# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45

## Tuesday, September 30

15:00 - 16:00 Benjamin Berkels (Bonn University): Extracting grain boundaries and macroscopic deformations from images on atomic scale

Abstract: Nowadays image acquisition in materials science allows the resolution of grains at atomic scale. Grains are material regions with different lattice orientation which are frequently in addition elastically stressed. At the same time, new microscopic simulation tools allow to study the dynamics of such grain structures. Single atoms are resolved experimentally as well as in simulation results on the data microscale, whereas lattice orientation and elastic deformation describe corresponding physical structures mesoscopically. A qualitative study of experimental images and simulation results and the comparison of simulation and experiment requires the robust and reliable extraction of mesoscopic properties from the microscopic image data. Based on a Mumford-Shah type functional, grain boundaries are described as free discontinuity sets at which the orientation parameter for the lattice jumps. The lattice structure itself is encoded in a suitable integrand depending on a local lattice orientation and an elastic displacement. For each grain a lattice orientation and one global elastic displacement function are considered as unknowns implicitly described by the image microstructure. In addition the approach incorporates solid-liquid interfaces. The resulting Mumford-Shah functional is approximated with a level set active contour model following the approach by Chan and Vese. The implementation is based on a finite element discretization in space and a step size controlled, regularized gradient descent algorithm.

## Tuesday, October 7

15:00 - 16:00 Prof. Riccardo March (Instituto per le Applicazioni deö Calcolo): A variational method for the classification of Magnetic Resonance Images

Abstract: We consider ordered sequences of digital images. At a given pixel a time course is observed which is related to the time courses at neighbor pixels. Useful information can be extracted from a set of such observations by classifying pixels in groups, according to some features of interest. We assume to observe a noisy version of a positive function depending on space and time, which is parameterized by a vector of unknown functions (depending on space) with discontinuities which separate regions with different features in the image domain. We propose a variational method which allows to estimate the parameter functions, to segment the image domain in regions, and to assign to each region a label according to the values that the parameters assume on the region. Approximation by Γ-convergence is used to design a numerical scheme. Numerical results are reported for a dynamic Magnetic Resonance imaging problem.

## Monday, October 13

15:00 - 16:00 Dr. Ha Quang Minh (Humboldt University of Berlin): Learning with kernels: supervised and semisupervised

Abstract: This talk will give an overview of supervised and semisupervised learning in the reproducing kernel Hilbert spaces (RKHS) setting. We will talk about the smoothing properties of the different RKHS norms. The semisupervised setting will involve the concept of the graph Laplacian from spectral graph theory. We will also discuss the design of two specific kernels that have been applied to problems in biology.

## Tuesday, October 14

15:00 - 16:00 Prof. Haomin Zhou (Georgia Institute of Technology): Images, PDEs and Wavelets

Abstract: It is well evident that both wavelets and PDEs have profound impacts on imaging sciences. The success relies on their remarkable mathematical properties, many of which are complimentary to each other. In this talk, I will present an overview of our work along the direction of merging them to further improve the performance and model new applications in image processing. A main goal is to reduce or even eliminate the well-known Gibbs’ oscillations, which are considered as one of the major obstacles in wavelet based image processing methods. Two different approaches have been used. One is to use PDE ideas to directly change wavelet transform algorithms so as to generate wavelet coefficients which can avoid oscillations in reconstructions when the high frequency coefficients are truncated. The other one stays with standard wavelet transforms and use variational PDE techniques to select and modify the coefficients in the thresholding process so that the less oscillatory reconstructions can be obtained. More precisely, the first part of the talk will be on an adaptive ENO wavelet transform designed by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet transforms retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining arbitrary high order accuracy uniformly and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have also shown the stability of the ENO-wavelet transform and obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. The second part of the talk is on using a variational framework, in particular the minimization of total variation (TV), to select and modify the retained standard wavelet coefficients in the wavelet thresholding procedure so that the reconstructed images have fewer oscillations near edges. Applications in image compression, denoising, wavelet inpainting will be mentioned. Connections with other related work such as compressed sensing (CS) will be commented.

## Tuesday, October 21

15:00 - 16:00 Prof. Giovanni Bellettini (Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata"): Completion of visible contours

Abstract: We show that the completion problem of reconstructing the hidden arcs of the contours of an image, given only the visible ones, has a solution. More precisely, given an oriented plane graph K having as vertices only T-junctions and nonexterior terminal points, there exists an apparent contour G such that K is the visible part of G. As a consequence, from the plane graph K we can reconstruct a solid shape E in three-dimensional space such that K coincides with the visible part of the apparent contour of E.

## Wednesday, October 29

15:00 - 16:00 Prof. Simon Masnou (Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie, Paris): Exemplar-based inpainting from a variational point of view

Abstract: Initially introduced in the context of texture synthesis, the so-called exemplar-based methods are also very efficient for the more general problem of inpainting, i.e. the task of recovering missing parts in a digital image. The results are often striking with a computational cost that can be significantly reduced. These methods have always been presented as algorithms in a discrete setting and very little can be said from a theoretical point of view to justify their efficiency, except a recent work by Bickel and Levina where probabilistic tools are used to explain why a texture can be so well reproduced. In a joint work with J.-F. Aujol (ENS Cachan, France) and S. Ladjal (Telecom Paris Tech, France) we propose a variational model as a counterpart, in the continuous domain in any dimension, to a generic exemplar-based algorithm. Our purpose was twofold: 1) showing that a well-posed model in a weak functional setting makes sense for these methods 2) understanding the ability to restore, at least locally, the geometry. I will discuss this variational model and several variants in the framework of BV functions and Caccioppoli partitions.

## Wednesday, November 5

14:00 - 16:00 Workshop on functional equations

Speakers: Szabolcs Baják, Attila Gilányi, Eszter Gselmann, Fruzsina Mészáros

## Wednesday, November 12

10:30 - 11:30 Richard Bamler (Princeton University): Dehn filling in dimension 3 and higher

## Wednesday, November 26

10:30 - 11:30 Alberto Della Vedova (Università degli Studi di Parma): CM-stability of blow-ups and canonical metrics

Abstract: We will address the problem of the existence of extremal Kahler metrics in certain classes near the boundary of the Kahler cone of a blown-up manifolds. We will show an obstruction to the existence given by a certain algebro-geometric instability of the center of blow-up. We will give new examples of classes without extremal metrics also in the case of the projective plane blown-up at a finite set of points.

## Tuesday, December 16

11:00 - 12:00 Zsolt Páles (University of Debrecen): Solution methods for two-variable functional inequalities

## Wednesday, December 17

14:00 - 15:00 Gyula Maksa (University of Debrecen): Non-negative information functions

14:45 - 15:45 Zsolt Páles (University of Debrecen): Minkowski sum of fractals