Abstract : The interplay of deep learning with inverse and imaging problems has seen a tremendous progress during the last years producing state-of-the-art results in most tasks. Apart from the availability of large data and the increased computing power, this progress has been mainly facilitated by the development of rigorous theoretical investigations. The purpose of this minisymposium is to bring together experts in data-driven inverse imaging problems who work both in theory and applications. The aim is to stimulate a fruitful knowledge exchange about how mathematical theories can contribute and further develop this field.
Organizer(s) : Guozhi Dong, Michael Hintermüller, Kostas Papafitsoros
00687 (1/3) : 2C @E605 [Chair: Michael Hintermueller]
[05273] Data-driven parameter optimization for some inverse problems with sparsity-based priors
Format : Talk at Waseda University
Author(s) :
Juan Carlos De los Reyes (MODEMAT)
Abstract : In recent years, novel ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different parameters of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based on a training set is used. When confronted with inverse problems with sparsity-based regularizers, however, the bilevel optimization problem structure becomes quite involved to be analyzed, as classical nonlinear or bilevel programming results cannot be directly utilized.
In this talk, I will discuss on a strategy to overcome these difficulties, leading to a reformulation of the bilevel problems as mathematical programs with complementarity constraints. This enables to obtain sharp first-order optimality conditions, but at the price of lifting the problems to a higher dimension. Some ideas on how to reduce the dimension of the problems back will also be presented, together with the different challenges that these problems pose, when dealing with large training sets.
[03185] Data-driven Joint Inversion for PDE Models
Format : Talk at Waseda University
Author(s) :
Kui Ren (Columbia University)
Abstract : The task of simultaneously reconstructing multiple physical coefficients in partial differential equations from observed data is ubiquitous in applications. In this work, we propose an integrated data-driven and model-based iterative reconstruction framework for such joint inversion problems where additional data on the unknown coefficients are supplemented for better reconstructions. Our method couples the supplementary data with the PDE model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We characterize the impact of learning uncertainty on the joint inversion results for two typical model inverse problems. This is based on a joint work with Lu ZHang.
[02972] A scalable deep learning approach for solving high-dimensional dynamic optimal transport
Format : Talk at Waseda University
Author(s) :
Zuoqiang Shi (Tsinghua University)
Abstract : The dynamic formulation of optimal transport has attracted growing interests. In this talk, we propose a deep learning based method to solve the dynamic optimal transport in high dimensional space based on carefully designed representation of the velocity field, the discretization along the characteristics, and the computation of high dimensional integral by Monte Carlo method. Numerical experiments show that our method could give more accurate results in high dimensional cases and has very good scalability.
[04688] Learning nonlinearities in time-dependent PDEs from data
Format : Talk at Waseda University
Author(s) :
Christian Aarset (University of Göttingen)
Martin Holler (University of Graz)
Tram Thi Ngoc Nguyen (Max-Planck Institute for Solar System Research, Göttingen)
Abstract : We introduce and analyze an all-at-once approach for learning parts of a partial-differential-equation-based model from data. More specifically, we consider the learning of a non-linearity in the model, which acts pointwise on the state, from indirect, noisy measurements. We provide a function-space analysis of the corresponding learning problem and of the resulting PDE with learned components in a general setting. Furthermore, we show numerical experiments that confirm the practical feasibility of the proposed method.
[04070] Conductivity imaging using deep neural networks
Format : Online Talk on Zoom
Author(s) :
Bangti Jin (Chinese University of Hong Kong)
Abstract : Conductivity imaging from various observational data represents one fundamental task in medical imaging. In this talk, we discuss numerical methods for identifying the conductivity parameters in elliptic PDEs. Commonly, a regularized formulation consists of a data fidelity and a regularizer is employed, and then it is discretized using finite difference method, finite element methods or deep neural networks in practical computation. One key issue is to establish a priori error estimates for the recovered conductivity distribution. In this talk, we discuss our recent findings on using deep neural networks for this class of problems, by effectively utilizing relevant stability results.
[04396] Model-corrected learned primal-dual models for fast photoacoustic tomography
Format : Talk at Waseda University
Author(s) :
Andreas Hauptmann (University of Oulu)
Abstract : Learned iterative reconstructions hold great promise to accelerate tomographic imaging with empirical robustness to model perturbations. Adoption for photoacoustic tomography is hindered by the computational expensive forward model. Computational feasibility can be obtained by the use of fast approximate models, but model errors need to be compensated.
In this talk we discuss conceptual difficulties and present methodological advances for model corrections in learned image reconstructions by embedding the model correction in a learned primal-dual framework.
[03165] Learning the Regularisation Parameter for Inverse Problems
Format : Online Talk on Zoom
Author(s) :
Sebastian Scott (University of Bath)
Matthias Ehrhardt (University of Bath)
Silvia Gazzola (University of Bath)
Abstract : Solving linear inverse problems via variational regularisation involves the use of unknown regularisation parameters. In order to attain a meaningful reconstruction, these parameters must be carefully chosen. This work will cover bilevel learning, a framework in which one is able to learn appropriate parameter values via a machine learning approach.
[04247] Lipschitz Training for Adversarially Robust Neural Networks
Format : Talk at Waseda University
Author(s) :
Tim Roith (Friedrich-Alexander-Universität Erlangen-Nürnberg)
Abstract : Adversarial examples have revealed the vulnerability of neural networks, making stability and robustness key concerns. To address this, we explore the role of the Lipschitz constant in adversarial machine learning. I will present an algorithm that employs an approximate Lipschitz constant as a regularizer. In each training step, we compute points that aim to maximize a difference quotient. Finally, I will discuss the conceptual limits of methods enforcing a low Lipschitz constant of neural networks.
[05448] The Unrolled Dynamics Modeling for Computed Tomography
Format : Talk at Waseda University
Author(s) :
Haimiao Zhang (Beijing Information Science and Technology University)
Abstract : Deep learning revolutionized the research paradigm of medical imaging in the last decades. In this talk, I will show our works on combining classical mathematical models in computed tomography (CT) imaging with the modern deep neural network. These new computational imaging techniques give us a broader way of dealing with the medical imaging problem. Furthermore, numerical results demonstrated that the proposed deep models could be generalized to different datasets, scanning geometries, and noise levels.
[04836] Dictionary learning for an inverse problem in quantitative MRI
Format : Talk at Waseda University
Author(s) :
Guozhi Dong (Central South University)
Michael Hintermueller (Weierstrass Institute Berlin)
Clemens Sirotenko (WIAS Berlin)
Abstract : The field of quantitative Magnetic Resonance Imaging aims at extracting physical tissue parameters from a sequence of highly under sampled MR images. Mathematically, this can be achieved by estimating a set of unknown parameters in an ODE model. We employ dictionary learning based approaches to regularize the reconstruction process and investigate iterative schemes to solve the resulting non-convex and non-smooth problems for stationarity. Moreover numerical results and open questions are presented.
[05295] Deformable volumetric Image registration based on unsupervised learning
Author(s) :
Ahsan Raza Siyal (University of Innsbruck)
Abstract : Deformable image registration has found its way into clinical routine, from image-guided adaptive radiotherapy to brain surgery. The traditional methods optimize an objective function independently for each pair of images, which is highly expensive in terms of time and computation. On the other hand, the deformable registration task can be defined as a parametric function and optimize its parameters on available image pairs and the function can be modeled as neural networks which learn the off-set displacement field through unsupervised loss function which contains data similarity term and a regularization term.
