Phase Retrieval

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.


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.



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.


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.