289,882 research outputs found
Autocalibration with the Minimum Number of Cameras with Known Pixel Shape
In 3D reconstruction, the recovery of the calibration parameters of the
cameras is paramount since it provides metric information about the observed
scene, e.g., measures of angles and ratios of distances. Autocalibration
enables the estimation of the camera parameters without using a calibration
device, but by enforcing simple constraints on the camera parameters. In the
absence of information about the internal camera parameters such as the focal
length and the principal point, the knowledge of the camera pixel shape is
usually the only available constraint. Given a projective reconstruction of a
rigid scene, we address the problem of the autocalibration of a minimal set of
cameras with known pixel shape and otherwise arbitrarily varying intrinsic and
extrinsic parameters. We propose an algorithm that only requires 5 cameras (the
theoretical minimum), thus halving the number of cameras required by previous
algorithms based on the same constraint. To this purpose, we introduce as our
basic geometric tool the six-line conic variety (SLCV), consisting in the set
of planes intersecting six given lines of 3D space in points of a conic. We
show that the set of solutions of the Euclidean upgrading problem for three
cameras with known pixel shape can be parameterized in a computationally
efficient way. This parameterization is then used to solve autocalibration from
five or more cameras, reducing the three-dimensional search space to a
two-dimensional one. We provide experiments with real images showing the good
performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
On first order Congruences of Lines in with irreducible fundamental Surface
In this article we study congruences of lines in , and in
particular of order one. After giving general results, we obtain a complete
classification in the case of in which the fundamental surface
is in fact a variety-i.e. it is integral-and the congruence is the
irreducible set of the trisecant lines of .Comment: 18 pages, AMS-LaTeX; submitte
On singular Luroth quartics
Plane quartics containing the ten vertices of a complete pentalateral and
limits of them are called L\"uroth quartics. The locus of singular L\"uroth
quartics has two irreducible components, both of codimension two in .
We compute the degree of them and we discuss the consequences of this
computation on the explicit form of the L\"uroth invariant. One important tool
are the Cremona hexahedral equations of the cubic surface. We also compute the
class in of the closure of the locus of nonsingular L\"uroth
quartics.Comment: Enlarged version. Computation of the class of the locus of Luroth
quartics in the moduli space adde
On quartics with lines of the second kind
We study the geometry of quartic surfaces in IP^3 that contain a line of the
second kind over algebraically closed fields of characteristic different from
2,3. In particular, we correct Segre's claims made for the complex case in
1943.Comment: 22 page
On the relationship between plane and solid geometry
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area
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