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.[-]

Parametric PDEs arise in key applications ranging from parameter optimization, inverse state estimation, to uncertainty quantification. Accurately solving these tasks requires an efficient treatment of the resulting sets of parametric PDE solutions that are generated when parameters vary in a certain range. These solution sets are difficult to handle since their are embedded in infinite dimensional spaces, and present a complex structure. They need to be approximated with numerically efficient reduction techniques, usually called Model Order Reduction methods. The techniques need to be adapted both to the nature of the PDE, and to the given application task. In this course, we will give an overview of linear and nonlinear model order reduction methods when applied to forward and inverse problems. We will particularly emphasize on the role played by nonlinear approximation and geometrical PDE properties to address classical bottlenecks.[-]

Parametric PDEs arise in key applications ranging from parameter optimization, inverse state estimation, to uncertainty quantification. Accurately solving these tasks requires an efficient treatment of the resulting sets of parametric PDE solutions that are generated when parameters vary in a certain range. These solution sets are difficult to handle since their are embedded in infinite dimensional spaces, and present a complex structure. They need to be approximated with numerically efficient reduction techniques, usually called Model Order Reduction methods. The techniques need to be adapted both to the nature of the PDE, and to the given application task. In this course, we will give an overview of linear and nonlinear model order reduction methods when applied to forward and inverse problems. We will particularly emphasize on the role played by nonlinear approximation and geometrical PDE properties to address classical bottlenecks.[-]

Parametric PDEs arise in key applications ranging from parameter optimization, inverse state estimation, to uncertainty quantification. Accurately solving these tasks requires an efficient treatment of the resulting sets of parametric PDE solutions that are generated when parameters vary in a certain range. These solution sets are difficult to handle since their are embedded in infinite dimensional spaces, and present a complex structure. They need to be approximated with numerically efficient reduction techniques, usually called Model Order Reduction methods. The techniques need to be adapted both to the nature of the PDE, and to the given application task. In this course, we will give an overview of linear and nonlinear model order reduction methods when applied to forward and inverse problems. We will particularly emphasize on the role played by nonlinear approximation and geometrical PDE properties to address classical bottlenecks.[-]

The determination of the shape of an obstacle from its effects on known acoustic waves is an important problem in many technologies such as sonar, geophysical exploration and medical imaging. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because of its ill-posed and nonlinear nature. Its investigation requires the understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods. The main goal of this work is the development of an efficient procedure for retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics of the surface of the obstacle. We propose a solution methodology based on a regularized Newton-type method. The solution of the considered IOP by the proposed iterative method incurs, at each iteration, the solution of a linear system whose entries are the Fréchet derivatives of the elasto-acoustic field with respect to the shape parameters. We prove that these derivatives are solutions of the same direct elasto-acoustic scattering problem that differs only in the transmission conditions on the surface of the scatterer. Furthermore, the computational efficiency of the IOP solver depends mainly on the computational efficiency of the solution of the forward problems that arise at each Newton iteration. We propose to solve the direct scattering-type problems using a finite-element method based on discontinuous Galerkin approximations equipped with curved element boundaries. Numerical results will be presented to illustrate the salient features of this computational methodology and highlight its performance characteristics.

acoustics - shape derivative - inverse obstacle problem - Fréchet derivatives - inverse elasto-acoustic scattering problems[-]