27 Marzo, 2009 15:00

STRATEGIE PER LA CLASSIFICAZIONE DI DATI FUNZIONALI E SPAZIO-FUNZIONALI: NUOVE PROPOSTE E PROBLEMI APERTI

Elvira Romano, Seconda Università degli Studi di Napoli

Aula Seminar F. Saleri VI Piano Mox - Dipartimento di Matematica

abstract

abstract

Gli ultimi anni hanno testimoniato nuovi importanti sviluppi nella teoria della classificazione di dati funzionali (Abraham et al., 2003; James and Sugar, 2003; Tarpey and Kinateder, 2003).

Spesso le metodologie proposte sono estremamente dipendenti dalla tipologia di trasformazione utilizzata per l approssimazione dalla forma funzionale grezza alla forma funzionale vera (Tarpey, 2007).

Si propone pertanto una strategia per la classificazione di dati funzionali che contemporaneamente stima e classifica le curve attraverso una generalizzazione dell algoritmo di classificazione dinamica (Romano, 2006, 2007). Diversamente dalle classiche metodologie utilizzate per il trattamento di dati funzionali, dove l individuazione della forma funzionale vera è effettuata al di fuori del processo di classificazione, si affronta qui il problema di individuare, congiuntamente, una struttura di classificazione ottimale che fornisce una funzione prototipo caratterizzante le classi a partire dall insieme delle singole osservazioni grezze nonché uno criterio per la stima della forma funzionale vera dei dati.
In tale contesto si pone l accento anche sull importanza della proprietà di robustezza di tale metodologia, cioè sulla sua capacità di fornire risultati stabili e attendibili anche in presenza di valori anomali (Balzanella, Romano, Verde, 2008).

Viene infine proposta un estensione della metodologia stessa qualora il dato funzionale inteso come curva temporale, presenti un ulteriore componente di natura spaziale. In particolare il problema che si affronta è la determinazione forme funzionali caratterizzanti delimitate regioni dello spazio (Romano, Balzanella, Verde, 2009 ).

27 Marzo, 2009 10:30

Sezione di Geometria, Algebra e loro applicazioni

Polarimetric SAR Tomography of Natural Scenarios: Current Achievements and Perspectives

Stefano Tebaldini, Politecnico di Milano, Dipartimento di Elettronica e Informazione

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

SAR imaging is a well established technology for the remote sensing of the Earth’s surface. The rationale of such technology is to synthesize a virtual sensor array as long as several kilometers by flying a Radar sensor onboard an airborne or spaceborne platform, resulting in the possibility to produce a Radar image of the illuminated scene with a spatial resolution in the order of few meters. By jointly processing several SAR images, acquired along different paths, the capabilities of SAR imaging get enhanced by one dimension, therefore producing a 3D Tomographic reconstruction of the illuminated scene. Furthermore, modern SAR sensors are capable of transmitting and receiving all the different vector components (or polarizations) of the Electric field, resulting in the possibility to discriminate among different targets basing on electromagnetic diversity.
In the last ten years, the availability of spatial and electromagnetic diversity within the data has been widely exploited in the analysis of forested areas, giving rise to the field of Polarimetric SAR Interferometry (PolInSAR). PolInSAR is today a well established technique, mostly used for the retrieval of the vegetation height above the ground basing on 2 multi-polarimetric SAR images.
In this paper we outline a new technique for the joint exploitation of several multi-polarimetric SAR images, to the aim of yielding a separate tomographic reconstruction of each of the different objects (often referred to as Scattering Mechanisms) that contribute to the received signal. Such technique extends PolInSAR, and will be referred to as Polarimetric SAR Tomography (PolT-SAR).
Under large hypotheses it will be shown that the data second order statistics can be expressed as a Sum of Kronecker Products (SKP) between two matrices, the first accounting for the electromagnetic properties and the second for the spatial structure of each of the Scattering Mechanism that contribute to the received signal. The key to the exploitation of the SKP structure is the existence of a technique for the decomposition of a matrix into a SKP. Such decomposition has the same formal properties as the SVD decomposition, the right and left singular vectors being replaced by two sets of matrices mutually orthogonal under the Frobenius inner product. As a consequence, the important result follows that, given the data covariance matrix, the K scattering mechanisms that contribute to the received signal are uniquely identified by K(K-1) real numbers.
The implications of this result will be discussed from both the theoretical and the experimental point of view, showing the current achievements and outlining the future research perspectives.

27 Marzo, 2009 11:30

Sezione di Geometria, Algebra e loro applicazioni

First results on material identification and imaging with a large-volume muon tomography prototype.

Paolo Checchia, Istituto Nazionale di Fisica Nucleare sez. di Padova

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

The muon tomography technique, based on the multiple Coulomb scattering of cosmic ray muons, has been proposed recently as a tool to perform non-destructive assays of large volume objects. Experimental results are reported from a large volume (about 11 m3) scanning system prototype, assembled using two large area CMS Muon Barrel drift chambers. The imaging capabilities of the system and the first measurements of the capability of the technique to discriminate among different materials are presented.

27 Marzo, 2009 15:00

Sezione di Geometria, Algebra e loro applicazioni

On the difference between solutions of discrete tomography problems.

Birgit van Dalen, Leiden University

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

We consider the problem of reconstructing binary images from their horizontal and vertical projections. It is well-known that the reconstruction may not be uniquely determined, and that it is in fact possible to have two disjoint reconstructions from the same set of projections. We will present a condition that the projections must necessarily satisfy in order for this to happen. More general, we can derive from given projections an upper bound on the symmetric difference between two reconstructions. We will also consider reconstructions from two different sets of projections, which gives us new stability results for the case of two directions.

27 Marzo, 2009 09:30

Sezione di Geometria, Algebra e loro applicazioni

Application of synchrotron radiation micro-CT to local morphological and numerical characterization of short fibre reinforced polymer composites.

Francesca Cosmi, Universita di Trieste

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

In components of short glass fibre reinforced polymer obtained by injection moulding, processing conditions produce complex orientation patterns that influence the mechanical properties of the component. Computed microtomography with synchrotron radiation constitutes the ideal analysis non destructive technique, being able to provide high resolution images of the internal structure of the material. In particular, the source high spatial coherence available at Elettra (Trieste) makes is possible to apply imaging techniques that exploit also information of the phase shifts induced by the sample inhomogeneities. These techniques are effective to detect even small details, such as short reinforce fibres within a polymeric matrix. Given fibre numerosity in our samples, instead of trying to isolate each single fibre and measure its geometrical properties from the 3D micro-CT reconstructions, we choose to compute global anisotropy parameters, commonly employed in other fields such as biomechanics or geology. Mean Intercept Length (MIL) and the related fabric tensor were used to assess local fibre orientation variations and to interpreter fatigue tests results in the light of different fibre orientation distributions. The problem of whether a morphological description can be predictive of the local elastic behaviour of these materials has also been addressed.

27 Marzo, 2009 12:00

Sezione di Geometria, Algebra e loro applicazioni

Variational approach to image segmentation.

Franco Tomarelli, Politecnico di Milano, Dipartimento di Matematica

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

This talk deals with free discontinuity problems related to image segmentation, focussing on the mathematical analysis of Blake & Zisserman functional.

Calculus of Variations is the framework where energy minimization and equilibrium notions find a precise language and formalizations by means of variational principles.

Image segmentation is a relevant problem both in digital image processing and in the understanding of biological vision.

There exist many different way to define the tasks of segmentation (template matching, component labelling, thresholding, boundary detection, quad-trees, texture matching, texture segmentation) and there is no universally accepted notion (optimality criteria for segmentation, analogies and differences between biological and automata perspective in segmentation): here the exposition is confined to some models for decomposing an image field, where is given a function describing the signal intensity associate to each point (typically the light intensity on a screen image). Such purpose has a clear connection with the problem of optimal partitions of a domain minimizing the length of the boundaries.

In simple words the segmentation we look for provides a cartoon of the given image satisfying some requirements: the decomposition of the image is performed by choosing a pattern of lines of steepest discontinuity for light intensity, and this pattern will be called segmentation of the image.

The variational formalizations of segmentation models provided deeper understanding of image analysis, produced intriguing mathematical questions (some of them still open) and entailed global estimates for geometric quantities in visual and automatic perception at both low and high level vision.

We discuss some recent results based on the innovative notion of free discontinuity problem introduced by Ennio De Giorgi. This approach balances carefully signal smoothing and segmentation length. In such framework, modern tools of Geometric Measure Theory and recent developments about minimal surfaces and regularity of extremals in Calculus of Variations allow the study of problems coupling bulk and surface terms: in such context discontinuous (in the mathematical sense) solutions are admissible and sometimes their discontinuities are the main features of the solution.

27 Marzo, 2009 14:00

Sezione di Geometria, Algebra e loro applicazioni

Using Tomography in Digital Plane to solve problems of Geometric Tomography.

Alain Daurat, Universite Louis Pasteur of Strasbourg

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

In this talk we will study the problem of determining in a constructive way a convex body in the plane from its tomographic projections. For this, we consider the similar problem in digital plane: reconstructing a lattice convex set from its discrete tomographic projection. We show that we can use a reconstruction algorithm for the discrete problem to solve the continuous reconstruction to any precision. The proof of this result uses stability properties of geometric tomography. An extension to point-source tomographic projections is also investigated.

27 Marzo, 2009 15:30

Sezione di Geometria, Algebra e loro applicazioni

Monitoring processes in soil laboratory samples with 3D electrical tomography

Guido Musso (sub for Cesare Comina), Politecnico di Torino

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

The talk will focus on the use of Electrical Resistivity Tomography as an imaging tool for the investigation of the hydro-mechanical behaviour of soil samples.
In soil mechanics laboratory measurements are mostly performed from the boundaries of the soil samples, or anyway on a portion of sample that as a first approximation is assumed to behave homogeneously. Nevertheless, due to the inherent structure of the soil and to the non linearity of hydro-chemo-mechanical processes, several different phenomena can occur perturbing the assumed homogeneity (localization of mechanical strains, hydro-chemical dispersion, saturation – desaturation processes and so on). The interpretation of experimental evidences involving such phenomena is therefore complicated if measurements are taken in the traditional way.
The use of 3D electrical tomography therefore has been investigated, in a dedicated advanced hydro-chemo-mechanical cell, in a number of experiments, both to reconstruct initial heterogeneities (e.g. due to local variations of porosity or mineralogy) or to monitor transient processes in homogeneous and heterogeneous soil conditions.
Good results have been obtained for different phenomena such as mechanical consolidation, chemical diffusion and saturation changes. Preliminary exercises would suggest that the technique can be used in the quantitative characterization of sample properties, provided its association with numerical simulations of ‘multiphysics’ processes. This can be done by relating the soil electrical conductivity to structural and environmental soil conditions (porosity, water saturation and salt concentration in the pore water) by means of known transport laws.

26 Marzo, 2009 16:30

Sezione di Geometria, Algebra e loro applicazioni

Solving some instances of the two color problem.

Stefano Brocchi, Universita di Firenze

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

The two color problem is an open problem in the field of discrete tomography, and it consists in determining a matrix, whose elements are of three different types, starting from its horizontal and vertical projections. It is known that the one color problem has a polynomial time reconstruction algorithm, while, with k > 2, the k-color problem is NP-complete. Thus, the two color problem constitutes an interesting example of a problem in the frontier between hard and easy problems.

In this talk we describe a linear time algorithm to solve a set of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to dimension of the problem. Our algorithm relies on classical studies for the solution of the one color problem.

26 Marzo, 2009 15:00

Sezione di Geometria, Algebra e loro applicazioni

On Some New Results in Discrete Tomography.

Peter Gritzmann, Technische Universität München

Dipartimento di Matematica, Sala Consiglio, VII piano

abstract

abstract

Discrete tomography deals with the reconstruction of finite sets from
knowledge about their interaction with certain query sets. The most
prominent example is that of the reconstruction of a finite subset $F$ of
$ mathbb{Z}^d$ from its X-rays (i.e., line sums) in a small positive
integer number $m$ of directions.
Applications of discrete tomography include quality control in
semiconductor industry, image processing, scheduling, and statistical data
security.
The reconstruction task is an ill-posed discrete inverse problem, depicting
(suitable variants of) all three Hadamard criteria for ill-posedness.


After a short introduction to the field of discrete tomography,
the first part of the talk addresses the following questions.
Does discrete tomography have the power of error correction?
Can noise be compensated by taking more X-ray images, and, if
so, what is the quantitative effect of taking one more X-ray?
Our main theoretical result gives the first nontrivial unconditioned
(and best possible) stability result. On the algorithmic side we show that
while there always is a certain inherent stability, the possibility of
making (worst-case) efficient use of it is rather limited.


The second part of the talk deals with the discrete tomography of
quasicrystals that live on finitely generated $ Z$-modules in some
$ R^s$.
Focussing on aspects in which the discrete tomography of quasicrystals
differs from that in the classical lattice case, we solve a basic
decomposition problem for the discrete tomography of quasicystals. More
generally, we study the problem of existence of pseudodiophantine
solutions to certain systems of linear equations over the reals and give a
complete characterization of when the index of Siegel grids is finite.


The results on stability are joint work with Andreas Alpers, that on
Siegel grids are joint work with Barbara Langfeld.