Stable super-resolution limit and smallest singular value of restricted Fourier matrices

Super-resolution refers to the process of recovering the locations and amplitudes of a collection of point sources, represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. The recovery process is highly sensitive to noise whenever the distance $\Delta$ between the two closest point sources is less than $1/M$. This paper studies the {\it fundamental difficulty of super-resolution} and the {\it performance guarantees of a subspace method called MUSIC} in the regime that $\Delta<1/M$. The most important quantity in our theory is the minimum singular value of the Vandermonde matrix whose nodes are specified by the source locations. Under the assumption that the nodes are closely spaced within several well-separated clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. This implies that, as the noise increases, the super-resolution capability of MUSIC degrades according to a power law where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the resolution limit of MUSIC. When there are $S$ point sources located on a grid with spacing $1/N$, the fundamental difficulty of super-resolution can be quantitatively characterized by a min-max error, which is the reconstruction error incurred by the best possible algorithm in the worst-case scenario. We show that the min-max error is closely related to the minimum singular value of Vandermonde matrices, and we provide a non-asymptotic and sharp estimate for the min-max error, where the dominant term is $(N/M)^{2S-1}$.Comment: 47 pages, 8 figure

A fast empirical method for galaxy shape measurements in weak lensing surveys

We describe a simple and fast method to correct ellipticity measurements of galaxies from the distortion by the instrumental and atmospheric point spread function (PSF), in view of weak lensing shear measurements. The method performs a classification of galaxies and associated PSFs according to measured shape parameters, and corrects the measured galaxy ellipticites by querying a large lookup table (LUT), built by supervised learning. We have applied this new method to the GREAT10 image analysis challenge, and present in this paper a refined solution that obtains the competitive quality factor of Q = 104, without any shear power spectrum denoising or training. Of particular interest is the efficiency of the method, with a processing time below 3 ms per galaxy on an ordinary CPU.Comment: 8 pages, 6 figures. Metric values updated according to the final GREAT10 analysis software (Kitching et al. 2012, MNRAS 423, 3163-3208), no qualitative changes. Associated code available at http://lastro.epfl.ch/megalu

A holistic approach to structure from motion

This dissertation investigates the general structure from motion problem. That is, how to compute in an unconstrained environment 3D scene structure, camera motion and moving objects from video sequences. We present a framework which uses concatenated feed-back loops to overcome the main difficulty in the structure from motion problem: the chicken-and-egg dilemma between scene segmentation and structure recovery. The idea is that we compute structure and motion in stages by gradually computing 3D scene information of increasing complexity and using processes which operate on increasingly large spatial image areas. Within this framework, we developed three modules. First, we introduce a new constraint for the estimation of shape using image features from multiple views. We analyze this constraint and show that noise leads to unavoidable mis-estimation of the shape, which also predicts the erroneous shape perception in human. This insight provides a clear argument for the need for feed-back loops. Second, a novel constraint on shape is developed which allows us to connect multiple frames in the estimation of camera motion by matching only small image patches. Third, we present a texture descriptor for matching areas of extended sizes. The advantage of this texture descriptor, which is based on fractal geometry, lies in its invariance to any smooth mapping (Bi-Lipschitz transform) including changes of viewpoint, illumination and surface distortion. Finally, we apply our framework to the problem of super-resolution imaging. We use the 3D motion estimation together with a novel wavelet-based reconstruction scheme to reconstruct a high-resolution image from a sequence of low-resolution images