Cristina Turrini

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.

Projective surfaces of small class

We classify smooth complex projective surfaces of degreed and class μ, satisfying either (i) μ−d≤16, or (ii) μ≤25. All these surfaces are rational or ruled. Indeed, we prove that the smallest value of the class μ of a non-ruled surface is 30 and in fact there are at least two surfacesS, both of degreed=10 and sectional genusg=6, with Kodaira dimension κ(S)=0 and class μ=30. Finally, we classify the smoothk-folds (k≥3) whose sectional surface has class μ≤23.

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.

Projective threefolds of small class

Projective threefolds whose discriminant locus has degree < 8 are classified. This is done by combining the class formula with many recent results on surfaces and threefolds with low sectional genus. Moreover classical results on surfaces concerning the class and the degree are partially extended to a special class of threefolds.

Applications of Multiview Tensors in Higher Dimensions

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.

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

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.

On the automorphisms of some line congruences in ?3

The automorphisms of line congruences in ℙ3 are studied via the analysis of the automorphisms of the associated focal loci. This study is applied to a Veronese surface (i.e. to a congruence of chords of a twisted cubic) and to the rational scrolls in the Grassmannian G(1, 3).

The Bordiga surface as critical locus for 3-view reconstructions

Abstract In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from P k to P 2 . The reconstruction problem consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a variety in P k containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P 4 to P 2 and of lines from P 3 to P 2 . In both cases, we consider 3 projections, minimal number for a uniquely determined reconstruction. In the case of projections of points, we declinate the Grassmann tensors introduced in Hartley and Schaffalitzky (2004) in our context, and we use them to compute the equations of the critical locus. Then, given the ideal that defines this locus, we prove that, in the general case, it defines a Bordiga surface, or a scheme in the same irreducible component of the associated Hilbert scheme. Furthermore, we prove that every Bordiga surface is actually the critical locus for the reconstruction for suitable projections. In the case of projections of lines, we compute the defining ideal of the critical locus, that is the union of 3 α-planes and a line congruence of bi-degree ( 3 , 6 ) and sectional genus 5 in the Grassmannian G ( 1 , 3 ) ⊂ P 5 . This last surface is biregular to a Bordiga surface ( Verra, 1988 ). We use this fact to link the two reconstruction problems by showing how to compute the projections of one of the two settings, given the projections of the other one. The link is effective, in the sense that we describe an algorithm to compute the projection matrices.