Fourier Analysis of Stochastic Sampling Strategies for Assessing Bias and Variance in Integration

Each pixel in a photorealistic, computer generated picture is calculated by approximately integrating all the light arriving at the pixel, from the virtual scene. A common strategy to calculate these high-dimensional integrals is to average the estimates at stochastically sampled locations. The strategy with which the sampled locations are chosen is of utmost importance in deciding the quality of the approximation, and hence rendered image. We derive connections between the spectral properties of stochastic sampling patterns and the first and second order statistics of estimates of integration using the samples. Our equations provide insight into the assessment of stochastic sampling strategies for integration. We show that the amplitude of the expected Fourier spectrum of sampling patterns is a useful indicator of the bias when used in numerical integration. We deduce that estimator variance is directly dependent on the variance of the sampling spectrum over multiple realizations of the sampling pattern. We then analyse Gaussian jittered sampling, a simple variant of jittered sampling, that allows a smooth trade-off of bias for variance in uniform (regular grid) sampling. We verify our predictions using spectral measurement, quantitative integration experiments and qualitative comparisons of rendered images.</jats:p

Combining Undersampled Dithered Images

Undersampled images, such as those produced by the HST WFPC-2, misrepresent fine-scale structure intrinsic to the astronomical sources being imaged. Analyzing such images is difficult on scales close to their resolution limits and may produce erroneous results. A set of ``dithered'' images of an astronomical source generally contains more information about its structure than any single undersampled image, however, and may permit reconstruction of a ``superimage'' with Nyquist sampling. I present a tutorial on a method of image reconstruction that builds a superimage from a complex linear combination of the Fourier transforms of a set of undersampled dithered images. This method works by algebraically eliminating the high order satellites in the periodic transforms of the aliased images. The reconstructed image is an exact representation of the data-set with no loss of resolution at the Nyquist scale. The algorithm is directly derived from the theoretical properties of aliased images and involves no arbitrary parameters, requiring only that the dithers are purely translational and constant in pixel-space over the domain of the object of interest. I show examples of its application to WFC and PC images. I argue for its use when the best recovery of point sources or morphological information at the HST diffraction limit is of interest.Comment: 22 pages, 9 EPS figures, submitted to PAS

Multifrequency Aperture-Synthesizing Microwave Radiometer System (MFASMR). Volume 1

Background material and a systems analysis of a multifrequency aperture - synthesizing microwave radiometer system is presented. It was found that the system does not exhibit high performance because much of the available thermal power is not used in the construction of the image and because the image that can be formed has a resolution of only ten lines. An analysis of image reconstruction is given. The system is compared with conventional aperture synthesis systems

ADAM: a general method for using various data types in asteroid reconstruction

We introduce ADAM, the All-Data Asteroid Modelling algorithm. ADAM is simple and universal since it handles all disk-resolved data types (adaptive optics or other images, interferometry, and range-Doppler radar data) in a uniform manner via the 2D Fourier transform, enabling fast convergence in model optimization. The resolved data can be combined with disk-integrated data (photometry). In the reconstruction process, the difference between each data type is only a few code lines defining the particular generalized projection from 3D onto a 2D image plane. Occultation timings can be included as sparse silhouettes, and thermal infrared data are efficiently handled with an approximate algorithm that is sufficient in practice due to the dominance of the high-contrast (boundary) pixels over the low-contrast (interior) ones. This is of particular importance to the raw ALMA data that can be directly handled by ADAM without having to construct the standard image. We study the reliability of the inversion by using the independent shape supports of function series and control-point surfaces. When other data are lacking, one can carry out fast nonconvex lightcurve-only inversion, but any shape models resulting from it should only be taken as illustrative global-scale ones.Comment: 11 pages, submitted to A&

Region-enhanced passive radar imaging

The authors adapt and apply a recently-developed region-enhanced synthetic aperture radar (SAR) image reconstruction technique to the problem of passive radar imaging. One goal in passive radar imaging is to form images of aircraft using signals transmitted by commercial radio and television stations that are reflected from the objects of interest. This involves reconstructing an image from sparse samples of its Fourier transform. Owing to the sparse nature of the aperture, a conventional image formation approach based on direct Fourier transformation results in quite dramatic artefacts in the image, as compared with the case of active SAR imaging. The regionenhanced image formation method considered is based on an explicit mathematical model of the observation process; hence, information about the nature of the aperture is explicitly taken into account in image formation. Furthermore, this framework allows the incorporation of prior information or constraints about the scene being imaged, which makes it possible to compensate for the limitations of the sparse apertures involved in passive radar imaging. As a result, conventional imaging artefacts, such as sidelobes, can be alleviated. Experimental results using data based on electromagnetic simulations demonstrate that this is a promising strategy for passive radar imaging, exhibiting significant suppression of artefacts, preservation of imaged object features, and robustness to measurement noise

py4DSTEM: a software package for multimodal analysis of four-dimensional scanning transmission electron microscopy datasets

Scanning transmission electron microscopy (STEM) allows for imaging, diffraction, and spectroscopy of materials on length scales ranging from microns to atoms. By using a high-speed, direct electron detector, it is now possible to record a full 2D image of the diffracted electron beam at each probe position, typically a 2D grid of probe positions. These 4D-STEM datasets are rich in information, including signatures of the local structure, orientation, deformation, electromagnetic fields and other sample-dependent properties. However, extracting this information requires complex analysis pipelines, from data wrangling to calibration to analysis to visualization, all while maintaining robustness against imaging distortions and artifacts. In this paper, we present py4DSTEM, an analysis toolkit for measuring material properties from 4D-STEM datasets, written in the Python language and released with an open source license. We describe the algorithmic steps for dataset calibration and various 4D-STEM property measurements in detail, and present results from several experimental datasets. We have also implemented a simple and universal file format appropriate for electron microscopy data in py4DSTEM, which uses the open source HDF5 standard. We hope this tool will benefit the research community, helps to move the developing standards for data and computational methods in electron microscopy, and invite the community to contribute to this ongoing, fully open-source project

Quantitative characterization of the x-ray imaging capability of rotating modulation collimators with laser light

We developed a method for making quantitative characterizations of bi-grid rotating modulation collimators (RMC ’s) that are used in a Fourier transform x-ray imager. With appropriate choices of the collimator spacings, this technique can be implemented with a beam-expanded He -Ne laser to simulate the plane wave produced by a point source at infinity even though the RMC ’s are diffraction limited at the He -Ne wavelength of 632.8 nm. The expanded beam passes through the grid pairs at a small angle with respect to their axis of rotation, and the modulated transmission through the grids as the RMC ’s rotate is detected with a photomultiplier tube. In addition to providing a quantitative characterization of the RMC ’s, the method also produces a measured point response function and provides an end-to-end check of the imaging system. We applied our method to the RMC ’s on the high-energy imaging device (HEIDI) balloon payload in its preflight configuration. We computed the harmonic ratios of the modulation time profile from the laser measurements and compared them with theoretical calculations, including the diffraction effects on irregular grids. Our results indicate the 25-in. (64-cm) x-ray imaging optics on HEIDI are capable of achieving images near the theoretical limit and are not seriously compromised by imperfections in the grids