30,492 research outputs found
A case against epipolar geometry
We discuss briefly a number of areas where epipolar geometry is currently central in carrying out visual tasks. In contrast we demonstrate configurations for which 3D projective invariants can be computed from perspective stereo pairs, but epipolar geometry (and full projective structure) cannot. We catalogue a number of these configurations which generally involve isotropies under the 3D projective group, and investigate the connection with camera calibration. Examples are given of the invariants recovered from real images. We also indicate other areas where a strong reliance on epipolar geometry should be avoided, in particular for image transfer
Cluster Complexes via Semi-Invariants
We define and study virtual representation spaces having both positive and
negative dimensions at the vertices of a quiver without oriented cycles. We
consider the natural semi-invariants on these spaces which we call virtual
semi-invariants and prove that they satisfy the three basic theorems: the First
Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition
Theorem. In the special case of Dynkin quivers with n vertices this gives the
fundamental interrelationship between supports of the semi-invariants and the
Tilting Triangulation of the (n-1)-sphere.Comment: 34 page
Towards non-reductive geometric invariant theory
We study linear actions of algebraic groups on smooth projective varieties X.
A guiding goal for us is to understand the cohomology of "quotients" under such
actions, by generalizing (from reductive to non-reductive group actions)
existing methods involving Mumford's geometric invariant theory (GIT). We
concentrate on actions of unipotent groups H, and define sets of stable points
X^s and semistable points X^{ss}, often explicitly computable via the methods
of reductive GIT, which reduce to the standard definitions due to Mumford in
the case of reductive actions. We compare these with definitions in the
literature. Results include (1) a geometric criterion determining whether or
not a ring of invariants is finitely generated, (2) the existence of a
geometric quotient of X^s, and (3) the existence of a canonical "enveloping
quotient" variety of X^{ss}, denoted X//H, which (4) has a projective
completion given by a reductive GIT quotient and (5) is itself projective and
isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.Comment: 37 pages, 1 figure (parabola2.eps), in honor of Bob MacPherson's 60th
birthda
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