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

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Comparitive assessment of the vulnerability and resilience of 10 deltas, synthesis report

The proposed framework for delta assessment and especially the scorecards are intended to enhance awareness raising, discussion and prioritization on most relevant delta issues, in each delta but also in comparison with other deltas. This should lead to more efficient and effective (multi-sectoral) policy formulation, management design and implementation, in concrete Delta plans, pilot-projects and (research) programmes. The target groups are all stakeholders who are involved in delta management at different levels and with different interests (government, private companies, NGOs, public), and who wish to contribute to the resilience of their own delta and other deltas worldwide

Compressive Wave Computation

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology.Comment: 45 pages, 4 figure

An Interdisciplinary Approach to Assessing, Planning and Managing Urban Rivers in the context of Greater London

PhDUrban rivers present complex management challenges due to the combined natural and anthropocentric factors affecting developed catchments. Planning urban river rehabilitation strategies and measures in parallel with green infrastructure initiatives requires the combined expertise of multi-disciplinary partnerships, encompassing river science and landscape engineering plus community engagement, to deliver integrated and sustainable outcomes. This thesis takes an interdisciplinary approach to investigate the assessment and management of urban rivers, focusing specifically upon the planning of integrated restoration projects for River Thames tributaries within Greater London. Comparisons of restored and unrestored sites on London tributary rivers at the reachand catchment-scale explore the versatility of the Urban River Survey method for assessing and communicating contrasts in the bio-physical condition and engineering:habitat associations of heavily modified rivers. A trial of the Ecosystem Services Assessment method for urban river restorations indicates the strengths and limitations of this approach and areas of research need. Urban river governance investigations and a review of changes in restoration practices over time confirm a decreasing emphasis on channel control and progressively lighter engineering, plus a greater social focus with urban river management becoming increasingly driven by awareness of the symbiosis between rivers and local communities. In some London boroughs partner organisations are developing new links through sustainable development objectives, but connections are geographically inconsistent and typically dependent upon key advocates. Findings indicate that integrated planning can facilitate interdisciplinary processes through the identification of cross-cutting themes (e.g. climate change) and open knowledge exchange when delivered with appropriate levels of detail. While some disciplinary boundaries are necessary (to define project scope and for task management), socio-ecological benefits may be achieved when these are flexible, permeable and managed responsively in relation to simple overarching goals; and by allowing time for different kinds of knowledge to merge and stimulate new creative and integrated interpretations