Minisymposia Abstracts

Speaker: Antonio Leitao
Federal University of St. Catarina, Brazil
Title: On new levelset type approaches for solving ill-posed problems
Abstract: We investigate level-set type approaches for solving ill-posed inverse problems, under the assumption that the solution is a piecewise constant function.

Our goal is to identify the level sets as well as the level values of the unknown parameter function.

Two distinct level-set frameworks are proposed for solving the inverse problem. In both of them the level-set function is assumed to be in L2.

Corresponding Tikhonov regularization approaches are derived and analyzed. Existence of minimizers for the Tikhonov functionals is proven. Moreover, convergence and stability results of the variational approaches are established, characterizing the Tikhonov approaches as regularization methods.

Speaker: Benedikt Wirth
Institute for Numerical Simulation, University of Bonn
Title: Fast Willmore flow time stepping for shape reconstruction from apparent contour
Abstract: Given a smooth three-dimensional object, its Huffman labeling is a two-dimensional image. Its value at each point is given by the number of times that the object surface is traversed by the ray emanating from the image point orthogonally to the image. To reconstruct from a given Huffman labeling the corresponding three-dimensional object, we apply a variational approach: We minimize the sum of a fitting energy (the squared L²-difference between the given and the actual Huffman labeling) and the Willmore energy of the object surface as a regularizer.

The implementation of the above energy is based on a novel phase field model for Willmore type functionals, which itself consists of a nested variational formulation. Instead of using a direct approximation of the involved mean curvature in space, a suitable approximation is extracted from a time-discrete curvature motion model. More precisely, the mean curvature is replaced by the approximate speed of mean curvature motion, which is computed via a fully implicit variational model. The minimization of the reconstruction functional is then performed in a nested fashion: The minimization of the actual functional is treated in an outer variational approach which contains the above approximation of the mean curvature as an inner variational problem. This results in a PDE-constrained optimization in which the actual surface geometry as well as the geometry resulting from the implicit curvature motion time step are represented by phase field functions.

Speaker: Wolfgang Ring
University of Graz
Title: Shape of optimally harmonic vibrating strings in tempered tuning
Abstract: Harmonic vibrating strings are characterized by harmonic eigenvalues or overtones λn = n λ0 with some fundamental frequency λ0 (the pitch of the string). Simultaneously sounding strings are perceived as consonant (as opposed to dissonant) if the overtones of the two strings coincide to a large extend. This concept has led—already in ancient Greece—to the definition of consonant intervals based on small integer ratios between the pitches of two strings. A problem arise if one aims to build up a coherent harmonic system from the fact that already the two most fundamental intervals, the octave 2:1 and the pure fifth 3:2 are incompatible in the sense that stacking pure fifth over a base tone never brings you exactly (only approximately) to an octave and the celebrated cycle of fifths does not close. As a remedy, well-tempered tuning was introduced in the Baroque age were the octave is divided into 12 equal intervals defined by the irrational factor 122. Doing so a consistent harmonic system without a distinguished central tone and the possibility to modulate between different keys can be build up, however at the expense of giving up purely consonant intervals with exactly matching overtone. In other words, tempered tuned instruments sound slightly out of tune.

By making the diameter of a vibrating string variable over the length of the string, the sequence of overtones can be shifted away from the harmonic sequence (n λ0)n ∈ N. We aim to find a diameter-profile of a string such that the overtones of an array of geometrically identical strings in tempered tuning satisfy—as much as possible—the requirements of consonant agreement of common overtones. We present numerical and acoustical experiments for a piano in tempered tuning with harmonically matching overtones.

Speaker: Esther Klann
Johannes Kepler University, Austria
Title: A Mumford–Shah like approach for limited tomography
Abstract: In this article the Mumford–Shah like method of Ramlau and Ring for complete tomographic data is generalized and applied to limited angle and region of interest tomography data. With the Mumford– Shah like method, one reconstructs a piecewise constant function and simultaneously a segmentation from its (complete) Radon transform data. For limited data, the ability of the Mumford–Shah like method to find a segmentation, and by that the singularity set of a function, is exploited. The method is applied to generated data from a torso phantom. The results demonstrate the performance of the method in reconstructing the singularity set, the density distribution itself for limited angle data and also some quantitative information about the density distribution for region of interest data. As a second example limited angle region of interest tomography is considered as a simplified model for electron tomography. For this problem we combine Lambda tomography and the Mumford-Shah like method. The combined method is applied to simulated ET data.

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