Manuel Guizar-Sicairos

Efficient subpixel image registration algorithms

Three new algorithms for 2D translation image registration to within a small fraction of a pixel that use nonlinear optimization and matrix-multiply discrete Fourier transforms are compared. These algorithms can achieve registration with an accuracy equivalent to that of the conventional fast Fourier transform upsampling approach in a small fraction of the computation time and with greatly reduced memory requirements. Their accuracy and computation time are compared for the purpose of evaluating a translation-invariant error metric.

Phase retrieval with transverse translation diversity: a nonlinear optimization approach

We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse far-field intensity measurements are taken after translating the object relative to a known illumination pattern. Analytical expressions for the gradient of a squared-error metric with respect to the object, illumination and translations allow joint optimization of the object and system parameters. This approach achieves superior reconstructions, with respect to a previously reported technique [H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004)], when the system parameters are inaccurately known or in the presence of noise. Applicability of this method for samples that are smaller than the illumination pattern is explored.

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

The method originally proposed by Yu et al. [Opt. Lett. 23, 409 (1998)] for evaluating the zero-order Hankel transform is generalized to high-order Hankel transforms. Since the method preserves the discrete form of the Parseval theorem, it is particularly suitable for field propagation. A general algorithm for propagating an input field through axially symmetric systems using the generalized method is given. The advantages and the disadvantages of the method with respect to other typical methods are discussed.

X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution

X-ray ptychography is a scanning variant of coherent diffractive imaging with the ability to image large fields of view at high resolution. It further allows imaging of non-isolated specimens and can produce quantitative mapping of the electron density distribution in 3D when combined with computed tomography. The method does not require imaging lenses, which makes it dose efficient and suitable to multi-keV X-rays, where efficient photon counting, pixelated detectors are available. Here we present the first highly resolved quantitative X-ray ptychographic tomography of an extended object yielding 16 nm isotropic 3D resolution recorded at 2 Å wavelength. This first-of-its-kind demonstration paves the way for ptychographic X-ray tomography to become a promising method for X-ray imaging of representative sample volumes at unmatched resolution, opening tremendous potential for characterizing samples in materials science and biology by filling the resolution gap between electron microscopy and other X-ray imaging techniques.

Maximum-likelihood refinement for coherent diffractive imaging

We introduce the application of maximum-likelihood (ML) principles to the image reconstruction problem in coherent diffractive imaging. We describe an implementation of the optimization procedure for ptychography, using conjugate gradients and including preconditioning strategies, regularization and typical modifications of the statistical noise model. The optimization principle is compared to a difference map reconstruction algorithm. With simulated data important improvements are observed, as measured by a strong increase in the signal-to-noise ratio. Significant gains in resolution and sensitivity are also demonstrated in the ML refinement of a reconstruction from experimental x-ray data. The immediate consequence of our results is the possible reduction of exposure, or dose, by up to an order of magnitude for a reconstruction quality similar to iterative algorithms currently in use.

Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging

We have employed ptychographic coherent diffractive imaging to completely characterize the focal spot wavefield and wavefront aberrations of a high-resolution diffractive X-ray lens. The ptychographic data from a strongly scattering object was acquired using the radiation cone emanating from a coherently illuminated Fresnel zone plate at a photon energy of 6.2 keV. Reconstructed images of the object were retrieved with a spatial resolution of 8 nm by combining the difference-map phase retrieval algorithm with a non-linear optimization refinement. By numerically propagating the reconstructed illumination function, we have obtained the X-ray wavefield profile of the 23 nm round focus of the Fresnel zone plate (outermost zone width, Δr = 20 nm) as well as the X-ray wavefront at the exit pupil of the lens. The measurements of the wavefront aberrations were repeatable to within a root mean square error of 0.006 waves, and we demonstrate that they can be related to manufacturing aspects of the diffractive optical element and to errors on the incident X-ray wavefront introduced by the upstream beamline optics.

Phase tomography from x-ray coherent diffractive imaging projections

Coherent diffractive imaging provides accurate phase projections that can be tomographically combined to yield detailed quantitative 3D reconstructions with a resolution that is not limited by imaging optics. We present robust algorithms for post-processing and alignment of these tomographic phase projections. A simple method to remove undesired constant and linear phase terms on the reconstructions is given. Also, we provide an algorithm for automatic alignment of projections that has good performance even for samples with no fiducial markers. Currently applied to phase projections, this alignment algorithm has proven to be robust and should also be useful for lens-based tomography techniques that pursue nanoscale 3D imaging. Lastly, we provide a method for tomographic reconstruction that works on phase projections that are known modulo 2π, such that the phase unwrapping step is avoided. We demonstrate the performance of these algorithms by 3D imaging of bacteria population in legume root-nodule cells.

Optical wavefront measurement using phase retrieval with transverse translation diversity

We demonstrate the use of transverse translation-diverse phase retrieval as a method for the measurement of wavefronts in situations where the detected intensity patterns would be otherwise undersampled. This technique involves using a smaller moving subaperture to produce a number of adequately sampled intensity patterns. The wavefront is then retrieved using an optimization jointly constrained by them. Expressions for the gradient of an error metric with respect to the optimization parameters are given. An experimental arrangement used to measure the transmitted wavefront of a plano-convex singlet using this technique is described. The results of these measurements were repeatable to within approximately lambda/100 RMS.

Reconstruction of an astigmatic hard X-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data.

We have used coherent X-ray diffraction experiments to characterize both the 1-D and 2-D foci produced by nanofocusing Kirkpatrick-Baez (K-B) mirrors, and we find agreement. Algorithms related to ptychography were used to obtain a 3-D reconstruction of a focused hard X-ray beam waist, using data measured when the mirrors were not optimally aligned. Considerable astigmatism was evident in the reconstructed complex wavefield. Comparing the reconstructed wavefield for a single mirror with a geometrical projection of the wavefront errors expected from optical metrology data allowed us to diagnose a 40 μrad misalignment in the incident angle of the first mirror, which had occurred during the experiment. Good agreement between the reconstructed wavefront obtained from the X-ray data and off-line metrology data obtained with visible light demonstrates the usefulness of the technique as a metrology and alignment tool for nanofocusing X-ray optics.

Holography with extended reference by autocorrelation linear differential operation

We introduce a generalization of Fourier transform holography that allows the use of the boundary waves of an extended object to act as a holographic-like reference. By applying a linear differential operator on the field autocorrelation, we use a sharp feature on the extended reference to reconstruct a complex-valued image of the object of interest in a single-step computation. We generalize the approach of Podorov et al. [Opt. Express 15, 9954 (2007)] to a much wider class of extended reference objects. Effects of apertures in Fourier domain and imperfections in the reference object are analyzed. Realistic numerical simulations show the feasibility of our approach and its robustness against noise.