- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 23:14:23
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=7&iss=2&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2005
2
Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision
Jeremy Yirmeyahu
Kaminski
Bar-Ilan University, RAMAT-GAN, ISRAEL
Michael
Fryers
Universität Hannover, HANNOVER, GERMANY
Mina
Teicher
Bar-Ilan University, RAMAT-GAN, ISRAEL
Plane and space curves, projections, machine vision, structure from motion
We study some geometric configurations related to the projection of an irreducible algebraic curve embedded in $\C \PP^3$ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve $X$ embedded in $\C \PP^3$, which degree is $d$ and genus $g$, can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in $\C \PP^3$ can be recovered modulo some action of the group of projective transformations of $\C \PP^3$. In particular in the case of two projections, we show how in a generic situation, a characteristic matrix of the pair of embeddings can be recovered. In the process we address dimensional issues and as a result establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degrees and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. For a generic situation, two projections define a curve with two irreducible components. One component has degree $d(d-1)$ and the other has degree $d$, being the original curve. Then we consider another problem. $N$ projections, with known projections operators and $N >> 1$, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in $\C \PP^3$. Those computations are respectively based on the dual variety and on the variety of intersecting lines. In both cases a simple lower bound for the number of necessary projections is given as a function of the degree and the genus. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a point. For each point the center of projection is different. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve, and of the degree of the curve of minimal degree generated by the centers of projection. Eventually we show how these questions were motivated by applications to static and dynamic computational vision. A second part of this work is devoted to several applications to this field. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.
Algebraic geometry
General
1
28
10.4171/JEMS/25
http://www.ems-ph.org/doi/10.4171/JEMS/25
The speed of propagation for KPP type problems. I: Periodic framework
Henry
Berestycki
Ecole des hautes études en sciences sociales, PARIS Cedex 13, FRANCE
François
Hamel
Université d'Aix-Marseille, MARSEILLE CEDEX 13, FRANCE
Nikolai
Nadirashvili
University of Chicago, CHICAGO, UNITED STATES
Reaction-diffusion equations, travelling fronts, propagation, periodic media, eigenvalue problems
This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article \cite{bh}. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.
Partial differential equations
General
173
213
10.4171/JEMS/26
http://www.ems-ph.org/doi/10.4171/JEMS/26
Singular principal G-bundles on nodal curves
Alexander
Schmitt
Freie Universität Berlin, BERLIN, GERMANY
Principal bundle, nodal curve, generalized parabolic bundle, moduli space
In the present paper, we give for the first time a general construction of compactified moduli spaces for semistable $G$-bundles on an irreducible complex projective curve $X$ with exactly one node, $G$ a semisimple linear algebraic group over the complex numbers.
Differential geometry
Group theory and generalizations
Manifolds and cell complexes
General
215
251
10.4171/JEMS/27
http://www.ems-ph.org/doi/10.4171/JEMS/27
Branching processes, and random-cluster measures on trees
Geoffrey
Grimmett
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Svante
Janson
Uppsala Universitet, UPPSALA, SWEDEN
Branching process, random-cluster measure, mean-field model
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of \rc\ measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?
Probability theory and stochastic processes
Statistical mechanics, structure of matter
General
253
281
10.4171/JEMS/28
http://www.ems-ph.org/doi/10.4171/JEMS/28