Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

A new 3D modelling paradigm for discrete model

Until few years ago, 3D modelling was a topic confined into a professional environment. Nowadays technological innovations, the 3D printer among all, have attracted novice users to this application field. This sudden breakthrough was not supported by adequate software solutions. The 3D editing tools currently available do not assist the non-expert user during the various stages of generation, interaction and manipulation of 3D virtual models. This is mainly due to the current paradigm that is largely supported by two-dimensional input/output devices and strongly affected by obvious geometrical constraints. We have identified three main phases that characterize the creation and management of 3D virtual models. We investigated these directions evaluating and simplifying the classic editing techniques in order to propose more natural and intuitive tools in a pure 3D modelling environment. In particular, we focused on freehand sketch-based modelling to create 3D virtual models, interaction and navigation in a 3D modelling environment and advanced editing tools for free-form deformation and objects composition. To pursuing these goals we wondered how new gesture-based interaction technologies can be successfully employed in a 3D modelling environments, how we could improve the depth perception and the interaction in 3D environments and which operations could be developed to simplify the classical virtual models editing paradigm. Our main aims were to propose a set of solutions with which a common user can realize an idea in a 3D virtual model, drawing in the air just as he would on paper. Moreover, we tried to use gestures and mid-air movements to explore and interact in 3D virtual environment, and we studied simple and effective 3D form transformations. The work was carried out adopting the discrete representation of the models, thanks to its intuitiveness, but especially because it is full of open challenges

Discrete Riemannian Calculus and A Posteriori Error Control on Shape Spaces

In this thesis, a novel discrete approximation of the curvature tensor on Riemannian manifolds is derived, efficient methods to interpolate and extrapolate images in the context of the time discrete metamorphosis model are analyzed, and an a posteriori error estimator for the binary Mumford–Shah model is examined. Departing from the variational time discretization on (possibly infinite-dimensional) Riemannian manifolds originally proposed by Rumpf and Wirth, in which a consistent time discrete approximation of geodesic curves, the logarithm, the exponential map and parallel transport is analyzed, we construct the discrete curvature tensor and prove its convergence under certain smoothness assumptions. To this end, several time discrete parallel transports are applied to suitably rescaled tangent vectors, where each parallel transport is computed using Schild’s ladder. The associated convergence proof essentially relies on multiple Taylor expansions incorporating symmetry and scaling relations. In several numerical examples we validate this approach for surfaces. The by now classical flow of diffeomorphism approach allows the transport of image intensities along paths in time, which are characterized by diffeomorphisms, and the brightness of each image particle is assumed to be constant along each trajectory. As an extension, the metamorphosis model proposed by Trouvé, Younes and coworkers allows for intensity variations of the image particles along the paths, which is reflected by an additional penalization term appearing in the energy functional that quantifies the squared weak material derivative. Taking into account the aforementioned time discretization, we propose a time discrete metamorphosis model in which the associated time discrete path energy consists of the sum of squared L2-mismatch functionals of successive square-integrable image intensity functions and a regularization functional for pairwise deformations. Our main contributions are the existence proof of time discrete geodesic curves in the context of this model, which are defined as minimizers of the time discrete path energy, and the proof of the Mosco-convergence of a suitable interpolation of the time discrete to the time continuous path energy with respect to the L2-topology. Using an alternating update scheme as well as a multilinear finite element respectively cubic spline discretization for the images and deformations allows to efficiently compute time discrete geodesic curves. In several numerical examples we demonstrate that time discrete geodesics can be robustly computed for gray-scale and color images. Taking into account the time discretization of the metamorphosis model we define the discrete exponential map in the space of images, which allows image extrapolation of arbitrary length for given weakly differentiable initial images and variations. To this end, starting from a suitable reformulation of the Euler–Lagrange equations characterizing the one-step extrapolation a fixed point iteration is employed to establish the existence of critical points of the Euler–Lagrange equations provided that the initial variation is small in L2. In combination with an implicit function type argument requiring H1-closeness of the initial variation one can prove the local existence as well as the local uniqueness of the discrete exponential map. The numerical algorithm for the one-step extrapolation is based on a slightly modified fixed point iteration using a spatial Galerkin scheme to obtain the optimal deformation associated with the unknown image, from which the unknown image itself can be recovered. To prove the applicability of the proposed method we compute the extrapolated image path for real image data. A common tool to segment images and shapes into multiple regions was developed by Mumford and Shah. The starting point to derive a posteriori error estimates for the binary Mumford–Shah model, which is obtained by restricting the original model to two regions, is a uniformly convex and non-constrained relaxation of the binary model following the work by Chambolle and Berkels. In particular, minimizers of the binary model can be exactly recovered from minimizers of the relaxed model via thresholding. Then, applying duality techniques proposed by Repin and Bartels allows deriving a consistent functional a posteriori error estimate for the relaxed model. Afterwards, an a posteriori error estimate for the original binary model can be computed incorporating a suitable cut-out argument in combination with the functional error estimate. To calculate minimizers of the relaxed model on an adaptive mesh described by a quadtree structure, we employ a primal-dual as well as a purely dual algorithm. The quality of the error estimator is analyzed for different gray-scale input images

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Proceedings, MSVSCC 2014

Proceedings of the 8th Annual Modeling, Simulation & Visualization Student Capstone Conference held on April 17, 2014 at VMASC in Suffolk, Virginia