Classification of totally real elliptic Lefschetz fibrations via necklace diagrams

We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an $S^1$-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz fibration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in $PSL(2,\Z)$. Using an algorithm to find such decompositions, we obtain an explicit list of necklace diagrams associated with certain classes of totally real elliptic Lefschetz fibrations. Moreover, we introduce refinements of necklace diagrams and show that refined necklace diagrams determine uniquely the isomorphism classes of the totally real elliptic Lefschetz fibrations which may not have a real section. By means of necklace diagrams we observe some interesting phenomena underlying special feature of real fibrations.Comment: 25 pages, 30 figure

Three-dimensional metamorphosis: a survey

International audienceA metamorphosis or a (3D) morphing is the process of continuously transforming one object into another. 2D and 3D morphing are popular in computer animation, industrial design, and growth simulation. Since there is no intrinsic solution to the morphing problem, user interaction can be a key component of a morphing software. Many morphing techniques have been proposed in recent years for 2D and 3D objects. We present a survey of the various 3D approaches, giving special attention to the user interface. We show how the approaches are intimately related to the object representations. We conclude by sketching some morphing strategies for the future

An interactive framework for component-based morphing

Ph.DDOCTOR OF PHILOSOPH

Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

In functional data analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called functions on surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries. Supplementary materials for this article are available online