Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Planning the Future of U.S. Particle Physics (Snowmass 2013): Chapter 4: Cosmic Frontier

These reports present the results of the 2013 Community Summer Study of the APS Division of Particles and Fields ("Snowmass 2013") on the future program of particle physics in the U.S. Chapter 4, on the Cosmic Frontier, discusses the program of research relevant to cosmology and the early universe. This area includes the study of dark matter and the search for its particle nature, the study of dark energy and inflation, and cosmic probes of fundamental symmetries.Comment: 61 page

Modeling and Compensating of Noise in Time-of-Flight Sensors

Three-dimensional (3D) sensors provide the ability to perform contactless measurements of objects and distances that are within their field of view. Unlike traditional two-dimensional (2D) cameras, which only provide RGB data about objects within a scene, 3D sensors are able to directly provide depth information for objects within a scene. Of these 3D sensing technologies, Time-of-Flight (ToF) sensors are becoming more compact which allows them to be more easily integrated with other devices and to find use in more applications. ToF sensors also provide several benefits over other 3D sensing technologies that increase the types of applications where ToF sensors can be used. For example, over the last decade, ToF sensors have become more widely used in applications such as 3D scanning, drone positioning, robotics, logistics, structural health monitoring, and road surveillance. To further extend the applications where ToF sensors can be employed, this work focuses on how to improve the performance of ToF sensors by suppressing and mitigating the effects of noise artifacts that are associated with ToF sensors. These issues include multipath interference, motion blur, and multicamera interference in 3D depth maps and point clouds