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To analyze defocus blur, we need to work in 3-D. We
use the 3-D cartesian frame where is the location of the effective
viewpoint, is the location of the effective pinhole,
is a unit vector in the direction
, the effective pinhole is located
at a distance *c* from the effective viewpoint, and the vectors
and are orthogonal unit vectors
in the plane *z*=0. As in Section 3, we also assume that
the conventional camera used in the catadioptric sensor has a frontal
image plane located at a distance *u* from the pinhole and that the
optical axis of the camera is aligned with the *z*-axis. In addition
to the previous assumptions, we assume that the effective pinhole of
the lens is located at the center of the lens, and that the lens has
a circular aperture. See Figure 9 for an illustration
of this configuration.
Consider a point on the mirror and a point
in the world, where
. Then, since the hyperboloid mirror satisfies
the fixed viewpoint constraint, a ray of light from which is
reflected by the mirror at passes directly through the
center of the lens (i.e. the effective pinhole.) This ray of light is
known as the *principal ray *[Hecht and Zajac, 1974]. Next, suppose
a ray of light from the world point is reflected at the point on
the mirror and then passes through the lens point . In general, this
ray of light will not be imaged at the same point on the image plane
as the principal ray. When this happens there is defocus blur.
The locus of the intersection of the incoming rays through
and the image plane as varies over the lens is
known as the *blur region* or *region of confusion *
[Hecht and Zajac, 1974]. For an ideal thin lens in isolation, the blur
region is circular and so is often referred to as the *blur circle *
[Hecht and Zajac, 1974].

If we know the points and , we can find
the point on the image plane where the ray of light through these
points is imaged. First, the line through in the
direction is extended to intersect
the *focused plane. * By the thin lens law [Hecht and Zajac, 1974]
the focused plane is:

| |
(40) |

where *f* is the focal length of the lens and *u* is the
distance from the focal plane to the image plane.
Since all points on the focused plane are perfectly focused, the
point of intersection on the focused plane can be mapped
onto the image plane using perspective projection. Hence, the *x* and
*y* coordinates of the intersection of the ray through and the image plane are the *x* and *y* coordinates of:
| |
(41) |

and the *z* coordinate is the *z* coordinate of the image plane *c*+*u*.
Given the lens point
and the world point
,there are three constraints on the point . First, must lie on the
mirror and so (for the hyperboloid) we have:

| |
(42) |

Secondly, the incident ray (), the
reflected ray (), and the normal to
the mirror at must lie in the same plane.
The normal to the mirror at lies in the
direction:
| |
(43) |

for the hyperboloid.
Hence, the second constraint is:
| |
(44) |

Finally, the angle of incidence must equal the angle of reflection
and so the third constraint on the point is:
| |
(45) |

These three constraints on are all multivariate
polynomials in *x*_{1}, *y*_{1}, and *z*_{1}: Equation (44)
and Equation (46) are both of order 2, and
Equation (47) is of order 5. We were unable to find a closed
form solution to these three equations (Equation (47) has 25
terms in general and so it is probable that none exists) but we did
investigate numerical solutions, the results of which are presented
in the following section.

** Next:** Numerical Results for the
** Up:** Defocus Blur of a
** Previous:** Defocus Blur of a
*Simon Baker*

*1/22/1998*