Luca BASSI lbassi@deis.unibo.it

DEIS, Università di Bologna

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Port-based modelling and control of complex dynamic systems

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- The port-Hamiltonian formalism: power, ports and interconnections
- Hamiltonian formulation for infinite-dimensional systems
- Modelling examples: flexible mechanical structures
- Control by interconnection and IDA-PBC

Andrea CERRI cerri@dm.unibo.it

ARCES & Dip. di Matematica, Università di Bologna

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Retrieval of trademark abstract images by means of Size Functions

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We propose a new, effective system for content-based trademark retrieval, which involves Size
Functions, a geometrical-topological tool for shape description and matching. Three different classes of shape descriptors are combined, for a total amount of 25 measuring functions. The evaluation has been performed on a benchmark database of 10,745 real trademark images, supplied by the UK Patent Office.

Giovanna CITTI citti@dm.unibo.it

Dip. di Matematica, Università di Bologna

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Subriemannian geometry for the description of the visual cortex

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We present a mathematical model of perceptual completion introduced together
with A. Sarti.
The structure of the visual cortex is modelled as the Lie group of rototranslations.
The initial image is lifted by the simple cells to a surface in this group
and the completion process is modeled via a diffusion driven motion by curvature.
This model can be alternatively considered as a lifting of the classical
elastica model.

Andrea Giovanni CUTTI agcutti@deis.unibo.it

Centro Protesi - INAIL, Bologna

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Application of geometric methods in motion analysis

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The presentation will briefly introduce to the mathematic background required for motion analysis. In particular two examples will addressed: the application of the Delaunay tessellation in computed aided surgery and the solution of the "hand-eye calibration problem" for motion systems "syncronisation".

Nadaniela EGIDI nadaniela.egidi@unicam.it

Università di Camerino

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Grid generation and the heat equation

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We consider the grid generation problem. For the solution of this problem we propose an iterative procedure resembling the explicit difference schemes for the heat equation, where, starting from a preliminary version of the grid under consideration a sequence of grid is considered. In particular, the solution of the grid generation problem is computed as the limit of such a sequence.

This approach gives an effective procedure to generate smooth grids when the domain is convex. For non-convex domains the procedure usually produces folded grids, since this iterative schemes does not take into account the curvature of the grid induced by the concavity of the domain, so we have to add a correction term that depends on the boundary of the domain under consideration.

Daniele FONTANELLI daniele.fontanelli@ing.unipi.it

Centro Interdip. di Ricerca "E. Piaggio", Università di Pisa

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A differential geometric approach for the control of nonholonomic mobile systems

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In the past, analysis and design of linear systems have taken advantage of results conveyed by many theoretical fields, such as the Laplace transform, the complex variable theory and the linear algebra. Differential geometry plays the same important role with respect to nonlinear control systems analysis. Indeed, from the point of view of the interaction between input and state and/or state and output, it allows the extension of some fundamental properties of linear systems like reachability and observability. From a feedback control design perspective, instead, it contributed with the local coordinates transformation for exact feedback linearization or state feedback via Lyapunov design. Differential geometric approach represents a fundamental tool also for the analysis of a class
of nonholonomic, input-affine systems, such as standard mobile platforms, whose control is hard to design due to the Brockett theorem.

Gregorio FRANZONI gregoriofranzoni@yahoo.it

Università di Cagliari

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MODELS OF SURFACES: WHY AND HOW TO BUILD THEM

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Mathematical objects have been represented, visualized
and animated in very effective ways in the last 20 years. Computer
graphics turned out to be a powerful tool to improve
understanding of Geometry of curves and surfaces and to attract people
to Mathematics. Physical modeling of geometric shapes can
be seen as the natural outgrowth of virtual modeling: material
models enhance our spatial perception of those objects by adding the
tactile experience to the visual one. In several occasions visual and material
representations of mathematical objects played a central role in research
development. We present some famous models of surfaces, from the
times of Galileo until now, and some models we built up by means
of several techniques and materials: plaster, wires, paper, metal, 3D printing.

Alessandra MICHELETTI alessandra.micheletti@unimi.it

Dip. di Matematica e Centro ADAMSS, Università di Milano

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Statistical Shape Analysis and Applications

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In applications, objects rarely have exactly the same shape within measurement error; hence the randomness of shapes need to be taken into account. Thanks to the development of information technologies, the last decade has seen a considerable growth of interest in the statistical theory of shape and its application to various scientific areas.
The solution to the problem of describing a “shape” via functions taking values in a finite dimensional space, without loosing relevant information, is needed for the mathematical and statistical analysis of the objects.
Recently new geometrical descriptors of shapes, the size functions, have been proposed. These functions are able to capture “globally” the geometric characteristics of an object, differently from landmarks (which usually are specific points, angles, distances, … on the objects, chosen by an expert), which are widely used in literature, but whose results in the statistical analysis are strongly dependent on their choice, leading to a sort of subjective quantitative analysis.
Size functions depend on the choice of a measuring function, and usually only a small number of choices lead to different statistical results; the measuring functions are chosen on the basis of the invariance properties that the geometrical descriptors must satisfy (e.g. invariance with respect to rotations, translations, scaling, etc.).
The theory of size functions has been developed mainly in a deterministic framework. A first attempt of joining this theory with randomness and to develop the related statistical analysis is here presented.
The approximation of size functions with their discrete counterpart leads to the formulation of suitable algorithms which may compute a graphical representation of the size function associated with a shape. The main features of the shape are thus described by a finite number of points and lines on a plane. The descriptor is robust, since small variations in the shape produce small variations in the location of such points and lines. Thus, by applying some cluster analysis techniques, it is possible to find 2D confidence regions for a family of shapes, and to detect the presence of outliers, i.e. of shapes not belonging to the family under study.
Applications to problems arising in microlithograpy and biomedicine will be presented.

Edie MIGLIO edie.miglio@polimi.it

MOX - Dip. di Matematica, Politecnico di Milano

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Shape reconstruction and fairing using univariate and bivariate integral spline operators

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In this talk we present a particular integral spline operators characterized by the presence of a shape parameter. Some applications concerning curve and surface reconstruction and fairing are presented.

Giuseppe PATANà© patane@ge.imati.cnr.it

Ist. di Matematica Applicata e Tecnologie Informatiche - CNR, Genova

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Global Parameterization of Triangulated Surafaces

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The global parameterization of a triangulated surface M consists of embedding the mesh on a planar domain by reducing it to a disc-like surface trough a cut-graph. Previous work has been focused on closed surfaces and the use of mesh transversal techniques for the evaluation of the geodesic metric has brought to build cut-graphs which are badly affected by the mesh connectivity and surface sampling. In this talk, we propose a simple method for finding a family of cut-graphs of M and guided by several criteria which have benefits on applications such as remeshing and texture mapping. We show that closed and bordered surfaces can be processed by a unique approach, which is stable with respect to the surface sampling and connectivity.

Gloria RINALDI rinaldi.gloria@unimore.it

Università di Modena e Reggio Emilia

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Applications of finite geometries to information security

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Many problems in information security can be resolved using combinatorial structures and geometries. I look at some models which give efficient solutions to the problem of key management in a cryptosystem.

Alfonso TORTORA Alfonso.Tortora@mat.unimi.it

Dip. di Matematica, Università di Milano

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A geometric approach to the trifocal tensor

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We give a full set of constraints for the trifocal tensor,
using tecniques of algebraic geometry.

CRISTINA TURRINI cristina.turrini@mat.unimi.it

Dip. di Matematica, Università di Milano

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Critical configurations for 1-view projections from P^k to P^2

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With some tools from algebraic geometry, critical configurations for 1-view projections from P^k to P^2 are described, together with their connections to the problem of reconstructing dynamical scenes.

Emanuela UGHI ughi@dipmat.unipg.it

Università di Perugia

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Geometry in a mathematical exhibition

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Geometers often “see” mathematical ideas through images in their minds, but sometimes it is hard to explain such images by words; I tried to avoid this difficulty by making “mathematical
models”,very simple, with poor materials, and offered them in an travelling exhibition to a large public.
My talk will be a report on this experience.

Andrea VIETRI vietri@dmmm.uniroma1.it

Dip. MeMoMat, Università di Roma "La Sapienza"

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From Phonetics to Combinatorics using Voronoi diagrams and graphs

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Vowels can be represented in a 2-dimensional space using their basic acoustic data. Such a representation, well known by phoneticians, lends itself to a suggestive interpretation in a Euclidean context. Every vowel determines indeed its own ''territory'', so that the acoustic plane is divided into regions - the Voronoi domains. Interactions between vowel inventories of different languages can then be easily described in a graphtheoretical fashion.