In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine learning.
imaging - image processing - sparsity - convex optimization - inverse problem - super-resolution[-]

One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This finding triggered an enormous research activity in recent years both in signal processing applications as well as their mathematical foundations. The present talk discusses connections of compressive sensing and time-frequency analysis (the sister of wavelet theory). In particular, we give on overview on recent results on compressive sensing with time-frequency structured random matrices.

Keywords: compressive sensing - time-frequency analysis - wavelets - sparsity - random matrices - $\ell_1$-minimization - radar - wireless communications[-]

Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality and optimality of approximations and related concepts from approximation the- ory. We will see that wavelet bases can serve as a basic ingredient, both for the theory as well as for algorithmic realizations. Particularly for situations where solutions exhibit singularities, wavelet concepts enable adaptive appproximations for which convergence and optimal algorithmic complexity can be established. I will describe corresponding implementations based on biorthogonal spline-wavelets.
Moreover, wavelet-related concepts have triggered new developments for efficiently solving complex systems of PDEs, as they arise from optimization problems with PDEs.[-]

The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, Poissonian, and multiplicative, and with several types of regularizers (TV, frame-based analysis, synthesis, or combinations thereof). I will then describe more recent work on using ADMM for other problems, namely blind deconvolution and image segmentation, as well as very recent work where ADMM is used with plug-in learned denoisers to achieve state-of-the-art results in class-specific image deconvolution. Finally, on the theoretical front, I will describe very recent work on tackling the infamous problem of how to adjust the penalty parameter of ADMM.[-]

Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by restricting our ability to observe biological structure and processes smaller than 100nm. Recent progress in sparsity-driven signal processing has created a powerful paradigm for increasing both the resolution and overall quality of imaging by promoting model-based image acquisition and reconstruction. This has led to multiple influential results demonstrating super-resolution in practical imaging systems. To date, however, the vast majority of work in signal processing has neglected the fundamental nonlinearity of the object-light interaction and its potential to lead to resolution enhancement. As a result, modern theory heavily focuses on linear measurement models that are truly effective only when object-light interactions are weak. Without a solid signal processing foundation for understanding such nonlinear interactions, we undervalue their impact on information transfer in the image formation. This ultimately limits our capability to image a large class of objects, such as biological tissue, that generally are in large-volumes and interact strongly and nonlinearly with light.
The goal of this talk is to present the recent progress in model-based imaging under multiple scattering. We will discuss several key applications including optical diffraction tomography, Fourier Ptychography, and large-scale Holographic microscopy. We will show that all these application can benefit from models, such as the Rytov approximation and beam propagation method, that take light scattering into account. We will discuss the integration of such models into the state-of-the-art optimization algorithms such as FISTA and ADMM. Finally, we will describe the most recent work that uses learned-priors for improving the quality of image reconstruction under multiple scattering.[-]

This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.[-]

This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.[-]