Title: Discretizations of the total variation for singular functions
Abstract: In this talk, based on joint work with Corentin Caillaud and Thomas Pock, I will discuss the merits of some finite differences or finite elements discretizations of the total variation functional. I will mention error bounds for the recovery of sharp discontinuities, inpainting straight lines in 2D, and trying to improve on some recent approaches proposed in the literature.
Title: Deep Learning-Based Solvability of Underdetermined Inverse Problems in Medical Imaging
Abstract: The significant recent developments in deep learning have made it relevant to solving underdetermined inverse problems, a major concern in medical imaging. Typical examples where deep learning techniques showed excellent performance include undersampled magnetic resonance imaging, local tomography, and sparse view computed tomography. Although these methods appear to overcome the limitations of existing mathematical methods in handling underdetermined problems, the rigorous mathematical foundations underpinning the success of deep learning remain unknown. This talk deals with the causal relationship between the structure of training data and the ability of deep learning to solve highly underdetermined inverse problems.
Title: Learning to Solve Inverse Problems in Imaging
Abstract: Traditional inverse problem solvers in imaging minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. In this talk, I will describe various classes of approaches to learned regularization, ranging from generative models to unrolled optimization perspectives, and explore their relative merits.
Title: Minimization based formulation and regularization of inverse problems
Abstract: The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter, is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to too restrictive conditions. Here the model and the observation are considered simultaneously as one large system with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing both the reduced and the all-at-once formulation, but also the well-known and highly versatile so-called variational approach (not to be mistaken with variational regularization) as special cases, is to formulate the inverse problem as a minimization problem (instead of an equation) for the parameter *and* state. Regularization can be incorporated via imposing constraints and/or adding regularization terms to the objective. In this talk we will provide convergence results as well as some new application examples of minimization based formulations, such as impedance tomography with the complete electrode model, identification of a nonlinear magnetic permeabilty from magnetic flux measurements, and localization of sound sources from microphone array measurements.
Title: From Compressed Sensing to Deep Learning: Tasks, Structures, and Models
Abstract: The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal and image processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting high rate digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution.
In this talk, we present a framework for sampling and processing a wide class of wideband analog signals at rates far below Nyquist by exploiting signal structure and the processing task and show several demos of real-time sub-Nyquist prototypes. We consider applications of these ideas to a variety of problems in imaging including fast and quantitative MRI, wireless ultrasound, fast Doppler imaging, and correlation based super-resolution in microscopy and ultrasound which combines high spatial resolution with short integration time. We then show how the ideas of exploiting the task, structure and model can be used to develop interpretable model-based deep learning methods that can adapt to existing structure and are trained from small amounts of data. These networks achieve a more favorable trade-off between increase in parameters and data and improvement in performance while remaining interpretable.
Title: Continuous relaxations for sparse l2-l0 constrained optimization problems
Abstract: Many inverse problems in signal and image processing are formulated as sparse optimization problems involving a least-square term (l2 data term) and an l0-pseudo norm as regularizing or constraint term. These NP-hard combinatorial problems are of fundamental importance in many applications such as coding, compressed sensing, source separation or variable selection. If the l1-norm can advantageously replace the l0-term for certain problems such as Compressed Sensing, this cannot be done for some other reconstruction problems.
Among several methods, a standard way to minimize such NP hard problems is to find a continuous relaxation for which standard minimization algorithms can be applied. In previous work, we have proposed a continuous relaxation for the penalized l2-l0 problem, which is exact in the sense that it does not change the global minimizers, while it removes some local minimizers. In this talk, we are interested in finding such a relaxation for the constrained l2-l0 problem. We propose a continuous relaxation for which we only have partial results on minimizers, but which allows to propose a new algorithm for the minimization of the constrained l2-l0 problem. This is of interest as fixing the constraint constant is often more intuitive than fixing the penalty parameter. We also introduce an alternative continuous relaxation with the associated algorithm, but which obliges to increase the size of the space of the unknowns. Finally comparative results on the problem of single molecule localization microscopy for super-resolution will be shown.
Title: Local and global structures of transmission eigenfunctions and applications to super-resolution imaging and electromagnetic mirage
Abstract: The interior transmission eigenvalue problems are a type of non-elliptic, non-self-adjoint and nonlinear spectral problems that arise in the wave scattering theory. It connects to many aspects of the direct and inverse scattering theory. In this talk, I shall discuss some of our recent discoveries on the global and local structures of the transmission eigenfunctions. We shall also consider their applications to super-resolution imaging and electromagnetic mirage.
Material: video, slides.
Title: On the efficient implementation of iterative methods for solving ill-posed problems: The range-relaxed strategy
Abstract: In this talk we investigate the iterated Tikhonov and the Levenberg-Marquardt iterations for solving ill-posed operator equations. More precisely we address the a posteriori choice of regularization parameters (or Lagrange multipliers) for these methods. We focus our attention on the range-relaxed strategy for choosing these parameters. Namely, the residual of the next iterate is prescribed to be in a range; consequently the set of feasible Lagrange multipliers, at each iteration, is a non-degenerate interval, which renders feasible their economical computation.
Title: Inverse problems for non-linear hyperbolic and elliptic equations.
Abstract: In the talk we give an overview on how inverse problems can be used solved using non-linear interaction of the solutions. This method can be used for several different inverse problems for non-linear hyperbolic or elliptic equations. In this approach one does not consider the non-linearity as a troublesome perturbation term, but as an effect that aids in solving the problem. Using it, one can solve inverse problems for non-linear equations for which the corresponding problem for linear equations is still unsolved.
For the hyperbolic equations, we consider the non-linear wave equation $\square_g u+u^m=f$ on a Lorentzian manifold $M\times R$ and the source-to-solution map $\Lambda_V:f\to u|_V$ that maps a source $f$, supported in an open domain $V\subset M\times R$, to the restriction of $u$ in $V$. Under suitable conditions, we show that the observations in $V$, that is, the map $\Lambda_V$, determine the metric $g$ in a larger domain which is the maximal domain where signals sent from $V$ can propagate and return back to $V$.
We apply non-linear interaction of solutions of the linearized equation also to study non-linear elliptic equations. For example, we consider $\Delta_g u+qu^m=0$ in $\Omega\subset R^n$ with the boundary condition $u|_{\partial \Omega}=f$. For this equation we define the Dirichlet-to-Neumann map $\Lambda_{\partial \Omega}:f\to u|_V$. Using the high-order interaction of the solutions, we consider various inverse problems for the metric $g$ and the potential $q$.
The presented results for the wave equations are obtained with Yaroslav Kurylev, Gunther Uhlmann, and Yiran Wang in the 3-dimensional case and with Ali Feizmohammadi and Lauri Oksanen in other dimensions. For elliptic equations the results are obtained with Tony Liimatainen, Yi-Hsuan Lin, and Mikko Salo.
Title: Deep CT Imaging by Unrolled Dynamics
Abstract: In this talk, I will start with a brief review of the dynamics and optimal control perspective on deep learning (including supervised learning, reinforcement learning, and meta-learning), especially the so-called unrolled dynamics approach and its applications in medical imaging. Then, I will present some of our recent studies on how this new approach may help us to advance CT imaging and image-based diagnosis further. Specifically, I will focus on our thoughts on how to combine the wisdom from mathematical modeling with ideas from deep learning. Such combination leads to new data-driven image reconstruction models and new data-driven scanning strategies for CT imaging, and with a potential to be generalized to other imaging modalities.
Title: Geometry of Deep Learning for Inverse Problems: A Signal Processing Perspective
Abstract: Recently, deep learning approaches have been extensively used for various inverse problems thanks to its excellent performance. However, it is still difficult to obtain coherent signal processing understanding why such deep learning architectures provide superior performance. In this talk, I will first discuss the limitations of the existing machine learning approaches and state the desirable properties that an ultimate machine learning approach should satisfy. Then, I will explain why deep neural networks, in particular CNN, satisfy the desiderata, using the novel theory of deep convolutional framelets and input manifold partition geometry by ReLUs. Then, our recent geometric understanding of unsupervised learning is explained in terms of optimal transport geometry. We also provide extensive experimental results for several biomedical imaging reconstruction problems to verify our geometric understanding of CNNs for inverse problems.
Title: Inverse problems arising in non-imaging optics
Abstract: The goal of non-imaging optics is to design optical components, such as mirrors or lenses, that transfer a given source light to a prescribed target. Several inverse problems arising in this field amount to solving Monge-Ampère type equations. In this talk, I will show how these equations are connected to optimal transport and can be solved using a geometric discretization called semi-discrete. I will also present the design of different kinds of mirrors or lenses that allow to transfer any pointwise or collimated source to any target.
This work involves Quentin Mérigot and Jocelyn Meyron.
Eric Bonnetier, Université Grenoble-Alpes, France.
Luca Calatroni, CNRS, Université Côte d'Azur, France.
Raymond Chan, CityU, Hong Kong.
Fadil Santosa, Johns Hopkins University, USA
Carola-Bibiane Schönlieb, University of Cambridge, UK.