Analytical
Forward
Projection
for
Axial
NonCentral Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam Mitsubishi
Electric
Research
Labs
(MERL) ECCV
2010
(oral
presentation) Given the 3D point and
perspective camera location, can we
analytically compute the point on the mirror (refractive
sphere/ball
lens) where
reflection (refraction) happens? Summary
Exact modeling of noncentral catadioptric cameras with rotationally symmetric mirror in axial configuration. Analytical equations for fast projection of 3D points observed via reflection through a quadric mirror or refraction through a sphere (ball lens). Abstract We
present a technique for modeling noncentral
catadioptric cameras
consisting of a perspective camera and a rotationally
symmetric conic
reflector. While previous approaches use a central
approximation and/or
iterative methods for forward projection, we present an
analytical
solution. This allows computation of the optical path
from a given 3D
point to the given viewpoint by solving a 6th degree
forward projection
equation for general conic mirrors. For a spherical
mirror, the forward
projection reduces to a 4th degree equation, resulting
in a
closed form solution. We also derive the forward
projection equation
for imaging through a refractive sphere (noncentral
dioptric camera)
and show that it is a 10th degree equation. While
central catadioptric
cameras lead to conic epipolar curves, we show the
existence of a
quartic
epipolar curve for catadioptric systems using a
spherical mirror. The
analytical forward projection leads to accurate and fast
3D
reconstruction via bundle adjustment. Simulations and
real results on
single image sparse 3D reconstruction are presented. We
demonstrate 100
times speed up using the analytical solution over
iterative forward
projection for
3D reconstruction using spherical mirrors. Paper 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 and refractive
sphere can be
considered as an extension of Alhazen problem. Our paper
provides the
analytical solutions and thus provide an exact
noncentral model when
catadioptric systems with rotationally symmetric mirrors
are used in
axial configuration. We also provide the analytical
solution for
looking through a glass sphere or a ball lens (crystal
ball
photography). 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. It allows fast 3D
reconstruction using bundle adjustment, similar to
perspective cameras
and offer a speed up of 100X over previous approaches. Our CVPR
2011 paper describes
the analytical projection model for general case
(offaxis camera
placement). References 1. Alhazen
problem,
Wikipedia
