamics from spatial ensembles
**Thanh Nguyen, Yujia Xue, Yunzhe Li, Lei Tian, George Nehmetallah

arXiv:1805.00334

Convolutional neural networks (CNNs) have gained tremendous success in solving complex inverse problems for both problems involving independent datasets from input-output pairs of static objects, as well as sequential datasets from dynamic objects. In order to learn the underlying temporal statistics, a video sequence is typically used at the cost of network complexity and computation. The aim of this work is to develop a novel CNN framework to reconstruct video sequence of dynamic live cells captured using a computational microscopy technique, Fourier ptychographic microscopy (FPM). The unique feature of the FPM is its capability to reconstruct images with both wide field-of-view (FOV) and high resolution, i.e. a large space-bandwidth-product (SBP), by taking a series of low resolution intensity images. For live cell imaging, a single FPM frame contains thousands of cell samples with different morphological features. Our idea is to fully exploit the statistical information provided by this large spatial ensembles so as to learn temporal information in a sequential measurement, without using any additional temporal dataset. Specifically, we show that it is possible to reconstruct high-SBP dynamic cell videos by a CNN trained only on the first FPM dataset captured at the beginning of a time-series experiment. Our CNN approach reconstructs a 12800×10800 pixels phase image using only ∼25 seconds, a 50× speedup compared to the model-based FPM algorithm. In addition, the CNN further reduces the required number of images in each time frame by ∼ 6×. Overall, this significantly improves the imaging throughput by reducing both the acquisition and computational times. The proposed CNN is based on the conditional generative adversarial network (cGAN) framework. We further propose a mixed loss function that combines the standard image domain loss and a weighted Fourier domain loss, which leads to improved reconstruction of the high frequency information. Our technique demonstrates a promising deep learning approach to continuously monitor large live-cell populations over an extended time and gather useful spatial and temporal information with sub-cellular resolution.

**High-throughput intensity diffraction tomography with a computational microscope
**Ruilong Ling, Waleed Tahir, Hsing-Ying Lin, Hakho Lee, and Lei Tian

Biomed. Opt. Express 9, 2130-2141 (2018).

We demonstrate a motion-free intensity diffraction tomography technique that enables the direct inversion of 3D phase and absorption from intensity-only measurements for weakly scattering samples. We derive a novel linear forward model featuring slice-wise phase and absorption transfer functions using angled illumination. This new framework facilitates flexible and efficient data acquisition, enabling arbitrary sampling of the illumination angles. The reconstruction algorithm performs 3D synthetic aperture using a robust computation and memory efficient slice-wise deconvolution to achieve resolution up to the incoherent limit. We demonstrate our technique with thick biological samples having both sparse 3D structures and dense cell clusters. We further investigate the limitation of our technique when imaging strongly scattering samples. Imaging performance and the influence of multiple scattering is evaluated using a 3D sample consisting of stacked phase and absorption resolution targets. This computational microscopy system is directly built on a standard commercial microscope with a simple LED array source add-on, and promises broad applications by leveraging the ubiquitous microscopy platforms with minimal hardware modifications.

**Structured illumination microscopy with unknown patterns and a statistical prior**

Li-Hao Yeh, Lei Tian, and Laura Waller

Biomed. Opt. Express 8, 695-711 (2017).

Structured illumination microscopy (SIM) improves resolution by down-modulating high-frequency information of an object to fit within the passband of the optical system. Generally, the reconstruction process requires prior knowledge of the illumination patterns, which implies a well-calibrated and aberration-free system. Here, we propose a new *algorithmic self-calibration* strategy for SIM that does not need to know the exact patterns *a priori*, but only their covariance. The algorithm, termed PE-SIMS, includes a pattern-estimation (PE) step requiring the uniformity of the sum of the illumination patterns and a SIM reconstruction procedure using a statistical prior (SIMS). Additionally, we perform a pixel reassignment process (SIMS-PR) to enhance the reconstruction quality. We achieve 2× better resolution than a conventional widefield microscope, while remaining insensitive to aberration-induced pattern distortion and robust against parameter tuning.

**Compressive holographic video**Zihao Wang, Leonidas Spinoulas, Kuan He, Lei Tian, Oliver Cossairt, Aggelos K. Katsaggelos, and Huaijin Chen

Opt. Express 25, 250-262 (2017).

Compressed sensing has been discussed separately in spatial and temporal domains. Compressive holography has been introduced as a method that allows 3D tomographic reconstruction at different depths from a single 2D image. Coded exposure is a temporal compressed sensing method for high speed video acquisition. In this work, we combine compressive holography and coded exposure techniques and extend the discussion to 4D reconstruction in space and time from one coded captured image. In our prototype, digital in-line holography was used for imaging macroscopic, fast moving objects. The pixel-wise temporal modulation was implemented by a digital micromirror device. In this paper we demonstrate 10× temporal super resolution with multiple depths recovery from a single image. Two examples are presented for the purpose of recording subtle vibrations and tracking small particles within 5 ms.

**3D differential phase contrast microscopy**

Michael Chen, Lei Tian, Laura Waller

Biomed. Opt. Express 7, 3940-3950 (2016).

We demonstrate 3D phase and absorption recovery from partially coherent intensity images captured with a programmable LED array source. Images are captured through-focus with four different illumination patterns. Using first Born and weak object approximations (WOA), a linear 3D differential phase contrast (DPC) model is derived. The partially coherent transfer functions relate the sample’s complex refractive index distribution to intensity measurements at varying defocus. Volumetric reconstruction is achieved by a global FFT-based method, without an intermediate 2D phase retrieval step. Because the illumination is spatially partially coherent, the transverse resolution of the reconstructed field achieves twice the NA of coherent systems and improved axial resolution.

**Nonlinear Optimization Algorithm for Partially Coherent Phase Retrieval and Source Recovery
**J. Zhong, L. Tian, P. Varma, L. Waller

IEEE Transactions on Computational Imaging 2 (3), 310 – 322 (2016).

We propose a new algorithm for recovering both complex field (phase and amplitude) and source distribution (illumination spatial coherence) from a stack of intensity images captured through focus. The joint recovery is formulated as a nonlinear least-square-error optimization problem, which is solved iteratively by a modified Gauss-Newton method. We derive the gradient and Hessian of the cost function and show that our second-order optimization approach outperforms previously proposed phase retrieval algorithms, for datasets taken with both coherent and partially coherent illumination. The method is validated experimentally in a commercial microscope with both Kohler illumination and a programmable LED dome.

**Relaxation of mask design for single-shot phase imaging with a coded aperture**

R. Egami, R. Horisaki, L. Tian, J. Tanida

Appl. Opt. 55, 1830-1837 (2016).

We present a method of relaxing the conditions of mask design in single-shot phase imaging with a coded aperture (SPICA), for extending the applications of SPICA. SPICA, based on compressive sensing, enables the acquisition of wide, high-resolution optical complex fields in a single exposure without the need for reference light. In our previous work on SPICA, a coded aperture (CA) was implemented with only amplitude modulation, resulting in a low transmission factor and low light efficiency because of the need for an independent phase retrieval process in the reconstruction. We attempt to alleviate these limitations by adapting a reconstruction algorithm to directly associate the phase-retrieval process with a sparsity-based reconstruction. With this approach, it is possible to realize SPICA with an amplitude-modulation-based CA having a high transmission factor, a phase-modulation-based CA, and a complex-amplitude (amplitude and phase)-modulation-based CA. We verified the effectiveness of these relaxed CA designs numerically and experimentally.

**Experimental robustness of Fourier Ptychography phase retrieval algorithms**

L. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, L. Waller

Opt. Express 23(26) 33212-33238 (2015).

Fourier ptychography is a new computational microscopy technique that provides gigapixel-scale intensity and phase images with both wide field-of-view and high resolution. By capturing a stack of low-resolution images under different illumination angles, an inverse algorithm can be used to computationally reconstruct the high-resolution complex field. Here, we compare and classify multiple proposed inverse algorithms in terms of experimental robustness. We find that the main sources of error are noise, aberrations and mis-calibration (i.e. model mis-match). Using simulations and experiments, we demonstrate that the choice of cost function plays a critical role, with amplitude-based cost functions performing better than intensity-based ones. The reason for this is that Fourier ptychography datasets consist of images from both brightfield and darkfield illumination, representing a large range of measured intensities. Both noise (e.g. Poisson noise) and model mis-match errors are shown to scale with intensity. Hence, algorithms that use an appropriate cost function will be more tolerant to both noise and model mis-match. Given these insights, we propose a global Newton’s method algorithm which is robust and accurate. Finally, we discuss the impact of procedures for algorithmic correction of aberrations and mis-calibration.

**Computational illumination for high-speed in vitro Fourier ptychographic microscopy**

L. Tian, Z. Liu, L. Yeh, M. Chen, J. Zhong, L. Waller

Optica 2(10), 904-911 (2015).

We demonstrate a new computational illumination technique that achieves a large space-bandwidth-time product, for quantitative phase imaging of unstained live samples in vitro. Microscope lenses can have either a large field of view (FOV) or high resolution, and not both. Fourier ptychographic microscopy (FPM) is a new computational imaging technique that circumvents this limit by fusing information from multiple images taken with different illumination angles. The result is a gigapixel-scale image having both a wide FOV and high resolution, i.e., a large space-bandwidth product. FPM has enormous potential for revolutionizing microscopy and has already found application in digital pathology. However, it suffers from long acquisition times (of the order of minutes), limiting throughput. Faster capture times would not only improve the imaging speed, but also allow studies of live samples, where motion artifacts degrade results. In contrast to fixed (e.g., pathology) slides, live samples are continuously evolving at various spatial and temporal scales. Here, we present a new source coding scheme, along with real-time hardware control, to achieve 0.8 NA resolution across a 4x FOV with subsecond capture times. We propose an improved algorithm and a new initialization scheme, which allow robust phase reconstruction over long time-lapse experiments. We present the first FPM results for both growing and confluent in vitro cell cultures, capturing videos of subcellular dynamical phenomena in popular cell lines undergoing division and migration. Our method opens up FPM to applications with live samples, for observing rare events in both space and time.

**Computational imaging: Machine learning for 3D microscopy**

L. Waller, L. Tian

Nature, 523, 416–417 (2015).

**Self-learning based Fourier ptychographic microscopy**

Y. Zhang, W. Jiang, L. Tian, L. Waller, Q. Dai

Opt. Express 23, 18471-18486 (2015).

Fourier Ptychographic Microscopy (FPM) is a newly proposed computational imaging method aimed at reconstructing a high-resolution wide-field image from a sequence of low-resolution images. These low-resolution images are captured under varied illumination angles and the FPM recovery routine then stitches them together in the Fourier domain iteratively. Although FPM has achieved success with static sample reconstructions, the long acquisition time inhibits real-time application. To address this problem, we propose here a self-learning based FPM which accelerates the acquisition and reconstruction procedure. We first capture a single image under normally incident illumination, and then use it to simulate the corresponding low-resolution images under other illumination angles. The simulation is based on the relationship between the illumination angles and the shift of the sample’s spectrum. We analyze the importance of the simulated low-resolution images in order to devise a selection scheme which only collects the ones with higher importance. The measurements are then captured with the selection scheme and employed to perform the FPM reconstruction. Since only measurements of high importance are captured, the time requirements of data collection as well as image reconstruction can be greatly reduced. We validate the effectiveness of the proposed method with simulation and experimental results showing that the reduction ratio of data size requirements can reach over 70%, without sacrificing image reconstruction quality.

**3D imaging in volumetric scattering media using phase-space measurements**

H. Liu, E. Jonas, L. Tian, J. Zhong, B. Recht, L. Waller

Opt. Express 23, 14461-14471 (2015).

We demonstrate the use of phase-space imaging for 3D localization of multiple point sources inside scattering material. The effect of scattering is to spread angular (spatial frequency) information, which can be measured by phase space imaging. We derive a multi-slice forward model for homogenous volumetric scattering, then develop a reconstruction algorithm that exploits sparsity in order to further constrain the problem. By using 4D measurements for 3D reconstruction, the dimensionality mismatch provides significant robustness to multiple scattering, with either static or dynamic diffusers. Experimentally, our high-resolution 4D phase-space data is collected by a spectrogram setup, with results successfully recovering the 3D positions of multiple LEDs embedded in turbid scattering media.

**Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective**

C. Zuo, Q. Chen, L. Tian, L. Waller, A. Asundi

Optics and Lasers in Engineering 71, 20-32 (2015).

The well-known transport of intensity equation (TIE) allows the phase of a coherent field to be retrieved non-interferometrically given positive defined intensity measurements and appropriate boundary conditions. However, in many cases like the optical microscopy, the imaging systems often involve extended and polychromatic sources for which the effect of the partial coherence is not negligible. In this work, we present a phase-space formulation for the TIE for analyzing phase retrieval under partially coherent illumination. The conventional TIE is reformulated in the joint space-spatial frequency domain using Wigner distribution functions. The phase-space formulation clarifies the physical meaning of the phase of partially coherent fields, and enables explicit account of partial coherence effects on phase retrieval. The correspondence between the Wigner distribution function and the light field in geometric optics limit further enables TIE to become a simple yet effective approach to realize high-resolution light field imaging for slowly varying phase specimens, in a purely computational way.

**Multi-contrast imaging and digital refocusing on a mobile microscope with a domed LED array**

Z. Phillips, M. D’Ambrosio, L. Tian, J. Rulison, H. Patel, N. Sadras, A. Gande, N. Switz, D. Fletcher, L. Waller

PLoS ONE 10, e0124938 (2015).

We demonstrate the design and application of an add-on device for improving the diagnostic and research capabilities of CellScope—a low-cost, smartphone-based point-of-care microscope. We replace the single LED illumination of the original CellScope with a programmable domed LED array. By leveraging recent advances in computational illumination, this new device enables simultaneous multi-contrast imaging with brightfield, darkfield, and phase imaging modes. Further, we scan through illumination angles to capture lightfield datasets, which can be used to recover 3D intensity and phase images without any hardware changes. This digital refocusing procedure can be used for either 3D imaging or software-only focus correction, reducing the need for precise mechanical focusing during field experiments. All acquisition and processing is performed on the mobile phone and controlled through a smartphone application, making the computational microscope compact and portable. Using multiple samples and different objective magnifications, we demonstrate that the performance of our device is comparable to that of a commercial microscope. This unique device platform extends the field imaging capabilities of CellScope, opening up new clinical and research possibilities.

**Quantitative differential phase contrast imaging in an LED array microscope**

L. Tian, L. Waller

Opt. Express 23, 11394-11403 (2015).

Illumination-based differential phase contrast (DPC) is a phase imaging method that uses a pair of images with asymmetric illumination patterns. Distinct from coherent techniques, DPC relies on spatially partially coherent light, providing 2× better lateral resolution, better optical sectioning and immunity to speckle noise. In this paper, we derive the 2D weak object transfer function (WOTF) and develop a quantitative phase reconstruction method that is robust to noise. The effect of spatial coherence is studied experimentally, and multiple-angle DPC is shown to provide improved frequency coverage for more stable phase recovery. Our method uses an LED array microscope to achieve real-time (10 Hz) quantitative phase imaging with in vitro live cell samples.

**Motion deblurring with temporally coded illumination in an LED array microscope**

C. Ma, Z. Liu, L. Tian, Q. Dai, L. Waller

Opt. Lett. 40, 2281-2284 (2015).

Motion blur, which results from time-averaging an image over the camera’s exposure time, is a common problem in microscopy of moving samples. Here, we demonstrate linear motion deblurring using temporally coded illumination in an LED array microscope. By illuminating moving objects with a well-designed temporal coded sequence that varies during each single camera exposure, the resulting motion blur is invertible and can be computationally removed. This scheme is implemented in an existing LED array microscope, providing benefits of being grayscale, fast, and adaptive, which leads to high-quality deblur performance and a flexible implementation with no moving parts. The proposed method is demonstrated experimentally for fast moving targets in a microfluidic environment.

**3D intensity and phase imaging from light field measurements in an LED array microscope**

Lei Tian, L. Waller

Optica 2, 104-111 (2015).

Realizing high resolution across large volumes is challenging for 3D imaging techniques with high-speed acquisition. Here, we describe a new method for 3D intensity and phase recovery from 4D light field measurements, achieving enhanced resolution via Fourier Ptychography. Starting from geometric optics light field refocusing, we incorporate phase retrieval and correct diffraction artifacts. Further, we incorporate dark-field images to achieve lateral resolution beyond the diffraction limit of the objective (5x larger NA) and axial resolution better than the depth of field, using a low magnification objective with a large field of view. Our iterative reconstruction algorithm uses a multi-slice coherent model to estimate the 3D complex transmittance function of the sample at multiple depths, without any weak or single-scattering approximations. Data is captured by an LED array microscope with computational illumination, which enables rapid scanning of angles for fast acquisition. We demonstrate the method with thick biological samples in a modified commercial microscope, indicating the technique’s versatility for a wide range of applications.

**Partially coherent phase imaging with unknown source shape**

J. Zhong, Lei Tian, J. Dauwels, L. Waller

Biomedical Optics Express 6, 257-265 (2015).

We propose a new method for phase retrieval that uses partially coherent illumination created by any arbitrary source shape in Kohler geometry. Using a stack of defocused intensity images, we recover not only the phase and amplitude of the sample, but also an estimate of the unknown source shape, which describes the spatial coherence of the illumination. Our algorithm uses a Kalman filtering approach which is fast, accurate and robust to noise. The method is experimentally simple and flexible, so should find use in optical, electron, X-ray and other phase imaging systems which employ partially coherent light. We provide an experimental demonstration in an optical microscope with various condenser apertures.

**Empirical concentration bounds for compressive holographic bubble imaging based on a Mie scattering model**

W. Chen, Lei Tian, S. Rehman, Z. Zhang, H. P. Lee, G. Barbastathis

Opt. Express 23, (2015).

We use compressive in–line holography to image air bubbles in water and investigate the effect of bubble concentration on reconstruction performance by simulation. Our forward model treats bubbles as finite spheres and uses Mie scattering to compute the scattered field in a physically rigorous manner. Although no simple analytical bounds on maximum concentration can be derived within the classical compressed sensing framework due to the complexity of the forward model, the receiver operating characteristic (ROC) curves in our simulation provide an empirical concentration bound for accurate bubble detection by compressive holography at different noise levels, resulting in a maximum tolerable concentration much higher than the traditional back-propagation method.

**Real-time brightfield, darkfield and phase contrast imaging in an LED array microscope**

Z. Liu, Lei Tian, S. Liu, L. Waller

Journal of Biomedical Optics, 19(10), 106002 (2014).

We demonstrate a single-camera imaging system that can simultaneously acquire brightfield, darkfield and phase contrast images in real-time. Our method uses computational illumination via a programmable LED array at the source plane, providing flexible patterning of illumination angles. Brightfield, darkfield and differential phase contrast (DPC) images are obtained by changing the LED patterns, without any moving parts. Previous work with LED array illumination was only valid for static samples because the hardware speed was not fast enough to meet real-time acquisition and processing requirements. Here, we time multiplex patterns for each of the three contrast modes in order to image dynamic biological processes in all three contrast modes simultaneously. We demonstrate multi-contrast operation at the maximum frame rate of our camera (50 Hz with 2160×2560 pixels).

**Multiplexed coded illumination for Fourier Ptychography with an LED array microscope**

Lei Tian, X. Li, K. Ramchandran, L. Waller

Biomedical Optics Express 5, 2376-2389 (2014).

Fourier Ptychography is a new computational microscopy technique that achieves gigapixel images with both wide field of view and high resolution in both phase and amplitude. The hardware setup involves a simple replacement of the microscope’s illumination unit with a programmable LED array, allowing one to flexibly pattern illumination angles without any moving parts. In previous work, a series of low-resolution images was taken by sequentially turning on each single LED in the array, and the data were then combined to recover a bandwidth much higher than the one allowed by the original imaging system. Here, we demonstrate a multiplexed illumination strategy in which multiple randomly selected LEDs are turned on for each image. Since each LED corresponds to a different area of Fourier space, the total number of images can be significantly reduced, without sacrificing image quality. We demonstrate this method experimentally in a modified commercial microscope. Compared to sequential scanning, our multiplexed strategy achieves similar results with approximately an order of magnitude reduction in both acquisition time and data capture requirements.

**3D differential phase contrast microscopy with computational illumination using an LED array**

Lei Tian, J. Wang, L. Waller

Optics Letters 39, 1326 – 1329 (2014).

We demonstrate 3D differential phase-contrast (DPC) microscopy, based on computational illumination with a programmable LED array. By capturing intensity images with various illumination angles generated by sequentially patterning an LED array source, we digitally refocus images through various depths via light field processing. The intensity differences from images taken at complementary illumination angles are then used to generate DPC images, which are related to the gradient of phase. The proposed method achieves 3D DPC with simple, inexpensive optics and no moving parts. We experimentally demonstrate our method by imaging a camel hair sample in 3D.

**Transport of Intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks**

A. Shanker, Lei Tian, M. Sczyrba, B. Connolly, A. Neureuther, L. Waller

Applied Optics, 53(34), J1 (2014).

We report theoretical and experimental results for imaging of electromagnetic phase edge effects in lithography photomasks. Our method starts from the transport of intensity equation (TIE), which solves for phase from through-focus intensity images. Traditional TIE algorithms make an implicit assumption that the underlying in-plane power flow is curl-free. Motivated by our current study, we describe a practical situation in which this assumption breaks down. Strong absorption gradients in mask features interact with phase edges to contribute a curl to the in-plane Poynting vector, causing severe artifacts in the phase recovered. We derive how curl effects are coupled into intensity measurements and propose an iterative algorithm that not only corrects the artifacts, but also recovers missing curl components.

**Low-noise phase imaging by hybrid uniform and structured illumination Transport of Intensity Equation**

Y. Zhu, A. Shanker, Lei Tian, L. Waller, G. Barbastathis

Optics Express, 22(22) 26696, (2014).

We demonstrate a new approach to the transport of intensity equation (TIE) phase retrieval method which uses structured illumination to improve low-frequency noise performance. In the hybrid scheme, two phase images are acquired: one with uniform and one with sinusoidal grating illumination intensity. The former preserves the high spatial frequency features of the phase best, whereas the latter dramatically increase the response at low spatial frequencies (where traditional TIE notoriously suffers). We then theoretically prove the design of a spectral filter that optimally combines the two phase results while suppressing noise. The combination of uniformly and structured illuminated TIE (hybrid TIE) phase imaging is experimentally demonstrated optically with a calibrated pure phase object.

**Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes**

J. Zhong, R. Claus, J. Dauwels, Lei Tian, L. Waller

Optics Express 22, 10661-10674 (2014).

We propose an alternative method for solving the Transport of Intensity equation (TIE) from a stack of through–focus intensity images taken by a microscope or lensless imager. Our method enables quantitative phase and amplitude imaging with improved accuracy and reduced data capture, while also being computationally efficient and robust to noise. We use prior knowledge of how intensity varies with propagation in the spatial frequency domain in order to constrain a fitting algorithm [Gaussian process (GP) regression] for estimating the axial intensity derivative. Solving the problem in the frequency domain inspires an efficient measurement scheme which captures images at exponentially spaced focal steps, significantly reducing the number of images required. Low–frequency artifacts that plague traditional TIE methods can be suppressed without an excessive number of captured images. We validate our technique experimentally by recovering the phase of human cheek cells in a brightfield microscope.

**Hamiltonian and phase-space representation of spatial solitons**

H. Gao, Lei Tian, G. Barbastathis

Optics Communications 318, 199-204 (2014).

We use Hamiltonian ray tracing and phase-space representation to describe the propagation of a single spatial soliton and soliton collisions in a Kerr nonlinear medium. Hamiltonian ray tracing is applied using the iterative nonlinear beam propagation method, which allows taking both wave effects and Kerr nonlinearity into consideration. Energy evolution within a single spatial soliton and the exchange of energy when two solitons collide are interpreted intuitively by ray trajectories and geometrical shearing of the Wigner distribution functions.

**Compressive holographic two-dimensional localization with 1/30 ^{2} subpixel accuracy**

Y. Liu, Lei Tian, C. Hsieh, G. Barbastathis

Optics Express 22, 9774-9782 (2014).

We propose the use of compressive holography for two–dimensional (2D) subpixel motion localization. Our approach is based on computational implementation of edge–extraction using a Fourier–plane spiral phase mask, followed by compressive reconstruction of the edge of the object. Using this technique and relatively low–cost computer and piezo motion stage to establish ground truth for the motion, we demonstrated localization within 1/30th of a camera pixel in each linear dimension.

**Compressive X-ray phase tomography based on the transport of intensity equation**

Lei Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, G. Barbastathis

Optics Letters 38, 3418-3421 (2013).

We develop and implement a compressive reconstruction method for tomographic recovery of refractive index distribution for weakly attenuating objects in a microfocus x-ray system. This is achieved through the development of a discretized operator modeling both the transport of intensity equation and the x-ray transform that is suitable for iterative reconstruction techniques.

**The transport of intensity equation for optical path length recovery using partially coherent illumination**

J. C. Petruccelli, Lei Tian, G. Barbastathis

Optics Express 21, 14430-14441 (2013).

***Highlighted in the OSA Spotlight on Optics**

We investigate the measurement of a thin sample’s optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample’s optical path length by the removal of illumination effects.

**Wigner function measurement using a lens let array**

Lei Tian, Z. Zhang, Jon. C. Petruccelli, G. Barbastathis

Optics Express 21, 10511-10525 (2013).

Geometrical–optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave–optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam’s WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross–talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell–model beam.

**Nonlinear diffusion regularization for transport of intensity phase imaging**

Lei Tian, J. C. Petruccelli, G. Barbastathis

Optics Letters 37, 4131-4133 (2012).

We demonstrate a nonlinear diffusion (NLD) regularization method to solve the transport of intensity equation (TIE). A novel NLD regularization function is proposed to enforce piecewise-constant priors and to remove low-frequency artifacts in the TIE solution.

**Wigner functions for evanescent waves**

J. C. Petruccelli, Lei Tian, S. B. Oh, G. Barbastathis

Journal of the Optical Society of America A 29, 1927-1938 (2012).

We propose phase space distributions, based on an extension of the Wigner distribution function, to describe fields of any state of coherence that contain evanescent components emitted into a half-space. The evanescent components of the field are described in an optical phase space of spatial position and complex-valued angle. Behavior of these distributions upon propagation is also considered, where the rapid decay of the evanescent components is associated with the exponential decay of the associated phase space distributions. To demonstrate the structure and behavior of these distributions, we consider the fields generated from total internal reflection of a Gaussian Schell-model beam at a planar interface.

**Scanning-free compressive holography for object localization with subpixel accuracy**

Y. Liu, Lei Tian, J. W. Lee, H. Y. H. Huang, M. S. Triantafyllou, G. Barbastathis

Optics Letters 37, 3357-3359 (2012).

We propose quantitative localization measurement of a known object with subpixel accuracy using compressive holography. We analyze the theoretical optimal solution in the compressive sampling framework and experimentally demonstrate localization accuracy of 1/45 pixel, in good agreement with the analysis.

**Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising**

H. Huang, Lei Tian, Z. Zhang, Y. Liu, Z. Chen, G. Barbastathis

Opt. Express 20, 14075 (2012).

Phase unwrapping is a challenging task for interferometry based techniques in the presence of noise. The majority of existing phase unwrapping techniques are path-following methods, which explicitly or implicitly define an intelligent path and integrate phase difference along the path to mitigate the effect of erroneous pixels. In this paper, a path-independent unwrapping method is proposed where the unwrapped phase gradient is determined from the wrapped phase and subsequently denoised by a TV minimization based method. Unlike the wrapped phase map where 2πphase jumps are present, the gradient of the unwrapped phase map is smooth and slowly-varying at noise-free areas. On the other hand, the noise is greatly amplified by the differentiation process, which makes it easier to separate from the smooth phase gradient. Thus an approximate unwrapped phase can be obtained by integrating the denoised phase gradient. The final unwrapped phase map is subsequently determined by adding the first few modes of the unwrapped phase. The proposed method is most suitable for unwrapping phase maps without abrupt phase changes. Its capability has been demonstrated both numerically and by experimental data from shearography and electronic speckle pattern interferometry (ESPI).

**Experimental compressive phase space tomography**

Lei Tian, J. Lee, S. B. Oh, G. Barbastathis

Optics Express 20, 8296-8308 (2012).

***Highlighted in the OSA Spotlight on Optics**

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two-point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low-entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground-truth estimate from the van Cittert-Zernike theorem and verify substantial quantitative improvements in the reconstruction.

**Wigner functions defined with Laplace transform kernels**

S. B. Oh, J. C. Petruccelli, L. Tian, G. Barbastathis

Optics Express 19, 21938-21944 (2011).

We propose a new Wigner–type phase–space function using Laplace transform kernels—Laplace kernel Wigner function. Whereas momentum variables are real in the traditional Wigner function, the Laplace kernel Wigner function may have complex momentum variables. Due to the property of the Laplace transform, a broader range of signals can be represented in complex phase–space. We show that the Laplace kernel Wigner function exhibits similar properties in the marginals as the traditional Wigner function. As an example, we use the Laplace kernel Wigner function to analyze evanescent waves supported by surface plasmon polariton.

**From Two-Dimensional Colloidal Self-Assembly to Three-Dimensional Nanolithography**

C.-H. Chang, L. Tian, W. R. Hesse, H. Gao, H. J. Choi, J.-G. Kim, M. Siddiqui, G. Barbastathis

Nano Letters 11, 2533-2537 (2011).

A number of “top-down” lithographic and “bottom-up” self-assembly methods have been developed to fabricate three-dimensional (3D) nanostructures to support the recent advances in nanotechnology. But they are limited by a number of factors such as fabrication cost, pattern resolution, and/or flexibility of geometry. Here we present a 3D nanolithography process that utilizes self-assembled nanospheres to create a periodic array of focal spots, which are then replicated across multiple depth in a transparent medium according to the Talbot effect. The Talbot field then exposes a pattern onto the underlying photoresist, recording the 3D intensity distribution. We have demonstrated designable complex 3D periodic structures with 80 nm minimum feature size, roughly one-fourth of the operating wavelength. This approach combines 2D colloidal self-assembly and 3D phase lithography, is robust, cost-effective, and widely applicable to nanoscale research and manufacturing.

**Aperiodic sub wavelength Lüneburg lens with nonlinear Kerr effect compensation**

H. Gao, S. Takahashi, L. Tian, G. Barbastathis

Optics Express 19, 2257-2265 (2011).

We introduce a Lüneburg lens design where Kerr nonlinearity is used to compensate for the focal point shift caused by diffraction of a Gaussian source. A computationally efficient iterative method introduced in [Opt. Lett. 35, 4148 (2010)] is used to provide ray diagrams in the nonlinear case and verify the focal shift compensation. We study the joint dependence of focal shift on waist size and intensity of Gaussian source, and show how to compensate spherical aberration caused by the nonlinearity by a small perturbation of the Lüneburg profile. Our results are specific to Lüneburg lens but our approach is applicable to more general cases of nonlinear nonperiodic metamaterials.

**Quantitative measurement of size and three-dimensional position of fast moving bubbles in air-water mixture flows using digital holography**

Lei Tian, N. Loomis, J. Dominguez-Caballero, G. Barbastathis

Applied Optics 49, 1549 (2010).

We present a digital in-line holographic imaging system for measuring the size and three-dimensional position of fast-moving bubbles in air–water mixture flows. The captured holograms are numerically processed by performing a two-dimensional projection followed by local depth estimation to quickly and efficiently obtain the size and position information of multiple bubbles simultaneously. Statistical analysis on measured bubble size distributions shows that they follow lognormal or gamma distributions.

**Transport of Intensity phase-amplitude imaging with higher order intensity derivatives**

L. Waller, Lei Tian, G. Barbastathis

Optics Express 18, 12552-12561 (2010).

We demonstrate a method for improving the accuracy of phase retrieval based on the Transport of Intensity equation by using intensity measurements at multiple planes to estimate and remove the artifacts due to higher order axial derivatives. We suggest two similar methods of higher order correction, and demonstrate their ability for accurate phase retrieval well beyond the ‘linear’ range of defocus that TIE imaging traditionally requires. Computation is fast and efficient, and sensitivity to noise is reduced by using many images.

**Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function**

H. Gao, Lei Tian, B. Zhang, G. Barbastathis

Optics Letters 35, 4148-4150 (2010).

We present an iterative method for simulating beam propagation in nonlinear media using Hamiltonian ray tracing. The Wigner distribution function of the input beam is computed at the entrance plane and is used as the initial condition for solving the Hamiltonian equations. Examples are given for the study of periodic self-focusing, spatial solitons, and Gaussian–Schell model in Kerr-effect media. Simulation results show good agreement with the split-step beam propagation method. The main advantage of ray tracing, even in the nonlinear case, is that ray diagrams are intuitive and easy to interpret in terms of traditional optical engineering terms, such as aberrations, ray-intercept plots, etc.