1,720,970 research outputs found
Equations of tensor Eigenschemes
No full textWe study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme
An explicit formula for a Bennequin-type invariant for apparent contours
No full textWe derive an explicit formula for the computation of a Vassiliev-type invariant of an apparent contour introduced recently by Ohmoto and Aicardi using appropriate linking numbers of its Legendrian lift. Our formula does not require the construction of the Legendrian lift and only takes into account the nodes, the cusps, the extremal points and the orientation of the apparent contour. In this way the computation can be implemented into a computer program; we demonstrate its use with some examples
Topological and variational properties of a model for the reconstruction of three-dimensional transparent images with self-occlusions
No full textWe introduce and study a two-dimensional model for the reconstruction of a smooth generic three-dimensional scene E, which may handle the self-occlusions and that can be considered as an improvement of the 2.1D sketch of Nitzberg and Mumford. We characterize from the topological viewpoint the apparent contour of E, namely, we characterize those planar graphs G that are apparent contours of some scene E. Moreover, we show that if E and F are two of these scenes, then E and F differ by a global homeomorphism which is strictly increasing on each fiber along the direction of the eye of the observer. These two topological theorems allow to find the domain of the functional F describing the model. Compactness, semicontinuity and relaxation properties of F are then studied, as well as connections of our model with the problem of completion of hidden contours
Eigenschemes of Ternary Tensors
No full textWe study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors, and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely on both classical and modern complex projective algebraic geometry
Completion of visible contours
No full textWe show that the completion problem of reconstructing the hidden arcs of the contours of an image, given only the visible ones, has a solution. More precisely we prove that, given an oriented plane graph K having as vertices only T-junctions and nonexterior terminal points, there exists an apparent contour G such that K is the visible part of G. This result is sharp, since the converse statement is easily seen to be satisfied. As a consequence, from K we can reconstruct a solid shape E in three-dimensional space such that K coincides with the visible part of the apparent contour of E. The main tools to prove our result are a Morse description of K and the Huffman labelling for apparent contours
A Characterization of Quasi-homogeneous Singularities of Free and Nearly Free Plane Curves
No full textThe goal of this paper is to establish a new characterization of quasi-homogeneous isolated singularities of free curves and nearly free curves C in P2C. The criterion will be in terms of a first syzygy matrix associated with the Jacobian ideal Jf of f, where f = 0 is the equation of the plane curve C
Canonical map of low codimensional subvarieties
No full textFix integers , and . We prove that for
certain projective varieties (e.g. certain
possibly singular complete intersections), there are only finitely
many components of the Hilbert scheme parametrizing irreducible,
smooth, projective, low codimensional subvarieties of such
that
h^0(X,\Cal O_X(aK_X-bH_X)) \leq \lambda
d^{\epsilon_1}+c\left(\sum_{1\leq h <
\epsilon_2}p_g(X^{(h)})\right),
where , and denote the degree, the canonical
divisor and the general hyperplane section of ,
denotes the geometric genus of the general linear section of
of dimension , and where , and
are suitable positive real numbers depending only on
the dimension of , on and on the ambient variety . In
particular, except for finitely many families of varieties, the
canonical map of any irreducible, smooth, projective, low
codimensional subvariety of , is birational
Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space
No full textIn this paper we are concerned with labelled apparent contours, namely with apparent contours of generic orthogonal projections of embedded surfaces in R3, endowed with a suitable depth information. We show that there exists a finite set of elementary moves (i.e. local topological changes) on labelled apparent contours such that the following theorem holds: two generic embeddings of a closed surface S in R3 are isotopic if and only if their apparent contours can be connected using only smooth isotopies and a finite sequence of moves
Shape Reconstruction from Apparent Contours. Theory and Algorithms
Full text linkMotivated by a variational model concerning the depth of the objects in a picture and the problem of hidden and illusory contours, this book investigates one of the central problems of computer vision: the topological and algorithmic reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour.
The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to diffeomorphic deformations of three dimensional scenes.
A large part of the book is devoted to the algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants. This book is addressed to theoretical and applied scientists working in the field of mathematical models of image segmentation
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