TY - JOUR

T1 - Recovering an algebraic curve using its projections from different points

AU - Kaminski, Jeremy Yirmeyahu

AU - Fryers, Michael

AU - Teicher, M.

PY - 2005

Y1 - 2005

N2 - We study some geometric configurations related to projections of an irreducible algebraic
curve embedded in CP3 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 CP3
, of
degree d and genus g, can be recovered using its projections from points onto embedded projective
planes. The embeddings are unknown. The only input is the defining equation of each projected
curve. We show how both the embeddings and the curve in CP3
can be recovered modulo some
action of the group of projective transformations of CP3
.
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 find 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. In 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 projection 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 CP3
. 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 projection
center is different. The projection operators are known. We show when and how the recovery of the
algebraic curve is possible, in terms of the degree of the curve, and of the degree of the curve of
minimal degree generated by the projection centers.

AB - We study some geometric configurations related to projections of an irreducible algebraic
curve embedded in CP3 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 CP3
, of
degree d and genus g, can be recovered using its projections from points onto embedded projective
planes. The embeddings are unknown. The only input is the defining equation of each projected
curve. We show how both the embeddings and the curve in CP3
can be recovered modulo some
action of the group of projective transformations of CP3
.
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 find 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. In 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 projection 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 CP3
. 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 projection
center is different. The projection operators are known. We show when and how the recovery of the
algebraic curve is possible, in terms of the degree of the curve, and of the degree of the curve of
minimal degree generated by the projection centers.

UR - http://u.math.biu.ac.il/~kaminsj/papers/jems.pdf

M3 - Article

SN - 1435-9855

VL - 7

SP - 145

EP - 172

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

ER -