Estimation of Translation, Rotation, and Scaling between Noisy Images Using the Fourier–Mellin Transform

In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problems, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation and scaling parameters to align a set of images. This criterion is a two step procedure which does not require the use of a reference template onto which aligning all the images. Our approach is based on M-estimation and we prove the consistency of the resulting estimators. A small scale simulation study and real examples are used to illustrate the numerical performances of our procedure

Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Superpixel-based spatial amplitude and phase modulation using a digital micromirror device

We present a superpixel method for full spatial phase and amplitude control of a light beam using a digital micromirror device (DMD) combined with a spatial filter. We combine square regions of nearby micromirrors into superpixels by low pass filtering in a Fourier plane of the DMD. At each superpixel we are able to independently modulate the phase and the amplitude of light, while retaining a high resolution and the very high speed of a DMD. The method achieves a measured fidelity $F=0.98$ for a target field with fully independent phase and amplitude at a resolution of $8\times 8$ pixels per diffraction limited spot. For the LG$_{10}$ orbital angular momentum mode the calculated fidelity is $F=0.99993$, using $768\times 768$ DMD pixels. The superpixel method reduces the errors when compared to the state of the art Lee holography method for these test fields by $50\%$ and $18\%$, with a comparable light efficiency of around $5\%$. Our control software is publicly available.Comment: 9 pages, 6 figure

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&