Imaging & inverse problems (IMAGINE) OneWorld seminars

Title: Imaging with unstructured adaptive meshes

Abstract: In many applications, imaging problems involve a numerical solution of a partial differential equation or an integral equation, requiring a discretization of the model. The method of choice for PDEs is to use finite element methods over a triangularization of the computational domain. To guarantee good resolution and mitigate the effect of the discretization error, dense meshes are preferred, which leads to an increased computational cost. In this talk, the problem is addressed from the point of view of mesh adaptation, the discretization being part of the inverse problem.

Title: Blind Image Deblurring: What is the Next Step?

Abstract: Blind image deblurring is a challenging task in imaging science where we need to estimate the latent image and blur kernel simultaneously. To get a stable and reasonable deblurred image, proper prior knowledge of the latent image and the blur kernel is urgently required. In this talk, we address several of our recent attempts related to image deblurring. Indeed, different from the recent works on the statistical observations of the difference between the blurred image and the clean one, we first report the surface-aware strategy arising from the intrinsic geometrical consideration. This approach facilitates the blur kernel estimation due to the preserved sharp edges in the intermediate latent image. Extensive experiments demonstrate that our method outperforms the state-of-the-art methods on deblurring the text and natural images. Moreover, we discuss the Quaternion-based method for color image restoration. After that, we extend the quaternion approach for blind image deblurring.

Title: Mean curvature flow, neural networks, and applications

Abstract: Many applications in image processing (denoising, segmentation), data science (point cloud smoothing, shape matching), material sciences (grain evolution in alloys, crystal growth) or biology (cell modeling) require the approximation of geometric interface evolution such as the emblematic mean curvature flow.

In this context, the phase field method is a particularly efficient tool to approximate the evolution of oriented surfaces, but things turn to be much more difficult for non-oriented surfaces. I will explain how interface evolutions can be approximated even in this case by training a neural network whose structure is derived from classical schemes associated with the Allen-Cahn equation. I will show applications of this new approach to the approximation of solutions to the Steiner and Plateau problems.

Title: Monotonicity method for extreme, singular and degenerate inclusions in electrical impedance tomography

Abstract: In electrical impedance tomography, the monotonicity method enables simultaneously reconstructing both definite and indefinite inclusions under only mild assumptions on the conductivity perturbations and the background conductivity. However, in its standard form the method requires the conductivity inside the inclusions to be bounded away from zero and infinity. This work generalizes the method for extreme inclusions that correspond to some parts of the domain being perfectly conducting or insulating. The result is also extended to allow the conductivity perturbation to be the restriction of an A2-Muckenhaupt weight in parts of the domain, thereby including singular and degenerate behavior in the governing elliptic equation.

Title: On the use of (Linear) Surrogate Models for Bayesian Inverse Problems

Abstract: In this talk we consider the use of surrogate (forward) models to efficiently solve Bayesian inverse problems. We particularly concern ourselves with the use of linear surrogates. The problems considered are from a range of applications but are all high dimensional problems resulting from the discretisation of partial differential equations. We adopt the Bayesian approach to account for the model discrepancy, which is treated as an additional stochastic error term. We prove a somewhat surprising result: under the assumption of a Gaussian prior and additive noise model the approximate posterior found by using a linear(-ised) surrogate is invariant to the choice of linear surrogate, so long as the model discrepancy is (approximately) taken into account. This is ongoing joint work with Noemi Petra, Umberto Villa, and Jari P. Kaipio.

Title : Analytic shape reconstruction of a planar conductivity inclusion

Abstract : A conductivity inclusion inserted in a homogeneous background induces a perturbation in the background potential. The perturbation admits a multipole expansion, whose coefficients are the so-called generalized polarization tensors (GPTs). For a simply connected planar domain, there exists uniquely an exterior Riemann mapping, which determines the shape of the domain. In this talk, I will present explicit expressions for the Riemann mapping coefficients of the inclusion in terms of the GPTs. The expression formulas are derived from matrix factorizations of the GPTs by using the concept of the Faber polynomial polarization tensors.

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Title: Learning Robust Imaging Models without Paired Data

Abstract: The observations in practical imaging systems always contain complex noise such that classical approaches are difficult to obtain satisfactory results. In recent years, deep neural networks directly learned a map between the noisy and clean images based on the training on paired data. Despite its promising results in various tasks, collecting the training data is difficult and time-consuming in practice. In this talk, in the unpaired data regime, we will discuss our recent progress for building AI-aided robust models and their applications in image processing. Leveraging the Bayesian inference framework, our model combines classical mathematical modeling and deep neural networks to improve interpretability. Experimental results on various real datasets validate the advantages of the proposed methods.

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