Sparse approximations of protein structure from noisy random projections

Single-particle electron microscopy is a modern technique that biophysicists employ to learn the structure of proteins. It yields data that consist of noisy random projections of the protein structure in random directions, with the added complication that the projection angles cannot be observed. In order to reconstruct a three-dimensional model, the projection directions need to be estimated by use of an ad-hoc starting estimate of the unknown particle. In this paper we propose a methodology that does not rely on knowledge of the projection angles, to construct an objective data-dependent low-resolution approximation of the unknown structure that can serve as such a starting estimate. The approach assumes that the protein admits a suitable sparse representation, and employs discrete $L^1$-regularization (LASSO) as well as notions from shape theory to tackle the peculiar challenges involved in the associated inverse problem. We illustrate the approach by application to the reconstruction of an E. coli protein component called the Klenow fragment.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS479 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Low-Complexity Modeling for Visual Data: Representations and Algorithms

With increasing availability and diversity of visual data generated in research labs and everyday life, it is becoming critical to develop disciplined and practical computation tools for such data. This thesis focuses on the low complexity representations and algorithms for visual data, in light of recent theoretical and algorithmic developments in high-dimensional data analysis. We first consider the problem of modeling a given dataset as superpositions of basic motifs. This model arises from several important applications, including microscopy image analysis, neural spike sorting and image deblurring. This motif-finding problem can be phrased as "short-and-sparse" blind deconvolution, in which the goal is to recover a short convolution kernel from its convolution with a sparse and random spike train. We normalize the convolution kernel to have unit Frobenius norm and then cast the blind deconvolution problem as a nonconvex optimization problem over the kernel sphere. We demonstrate that (i) in a certain region of the sphere, every local optimum is close to some shift truncation of the ground truth, when the activation spike is sufficiently sparse and long, and (ii) there exist efficient algorithms that recover some shift truncation of the ground truth under the same conditions. In addition, the geometric characterization of the local solution as well as the proposed algorithm naturally extend to more complicated sparse blind deconvolution problems, including image deblurring, convolutional dictionary learning. We next consider the problem of modeling physical nuisances across a collection of images, in the context of illumination-invariant object detection and recognition. Illumination variation remains a central challenge in object detection and recognition. Existing analyses of illumination variation typically pertain to convex, Lambertian objects, and guarantee quality of approximation in an average case sense. We show that it is possible to build vertex-description convex cone models with worst-case performance guarantees, for nonconvex Lambertian objects. Namely, a natural detection test based on the angle to the constructed cone guarantees to accept any image which is sufficiently well approximated with an image of the object under some admissible lighting condition, and guarantees to reject any image that does not have a sufficiently approximation. The cone models are generated by sampling point illuminations with sufficient density, which follows from a new perturbation bound for point images in the Lambertian model. As the number of point images required for guaranteed detection may be large, we introduce a new formulation for cone preserving dimensionality reduction, which leverages tools from sparse and low-rank decomposition to reduce the complexity, while controlling the approximation error with respect to the original cone. Preliminary numerical experiments suggest that this approach can significantly reduce the complexity of the resulting model

Interference Removal for Radar/Communication Co-existence: the Random Scattering Case

In this paper we consider an un-cooperative spectrum sharing scenario, wherein a radar system is to be overlaid to a pre-existing wireless communication system. Given the order of magnitude of the transmitted powers in play, we focus on the issue of interference mitigation at the communication receiver. We explicitly account for the reverberation produced by the (typically high-power) radar transmitter whose signal hits scattering centers (whether targets or clutter) producing interference onto the communication receiver, which is assumed to operate in an un-synchronized and un-coordinated scenario. We first show that receiver design amounts to solving a non-convex problem of joint interference removal and data demodulation: next, we introduce two algorithms, both exploiting sparsity of a proper representation of the interference and of the vector containing the errors of the data block. The first algorithm is basically a relaxed constrained Atomic Norm minimization, while the latter relies on a two-stage processing structure and is based on alternating minimization. The merits of these algorithms are demonstrated through extensive simulations: interestingly, the two-stage alternating minimization algorithm turns out to achieve satisfactory performance with moderate computational complexity