Beyond
Alhazen's
Problem: Analytical
Projection
Model for NonCentral Catadioptric
Cameras with Quadric Mirrors Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam Mitsubishi
Electric
Research
Labs
(MERL) CVPR
2011
(oral
presentation) Given the 3D point and
perspective camera location (possibly
offaxis), can we
analytically compute the point on the quadric mirror
where
reflection happens? Summary
Exact modeling of noncentral catadioptric cameras with rotationally symmetric quadric mirrors. Our model allows to compute the projection of a 3D point analytically even when the camera is offaxis. Abstract Catadioptric cameras are widely
used to
increase the field of view using mirrors. Central
catadioptric systems
having an effective single viewpoint are easy to model
and use, but
severely constraint the camera positioning with respect
to the mirror.
On the other hand, noncentral catadioptric systems
allow greater
flexibility in camera placement, but are often
approximated using
central or linear models due to the lack of an exact
model. We bridge
this gap and describe an exact projection model for
noncentral
catadioptric systems. We derive an analytical `forward
projection'
equation for the projection of a 3D point reflected by a
quadric mirror
on the imaging plane of a perspective camera, with no
restrictions on
the camera placement, and show that it is an 8th degree
equation in a
single unknown. Paper pdf, Supplementary pdf, Talk
Slides
Forward
projection
for
spherical
mirror is a classical problem in geometry
known as the Alhazen problem or the circular billiard
problem. The
problem can be traced back to ancient Greeks and is
described by
Ptolemy. Alhazen, who is widely regarded as the father
of optics,
talked extensively about this problem in his Book of
Optics around 1000
A.D. Several mathematicians have formulated algebraic,
trigonometric
and
analytical solutions to this problem and have shown that
there exist
four solutions. However, Alhazen only considered
spherical and
cylindrical
mirrors. The problem of computing analytical solutions
of forward
projection for general conic mirrors can be
considered as an extension of Alhazen problem. In
this paper, we solve this fundamental
problem for quadric mirrors where the camera can be
placed anywhere
(offaxis). Our ECCV 2010 paper provides the analytical
model only for
the axial case. Advantages The
analytical
equations
allow
us
to
use
exact
noncentral
model
without
using
any approximations such as (a) central approximation and
(b)
General Linear Cameras (GLC) approximation when using
these
nonperspective cameras. Most of the catadioptric
imaging techniques
minimize the 3D error instead of reprojection error for
3D
reconstruction, which is
suboptimal. Using our model, one can minimize the
reprojection error.
Thus, we can use the familiar bundle
adjustment pipeline for 3D reconstruction for
noncentral catadioptric
cameras, by simply replacing the perspective
projection equation with our analytical model. References 1. Alhazen
problem,
Wikipedia
