Marina Bertolini

Projective normality of varieties of small degree

The projective normality of linearly normal smooth complex varieties of degree d ≤ 8 is investigated. The complete list of non projectively normal such manifolds is given; all of them are shown to be not 2-normal.

Linear pose estimate from corresponding conics

We propose here a new method to recover the orientation and position of a plane by matching at least three projections of a conic lying on the plane itself. The procedure is based on rearranging the conic projection equations such that the non linear terms are eliminated. It works with any kind of conic and does not require that the shape of the conic is known a-priori. The method was extensively tested using ellipses, but it can also be used for hyperbolas and parabolas. It was further applied to pairs of lines, which can be viewed as a degenerate case of hyperbola, without requiring the correspondence problem to be solved first. Critical configurations and numerical stability have been analyzed through simulations. The accuracy of the proposed algorithm was compared to that of traditional algorithms and of a trinocular vision system using a set of landmarks.

Critical Configurations for 1-View in Projections from ℙk → ℙ2

In this paper we describe, from a theoretical point of view, critical configurations for the projective reconstruction of a set of points, for a single view, i.e. for calibration of a camera, in the case of projections from ℙk to ℙ2 for k ≥ 4. We give first a general result describing these critical loci in ℙk, which, if irreducible, are algebraic varieties of dimension k−2 and degree 3. If k=4 they can be either a smooth ruled surface or a cone and if k = 5 they can be a smooth three dimensional variety, ruled in planes, or a cone. If k≥ 6, the variety is always a cone, the vertex of which has dimension at least k − 6. The reducible cases are studied in Appendix A.These results are then applied to determine explicitly the critical loci for the projections from ℙk which arise from the dynamic scenes in ℙ3 considered in [13].

Congruences of small degree in G(1,4)

We give the list of all possible congruences in G(1,4) of degree d less than or equal to 10 and we explicitely construct most of them.

Instability of Projective Reconstruction of Dynamic Scenes near Critical Configurations

In the context of multiple view geometry in any dimension, we compute the minimum number of views necessary for projective reconstruction of both the set of cameras and of scenes. Within a unified approach to critical configurations and their loci, the paper focuses on the case of dynamic scenes of multiple bodies traveling along parallel straight-line trajectories with constant velocities, in the framework of higher dimensional projections introduced by Shashua and Wolf. Critical loci in this case are explicitly determined. A stratification of the resulting locus in terms of fixed common velocities is presented and leveraged to show, via a number of simulated experiments, instability of the reconstruction near critical configurations.

Numerical criteria for very ampleness of divisors on projective bundles over an elliptic curve

In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a sufficient condition for a line bundle associated with a divisor D to be normally generated on

Applications of Multiview Tensors in Higher Dimensions

X=P(E)

Reconstruction of Some Segmented and Dynamic Scenes: Trifocal Tensors in P 4 Theoretical Set Up for Critical Loci, and Instability

where E is a vector bundle over a smooth curve C. A line bundle which is ample and normally generated is automatically very ample. Therefore the condition found in Butlers work, together with Miyaokas well known ampleness criterion, give a sufficient condition for the very ampleness of D on X. This work is devoted to the study of numerical criteria for very ampleness of divisors D which do not satisfy the above criterion, in the case of C elliptic. Numerical conditions for the very ampleness of D are proved,improving existing results. In some cases a complete numerical characterization is found.

Two dimensional scrolls contained in quadric cones in ℙ5

This chapter is devoted to applications of multiview tensors, in higher dimension, to projective recostruction of segmented or dynamic scenes. Particular emphasis is placed on the analysis of critical configurations and their loci in this context, i.e. configurations of chosen scene-points and cameras that turn out to prevent successful reconstruction or allow for multiple possible solutions giving rise to ambiguities. A general geometric set up for higher dimensional spaces ad projections is firstly recalled. Examples of segmented and dynamic scenes, interpreted as static scenes in higher dimensional projective spaces, are then considered, following Shashua and Wolf. A theoretical approach to multiview tensors in higher dimension is presented, according to Hartley and Schaffalitzky. Using techniques of multilinear algebra and proper formalized language of algebraic geometry, a complete description of the geometric structure of the loci of critical configurations in any dimension is given. Supporting examples are supplied, both for reconstruction from one view and from multiple views. In an experimental context, the following two cases are realized as static scenes in P4: 3D points lying on two bodies moving relatively to each other by pure translation and 3D points moving independently along parallel straight lines with constant velocities. More explicitly, algorithms to determine suitable tensors used to reconstruct a scene in P4: from three views are implemented with MATLAB. A number of simulated experiments are finally performed in order to prove instability of reconstruction near critical loci in both cases described above.

On the automorphisms of some line congruences in ?3

The context of this work is projective reconstruction ofsegmented or dynamic scenes from multiple views. More explicitly,the following two cases are considered: 3D points lying ontwo bodies moving relatively to each other by pure translation and3D points moving independently along parallel straightlines with constant velocities. These situations are interpreted asstatic scenes in ℙ4 following Shashua and Wolf,(IJCV 48, 2002). Algorithms to determine suitable tensors used toreconstruct a scene in ℙ4 from three views areimplemented in MATLAB®. From the theoretical pointof view, a detailed description, with proof, of critical loci inthe general context of projections ℙ4 →ℙ2 is given. A number of simulated experiments areperformed to prove instability of reconstruction near critical lociin both cases described above. Algorithms for tensor reconstructionallow us to bypass the stratification approach followed in ourprevious work (ICCV, 2007), thus offering more comprehensiveevidence of instability in higher dimension.