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67
Algebraic Properties of Multilinear Constraints
, 1996
"... In this paper the dioeerent algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, Vn , to work with is the image of IP 3 in IP 2 \Theta IP 2 \Theta \Delta \Delta \Delta \Theta IP 2 under a corresponding product ..."
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Cited by 33 (4 self)
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In this paper the dioeerent algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, Vn , to work with is the image of IP 3 in IP 2 \Theta IP 2 \Theta \Delta \Delta \Delta \Theta IP 2 under a corresponding product of projections, (A1 \Theta A2 \Theta : : : \Theta Am). Another descriptor, the variety Vb , is the one generated by all bilinear forms between pairs of views, which consists of all points in IP 2 \Theta IP 2 \Theta \Delta \Delta \Delta \Theta IP 2 where all bilinear forms vanish. Yet another descriptor, the variety V t , is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, Vb is a reducible variety with one component corresponding to V t and another corresponding to the trifocal plane. Furthermore, when m = 3, V t is generated by the three bilinearities and one trilinearity, when m = 4, V t is generated by the six bil...
Implicit Representation of Rational Parametric Surfaces
 J. SYMBOLIC COMPUTATION
, 1992
"... this paper we present algorithms to implicitize rational parametric surfaces with and without base points. One of the strength of the algorithms lies in the fact that we do not use multivariate factorization. The base points blow up to rational curves on the surface and we present techniques to comp ..."
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Cited by 24 (6 self)
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this paper we present algorithms to implicitize rational parametric surfaces with and without base points. One of the strength of the algorithms lies in the fact that we do not use multivariate factorization. The base points blow up to rational curves on the surface and we present techniques to compute the rational parametrization of the blow up curves.
On Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views
 In Proceedings of the 2nd European Workshop on Invariants, Ponta Delagada, Azores
, 1993
"... Part I of this paper investigates the differences  conceptually and algorithmically  between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a ..."
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Cited by 23 (8 self)
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Part I of this paper investigates the differences  conceptually and algorithmically  between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. The projective extension is then derived, showing that it is necessary only for tasks for which a reference view is not available  such as happens when updating scene structure from a moving stereo rig. The geometric difference between the two proposed invariants are that the affine invariant measures the relative deviation from a single reference plane, whereas the projective invariant measures the relative deviation from two reference planes. The affine invariant can be computed from three correspondin...
Nonnormal del Pezzo surfaces
 Publications of RIMS, Kyoto
, 1994
"... 0.1 Throughout this paper, a del Pezzo surface is by definition a connected, 2dimensional, projective kscheme X, OX(1) that is Gorenstein and anticanonically polarised; in other words, X is Cohen–Macaulay, and the dualising sheaf is invertible and antiample: ωX ∼ = OX(−1). For example, X = X3 ⊂ P ..."
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Cited by 21 (2 self)
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0.1 Throughout this paper, a del Pezzo surface is by definition a connected, 2dimensional, projective kscheme X, OX(1) that is Gorenstein and anticanonically polarised; in other words, X is Cohen–Macaulay, and the dualising sheaf is invertible and antiample: ωX ∼ = OX(−1). For example, X = X3 ⊂ P 3 an arbitrary
Algorithm for Implicitizing Rational Parametric Surfaces
 In IMA Conf. on Mathematics of Surfaces
, 1992
"... : Many current geometric modeling systems use the rational parametric form to represent surfaces. Although the parametric representation is useful for tracing, rendering and surface fitting, many operations like surface intersection desire one of the surfaces to be represented implicitly. Moreover, ..."
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Cited by 21 (3 self)
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: Many current geometric modeling systems use the rational parametric form to represent surfaces. Although the parametric representation is useful for tracing, rendering and surface fitting, many operations like surface intersection desire one of the surfaces to be represented implicitly. Moreover, the implicit representation can be used for testing whether a point lies on the surface boundary and to represent an object as a semialgebraic set. Previously resultants and Grobner basis have been used to implicitize parametric surfaces. In particular, different formulations of resultants have been used to implicitize tensor product surfaces and triangular patches and in many cases the resulting expression contains an extraneous factor. The separation of these extraneous factors can be a time consuming task involving multivariate factorization. Furthermore, these algorithms fail altogether if the given parametrization has base points. In this paper we present an algorithm to implicitize pa...
CalabiYau threefolds and moduli of abelian surfaces
"... be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in ..."
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Cited by 17 (2 self)
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be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in the sequel definitions and notation as in [GP1], [GP2]; see also [Mu1], [LB] and [HKW] for basic facts concerning abelian varieties and their moduli. Throughout the paper the base field will be C. Let Ad denote the moduli space of polarized abelian surfaces of type (1,d), and let A lev d The main goal of this paper, which is a continuation of [GP1] and [GP2], is to describe birational models for moduli spaces of these types for small values of d. Since the Kodaira dimension is a birational invariant, thus independent of the chosen compactification, we can decide the uniruledness, unirationality or rationality of nonsingular models of (any) compactifications of these moduli spaces.
The Critical Sets of Lines for Camera Displacement Estimation: A Mixed EuclideanProjective and Constructive Approach
 IN PROC. INTERNATIONAL CONFERENCE ON COMPUTER VISION
, 1994
"... The problem of the recovery of the motion, and the structure from motion is relevant to many computer vision applications. Many algorithms have been proposed to solve this problem. Some of these use line correspondences. For obvious practical reasons, it is important to study the limitation of such ..."
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Cited by 14 (3 self)
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The problem of the recovery of the motion, and the structure from motion is relevant to many computer vision applications. Many algorithms have been proposed to solve this problem. Some of these use line correspondences. For obvious practical reasons, it is important to study the limitation of such algorithms. In this paper, we are concerned with the problem of recovering the relative displacements of a camera by using line matches in three views. In particular we want to know whether there exist sets of 3D lines such that no matter how many lines we observe there will always be several solutions to the relative displacement estimation problem. Such sets of lines may be called critical in the sense that they defeat the corresponding algorithm. This question has been studied in detail in the case of point matches by earlycentury Austrian photogrammeters and, independently, in the midseventies and earlyeighties by computer vision scientists. The answer lies in the idea of a critical...
Equations of (1; d)polarized abelian surfaces
 Math. Ann
, 1998
"... §0. Introduction. In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and ..."
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Cited by 13 (5 self)
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§0. Introduction. In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A
Visual Space Distortion
 Biological Cybernetics
, 1997
"... We are surrounded by surfaces that we perceive by visual means. Understanding the basic principles behind this perceptual process is a central theme in visual psychology, psychophysics and computational vision. In many of the computational models employed in the past, it has been assumed that a metr ..."
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Cited by 12 (11 self)
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We are surrounded by surfaces that we perceive by visual means. Understanding the basic principles behind this perceptual process is a central theme in visual psychology, psychophysics and computational vision. In many of the computational models employed in the past, it has been assumed that a metric representation of physical space can be derived by visual means. Psychophysical experiments, as well as computational considerations, can convince us that the perception of space and shape has a much more complicated nature, and that only a distorted version of actual, physical space can be computed. This paper develops a computational geometric model that explains why such distortion might take place. The basic idea is that, both in stereo and motion, we perceive the world from multiple views. Given the rigid transformation between the views and the properties of the image correspondence, the depth of the scene can be obtained. Even a slight error in the rigid transformation parameters c...