A Theory of Single-Viewpoint Catadioptric Image Formation.
Conventional video cameras have limited fields of view which make them restrictive for certain applications in computational vision. A catadioptric sensor uses a combination of lenses and mirrors placed in a carefully arranged configuration to capture a much wider field of view. One important design goal for catadioptric sensors is choosing the shapes of the mirrors in a way that ensures that the complete catadioptric system has a single effective viewpoint. The reason a single viewpoint is so desirable is that it is a requirement for the generation of pure perspective images from the sensed images. In this paper, we derive the complete class of single-lens single-mirror catadioptric sensors that have a single viewpoint. We describe all of the solutions in detail, including the degenerate ones, with reference to many of the catadioptric systems that have been proposed in the literature. In addition, we derive a simple expression for the spatial resolution of a catadioptric sensor in terms of the resolution of the cameras used to construct it. Moreover, we include detailed analysis of the defocus blur caused by the use of a curved mirror in a catadioptric sensor.
Introduction
Many applications in computational vision require that a large field of view is imaged.
Examples include surveillance, teleconferencing, and model acquisition for virtual reality.
A number of other applications, such as ego-motion estimation and tracking, would also
benefit from enhanced fields of view. Unfortunately, conventional imaging systems are
severely limited in their fields of view. Both researchers and practitioners have therefore
had to resort to using either multiple or rotating cameras in order to image the entire scene.
One effective way to enhance the field of view is to use mirrors in conjunction
with lenses. See, for example, [Rees, 1970], [Charles et al., 1987], [Nayar, 1988], [Yagi
and Kawato, 1990], [Hong, 1991], [Goshtasby and Gruver, 1993], [Yamazawa et al., 1993],
[Bogner, 1995], [Nalwa, 1996], [Nayar, 1997a] , and [Chahl and Srinivassan, 1997]. We refer
to the approach of using mirrors in combination with conventional imaging systems as catadioptric
image formation. Dioptrics is the science of refracting elements (lenses) whereas
catoptrics is the science of reflecting surfaces (mirrors) [Hecht and Zajac, 1974]. The combination
of refracting and reflecting elements is therefore referred to as catadioptrics.
As noted in [Rees, 1970], [Yamazawa et al., 1995], [Nalwa, 1996], and [Nayar and
Baker, 1997], it is highly desirable that a catadioptric system (or, in fact, any imaging
system) have a single viewpoint (center of projection). The reason a single viewpoint is so
desirable is that it permits the generation of geometrically correct perspective images from
the images captured by the catadioptric cameras. This is possible because, under the single
viewpoint constraint, every pixel in the sensed images measures the irradiance of the light
passing through the viewpoint in one particular direction. Since we know the geometry of
the catadioptric system, we can precompute this direction for each pixel. Therefore, we
can map the irradiance value measured by each pixel onto a plane at any distance from the
viewpoint to form a planar perspective image. These perspective images can subsequently
be processed using the vast array of techniques developed in the field of computational
vision that assume perspective projection. Moreover, if the image is to be presented to
a human, as in [Peri and Nayar, 1997], it needs to be a perspective image so as not to
appear distorted. Naturally, when the catadioptric imaging system is omnidirectional in
its field of view, a single effective viewpoint permits the construction of geometrically correct
panoramic images as well as perspective ones.
In this paper, we take the view that having a single viewpoint is the primary design
goal for the catadioptric sensor and restrict attention to catadioptric sensors with a single
effective viewpoint [Baker and Nayar, 1998]. However, for many applications, such as robot
navigation, having a single viewpoint may not be a strict requirement [Yagi et al., 1994].
In these cases, sensors that do not obey the single viewpoint requirement can also be
used. Then, other design issues become more important, such as spatial resolution, sensor
size, and the ease of mapping between the catadioptric images and the scene [Yamazawa
et al., 1995]. Naturally, it is also possible to investigate these other design issues. For
example, Chahl and Srinivassan recently studied the class of mirror shapes that yield a
linear relationship between the angle of incidence onto the mirror surface and the angle of
reflection into the camera [Chahl and Srinivassan, 1997].
We begin this paper in Section 2 by deriving the entire class of catadioptric systems
with a single effective viewpoint, and which can be constructed using just a single
conventional lens and a single mirror. As we will show, the 2-parameter family of mirrors
that can be used is exactly the class of rotated (swept) conic sections. Within this class
of solutions, several swept conics are degenerate solutions that cannot, in fact, be used to
construct sensors with a single effective viewpoint. Many of these solutions have, however,
been used to construct wide field of view sensors with non-constant viewpoints. For these
mirror shapes, we derive the loci of the viewpoint. Some, but not all, of the non-degenerate
solutions have been used in sensors proposed in the literature. In these cases, we mention
all of the designs that we are aware of. A different, coordinate free, derivation of the fact
that only swept conic sections yield a single effective viewpoint was recently suggested by
Drucker and Locke [1996].
A very important property of a sensor that images a large field of view is its reso-
lution. The resolution of a catadioptric sensor is not, in general, the same as that of any
of the sensors used to construct it. In Section 3, we study why this is the case, and derive
a simple expression for the relationship between the resolution of a conventional imaging
system and the resolution of a derived single-viewpoint catadioptric sensor. We specialize
this result to the mirror shapes derived in the previous section. This expression should be
carefully considered when constructing a catadioptric imaging system in order to ensure
that the final sensor has sufficient resolution. Another use of the relationship is to design
conventional sensors with non-uniform resolution, which when used in an appropriate
catadioptric system have a specified (e.g. uniform) resolution.
Another optical property which is affected by the use of a catadioptric system is
focusing. It is well known that a curved mirror increases image blur [Hecht and Zajac,
1974]. In Section 4, we analyze this effect for catadioptric sensors. Two factors combine to
cause additional blur in catadioptric systems: (1) the finite size of the lens aperture, and
(2) the curvature of the mirror. We first analyze how the interaction of these two factors
causes defocus blur and then present numerical results for three different mirror shapes: the
hyperboloid, the ellipsoid, and the paraboloid. The results show that the focal setting of a
catadioptric sensor using a curved mirror may be substantially different from that needed in
a conventional sensor. Moreover, even for a scene of constant depth, significantly different
focal settings may be needed for different points in the scene. This effect, known as field
curvature, can be partially corrected using additional lenses [Hecht and Zajac, 1974].
2 The Fixed Viewpoint Constraint
The fixed viewpoint constraint is the requirement that a catadioptric sensor only measure
the intensity of light passing through a single point in 3-D space. The direction of the light
passing through this point may vary, but that is all. In other words, the catadioptric sensor
must sample the 5-D plenoptic function [Adelson and Bergen, 1991] [Gortler et al., 1996]
at a single point in 3-D space. The fixed 3-D point at which a catadioptric sensor samples
the plenoptic function is known as the effective viewpoint.
Suppose we use a single conventional camera as the only sensing element and a
single mirror as the only reflecting surface. If the camera is an ideal perspective camera
and we ignore defocus blur, it can be modeled by the point through which the perspective
projection is performed; i.e. the effective pinhole. Then, the fixed viewpoint constraint
requires that each ray of light passing through the effective pinhole of the camera (that
was reflected by the mirror) would have passed through the effective viewpoint if it had
not been reflected by the mirror. We now derive this constraint algebraically.
2.1 Derivation of the Fixed Viewpoint Constraint Equation
Without loss of generality we can assume that the effective viewpoint v of the catadioptric
system lies at the origin of a Cartesian coordinate system. Suppose that the effective
pinhole is located at the point p. Then, again without loss of generality, we can assume that
the z-axis - z lies in the direction ~
vp. Moreover, since perspective projection is rotationally
symmetric about any line through p, the mirror can be assumed to be a surface of revolution
about the z-axis - z. Therefore, we work in the 2-D Cartesian frame (v; - r is a unit
vector orthogonal to - z, and try to find the 2-dimensional profile of the mirror
Finally, if the distance from v to p is denoted by the parameter c, we
have

Figure

1 for an illustration 1 of the coordinate frame.
We begin the translation of the fixed viewpoint constraint into symbols by denoting
the angle between an incoming ray from a world point and the r-axis by '. Suppose that
this ray intersects the mirror at the point (z; r). Then, since we assume that it also passes
through the origin we have the relationship:
If we denote the angle between the reflected ray and the (negative) r-axis by ff, we also
tan
r
(2)
since the reflected ray must pass through the pinhole is the angle
between the z-axis and the normal to the mirror at the point (r; z), we have:
dz
Our final geometric relationship is due to the fact that we can assume the mirror to be
specular. This means that the angle of incidence must equal the angle of reflection. So,
if fl is the angle between the reflected ray and the z-axis, we have
Figure 1 for an illustration of this constraint.) Eliminating
fl from these two expressions and rearranging gives:
In

Figure

1 we have drawn the image plane as though it were orthogonal to the z-axis - z indicating
that the optical axis of the camera is (anti) parallel to - z. In fact, the effective viewpoint v and the axis of
symmetry of the mirror profile z(r) need not necessarily lie on the optical axis. Since perspective projection
is rotationally symmetric with respect to any ray that passes through the pinhole p, the camera could be
rotated about p so that the optical axis is not parallel to the z-axis. Moreover, the image plane can be
rotated independently so that it is no longer orthogonal to - z. In this second case, the image plane would
be non-frontal. This does not pose any additional problem since the mapping from a non-frontal image
plane to a frontal image plane is one-to-one.
effective viewpoint, v=(0,0) -
r
effective pinhole, p=(0,c)
c
image plane
z -
world point
image of world point
normal
a
mirror point, (r,z)
z
Figure

1: The geometry used to derive the fixed viewpoint constraint equation. The viewpoint
is located at the origin of a 2-D coordinate frame (v; - and the pinhole of the camera
c) is located at a distance c from v along the z-axis - z. If a ray of light, which was about
to pass through v, is reflected at the mirror point (r; z), the angle between the ray of light and - r
is
r
. If the ray is then reflected and passes through the pinhole p, the angle it makes
with - r is
r , and the angle it makes with - z is Finally, if
dr
is the angle between the normal to the mirror at (r; z) and - z, then by the fact that the angle of
incidence equals the angle of reflection, we have the constraint that ff
Then, taking the tangent of both sides and using the standard rules for expanding the
tangent of a sum:
we have:
Substituting from Equations (1), (2), and (3) yields the fixed viewpoint constraint equation:
\Gamma2 dz
dr
dz
dr
which when rearranged is seen to be a quadratic first-order ordinary differential equation:
dz
dr
dr
2.2 General Solution of the Constraint Equation
The first step in the solution of the fixed viewpoint constraint equation is to solve it as a
quadratic to yield an expression for the surface slope:
dz
The next step is to substitute cwhich yields:
dy
Then, we substitute differentiated gives:
2y dy
and so we have:
dr
Rearranging this equation yields:p
dx
r
Integrating both sides with respect to r results in:
where C is the constant of integration. Hence,
constant. By back substituting, rearranging, and simplifying
we arrive at the two equations which comprise the general solution of the fixed viewpoint
constraint equation:
In the first of these two equations, the constant parameter k is constrained by k - 2 (rather
leads to complex solutions.
2.3 Specific Solutions of the Constraint Equation
Together, Equations (16) and (17) define the complete class of mirrors that satisfy the fixed
viewpoint constraint. A quick glance at the form of these equations reveals that the mirror
profiles form a 2-parameter (c and family of conic sections. Hence, the shapes of the
3-D mirrors are all swept conic sections. As we shall see, however, although every conic
section is theoretically a solution of one of the two equations, a number of the solutions are
degenerate and cannot be used to construct real sensors with a single effective viewpoint.
We will describe the solutions in detail in the following order:
Planar Solutions: Equation (16) with
Conical Solutions: Equation (16) with k - 2 and
Spherical Solutions: Equation (17) with k ? 0 and
Ellipsoidal Solutions: Equation (17) with k ? 0 and c ? 0.
Hyperboloidal Solutions: Equation (16) with k ? 2 and c ? 0.
For each solution, we demonstrate whether it is degenerate or not. Some of the
non-degenerate solutions have actually been used in real sensors. For these solutions, we
mention all of the existing designs that we are aware of which use that mirror shape. Several
of the degenerate solutions have also been used to construct sensors with a wide field of
view, but with no fixed viewpoint. In these cases we derive the loci of the viewpoint.
There is one conic section that we have not mentioned: the parabola. Although the
parabola is not a solution of either equation for finite values of c and k, it is a solution of
Equation (16) in the limit that c ! 1,
h, a constant. These limiting
conditions correspond to orthographic projection. We briefly discuss the orthographic case
and the corresponding paraboloid solution in Section 2.4.
2.3.1 Planar Mirrors
In Solution (16), if we set we get the cross-section of a planar mirror:
As shown in Figure 2, this plane is the one which bisects the line segment ~
vp joining the
viewpoint and the pinhole.
ceffective viewpoint,
effective pinhole, p=(0,c)
image plane
z -
world point
image of world point
c
Figure

2: The plane
2 is a solution of the fixed viewpoint constraint equation. Conversely,
it is possible to show that, given a fixed viewpoint and pinhole, the only planar solution is the
perpendicular bisector of the line joining the pinhole to the viewpoint. Hence, for a fixed pinhole,
two different planar mirrors cannot share the same effective viewpoint. For each such plane the
effective viewpoint is the reflection of the pinhole in the plane. This means that it is impossible
to enhance the field of view using a single perspective camera and an arbitrary number of planar
mirrors, while still respecting the fixed viewpoint constraint. If multiple cameras are used then
solutions using multiple planar mirrors are possible [ Nalwa, 1996 ] .
The converse of this result is that for a fixed viewpoint v and pinhole p, there is
only one planar solution of the fixed viewpoint constraint equation. The unique solution is
the perpendicular bisector of the line joining the pinhole to the viewpoint:
To prove this, it is sufficient to consider a fixed pinhole p, a planar mirror with unit normal
- n, and a point q on the mirror. Then, the fact that the plane is a solution of the fixed
viewpoint constraint implies that there is a single effective viewpoint q). To be
more precise, the effective viewpoint is the reflection of the pinhole p in the mirror; i.e. the
single effective viewpoint is:
n: (20)
Since the reflection of a single point in two different planes is always two different points,
the perpendicular bisector is the unique planar solution.
An immediate corollary of this result is that for a single fixed pinhole, no two different
planar mirrors can share the same viewpoint. Unfortunately, a single planar mirror does
not enhance the field of view, since, discounting occlusions, the same camera moved from
p to v and reflected in the mirror would have exactly the same field of view. It follows
that it is impossible to increase the field of view by packing an arbitrary number of planar
mirrors (pointing in different directions) in front of a conventional imaging system, while
still respecting the fixed viewpoint constraint. On the other hand, in applications such
as stereo where multiple viewpoints are a necessary requirement, the multiple views of a
scene can be captured by a single camera using multiple planar mirrors. See, for example,
[Goshtasby and Gruver, 1993], [Inaba et al., 1993], and [Nene and Nayar, 1998].
This brings us to the panoramic camera proposed by Nalwa [1996]. To ensure a
single viewpoint while using multiple planar mirrors, Nalwa [1996] arrived at a design that
uses four separate imaging systems. Four planar mirrors are arranged in a square-based
pyramid, and each of the four cameras is placed above one of the faces of the pyramid.
The effective pinholes of the cameras are moved until the four effective viewpoints (i.e.
the reflections of the pinholes in the mirrors) coincide. The result is a sensor that has a
single effective viewpoint and a panoramic field of view of approximately 360 ffi \Theta 50 ffi . The
panoramic image is of relatively high resolution since it is generated from the four images
captured by the four cameras. This sensor is straightforward to implement, but requires
four of each component: i.e. four cameras, four lenses, and four digitizers. (It is, of course,
possible to use only one digitizer but at a reduced frame rate.)
2.3.2 Conical Mirrors
In Solution (16), if we set we get a conical mirror with circular cross
section:
s

Figure

3 for an illustration of this solution. The angle at the apex of the cone is 2-
where:
This might seem like a reasonable solution, but since the pinhole of the camera must
be at the apex of the cone. This implies that the only rays of light entering the pinhole
from the mirror are the ones which graze the cone and so do not originate from (finite
extent) objects in the world (see Figure 3.) Hence, the cone with the pinhole at the vertex
is a degenerate solution that cannot be used to construct a wide field of view sensor with
a single viewpoint.
In spite of this fact, the cone has been used in wide-angle imaging systems several
times [Yagi and Kawato, 1990] [Yagi and Yachida, 1991] [Bogner, 1995]. In these imple-
effective viewpoint, v=(0,0)
effective pinhole, p=(0,0)
image plane
z -
mirror
world point
imaged world point

Figure

3: The conical mirror is a solution of the fixed viewpoint constraint equation. Since the
pinhole is located at the apex of the cone, this is a degenerate solution that cannot be used to
construct a wide field of view sensor with a single viewpoint. If the pinhole is moved away from
the apex of the cone (along the axis of the cone), the viewpoint is no longer a single point but
rather lies on a circular locus. If 2- is the angle at the apex of the cone, the radius of the circular
locus of the viewpoint is e \Delta cos 2- , where e is the distance of the pinhole from the apex along the
axis of the cone. If - ? the circular locus lies inside (below) the cone, if - ! 60 ffi the circular
locus lies outside (above) the cone, and if the circular locus lies on the cone.
mentations the pinhole is placed some distance from the apex of the cone. It is easy to
show that in such cases the viewpoint is no longer a single point [Nalwa, 1996]. If the
pinhole lies on the axis of the cone at a distance e from the apex of the cone, the locus of
the effective viewpoint is a circle. The radius of the circle is easily seen to be:
the circular locus lies inside (below) the cone, if - ! 60 ffi the circular locus
lies outside (above) the cone, and if the circular locus lies on the cone. In some
applications such as robot navigation, the single viewpoint constraint is not vital. Conical
mirrors can be used to build practical sensors for such applications. See, for example, the
designs in [Yagi et al., 1994] and [Bogner, 1995].
2.3.3 Spherical Mirrors
In Solution (17), if we set we get the spherical mirror:
Like the cone, this is a degenerate solution which cannot be used to construct a wide field of
view sensor with a single viewpoint. Since the viewpoint and pinhole coincide at the center
of the sphere, the observer would see itself and nothing else, as is illustrated in Figure 4.
The sphere has also been used to build wide field of view sensors several times
[Hong, 1991] [Bogner, 1995] [Murphy, 1995]. In these implementations, the pinhole is placed
outside the sphere and so there is no single effective viewpoint. The locus of the effective
viewpoint can be computed in a straightforward manner using a symbolic mathematics
package. Without loss of generality, suppose that the radius of the mirror is 1:0. The
first step is to compute the direction of the ray of light which would be reflected at the
mirror point
pass through the pinhole. This computation is
viewpoint, v
pinhole, p

Figure

4: The spherical mirror satisfies the fixed viewpoint constraint when the pinhole lies at
the center of the sphere. (Since the viewpoint also lies at the center of the sphere.) Like the
conical mirror, the sphere cannot actually be used to construct a wide field of view sensor with a
single viewpoint because the observer can only see itself; rays of light emitted from the center of
the sphere are reflected back at the surface of the sphere directly towards the center of the sphere.
then repeated for the neighboring mirror point (r dz). Next, the intersection of
these two rays is computed, and finally the limit dr ! 0 is taken while constraining dz by
1. The result of performing this derivation is that the locus of the
effective viewpoint is:@
c
as r varies from \Gamma
to
. The locus of the effective viewpoint is plotted for
various values of c in Figure 5. As can be seen, for all values of c the locus lies within

Figure

5: The locus of the effective viewpoint of a circular mirror of radius 1:0 (which is also
shown) plotted for (d). For all values of c, the
locus lies within the mirror and is of comparable size to the mirror.
the mirror and is of comparable size to it. Like multiple planes, spheres have also been
used to construct stereo rigs [Nayar, 1988] [Nene and Nayar, 1998] , but as described before,
multiple viewpoints are a requirement for stereo.
2.3.4 Ellipsoidal Mirrors
In Solution (17), when k ? 0 and c ? 0; we get the ellipsoidal mirror:a 2
e
e
where:
a
s
s
The ellipsoid is the first solution that can actually be used to enhance the field of view of a
camera while retaining a single effective viewpoint. As shown in Figure 6, if the viewpoint
and pinhole are at the foci of the ellipsoid and the mirror is taken to be the section of the
ellipsoid that lies below the viewpoint (i.e. z ! 0), the effective field of view is the entire
upper hemisphere z - 0.
2.3.5 Hyperboloidal Mirrors
In Solution (16), when k ? 2 and c ? 0, we get the hyperboloidal mirror:a 2
where:
a
As seen in Figure 7, the hyperboloid also yields a realizable solution. The curvature of
the mirror and the field of view both increase with k. In the other direction (in the limit
the hyperboloid flattens out to the planar mirror of Section 2.3.1.
p=(0,c)
image plane
z -
world point
image of world point
c

Figure

The ellipsoidal mirror satisfies the fixed viewpoint constraint when the pinhole and
viewpoint are located at the two foci of the ellipsoid. If the ellipsoid is terminated by the horizontal
plane passing through the viewpoint z = 0, the field of view is the entire upper hemisphere z ? 0.
It is also possible to cut the ellipsoid with other planes passing through v, but it appears there is
little to be gained by doing so.
effective pinhole, p=(0,c)
image plane
z -
c
world point
image of world point

Figure

7: The hyperboloidal mirror satisfies the fixed viewpoint constraint when the pinhole and
the viewpoint are located at the two foci of the hyperboloid. This solution does produce the
desired increase in field of view. The curvature of the mirror and hence the field of view increase
with k. In the limit k ! 2, the hyperboloid flattens to the planar mirror of Section 2.3.1.
Rees [1970] appears to have been first to use a hyperboloidal mirror with a perspective
lens to achieve a large field of view camera system with a single viewpoint. Later,
Yamazawa et al. [1993] [1995] also recognized that the hyperboloid is indeed a practical
solution and implemented a sensor designed for autonomous navigation.
2.4 The Orthographic Case: Paraboloidal Mirrors
Although the parabola is not a solution of the fixed viewpoint constraint equation for finite
values of c and k, it is a solution of Equation (16) in the limit that c ! 1,
c
h, a constant. Under these limiting conditions, Equation (16) tends to:
As shown in [Nayar, 1997b] and Figure 8, this limiting case corresponds to orthographic
projection. Moreover, in that setting the paraboloid does yield a practical omnidirectional
sensor with a number of advantageous properties [Nayar, 1997b].
One advantage of using an orthographic camera is that it can make the calibration of
the catadioptric system far easier. Calibration is simpler because, so long as the direction of
orthographic projection remains parallel to the axis of the paraboloid, any size of paraboloid
is a solution. The paraboloid constant and physical size of the mirror therefore do not need
to be determined during calibration. Moreover, the mirror can be translated arbitrarily
and still remain a solution. Implementation of the sensor is therefore also much easier
because the camera does not need to be positioned precisely. By the same token, the fact
that the mirror may be translated arbitrarily can be used to set up simple configurations
where the camera zooms in on part of the paraboloid mirror to achieve higher resolution
(with a reduced field of view), but without the complication of having to compensate for
the additional non-linear distortion caused by the rotation of the camera that would be
needed to achieve the same effect in the perspective case.
focus
image plane
z -
world point
image of world point
direction of
orthographic projection

Figure

8: Under orthographic projection, the only solution is a paraboloid with the effective
viewpoint at the focus of the paraboloid. One advantage of this solution is that the camera
can be translated arbitrarily and remain a solution. This property can greatly simplify sensor
calibration [ Nayar, 1997b ] . The assumption of orthographic projection is not as restrictive a
solution as it may sound since there are simple ways to convert a standard lens and camera from
perspective projection to orthographic projection. See, for example, [ Nayar, 1997b ] .
3 Resolution of a Catadioptric Sensor
In this section, we 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 axis of symmetry of the mirror. See Figure 9 for
an illustration of this scenario. Then, the definition of resolution that we will use is the
following. Consider an infinitesimal area dA on the image plane. If this infinitesimal pixel
images an infinitesimal solid angle d- of the world, the resolution of the sensor as a function
of the point on the image plane at the center of the infinitesimal area dA is:
dA
If / is the angle made between the optical axis and the line joining the pinhole to
the center of the infinitesimal area dA (see Figure 9), the solid angle subtended by the
infinitesimal area dA at the pinhole is:
Therefore, the resolution of the conventional camera is:
dA
Then, the area of the mirror imaged by the infinitesimal area dA is:
d!
cos OE cos 2
dA
where OE is the angle between the normal to the mirror at (r; z) and the line joining the
pinhole to the mirror point (r; z). Since reflection at the mirror is specular, the solid angle
of the world imaged by the catadioptric camera is:
dA
viewpoint, v=(0,0)
f
pinhole, p=(0,c)
c
image plane
focal plane
z -
optical axis
mirror area, dS
normal
pixel area, dA
f
solid angle, dw
solid angle, du
solid angle, dw
y
mirror point, (r,z)
world point
image of world point

Figure

9: The geometry used to derive the spatial resolution of a catadioptric sensor. Assuming
the conventional sensor has a frontal image plane which is located at a distance u from the pinhole
and the optical axis is aligned with the z-axis - z, the spatial resolution of the conventional sensor
is dA
. Therefore the area of the mirror imaged by the infinitesimal image plane area dA
is
dA. So, the solid angle of the world imaged by the infinitesimal area dA on
the image plane is
Hence, the spatial resolution of the catadioptric sensor
is dA
d! since cos 2
Therefore, the resolution of the catadioptric camera is:
dA
dA
But, since:
we have:
dA
dA
Hence, the resolution of the catadioptric camera is the resolution of the conventional camera
used to construct it multiplied by a factor of:
where (r; z) is the point on the mirror being imaged.
The first thing to note from Equation (38) is that for the planar mirror
the resolution of the catadioptric sensor is the same as that of the conventional sensor
used to construct it. This is as expected by symmetry. Secondly, note that the factor in
Equation (39) is the square of the distance from the point (r; z) to the effective viewpoint
divided by the square of the distance to the pinhole
the distance from the viewpoint to (r; z) and d p the distance of (r; z) from the pinhole.
Then, the factor in Equation (39) is d 2
: For the ellipsoid, d
. Therefore, for the ellipsoid the factor is:
which increases as d p decreases and d v increases. For the hyperboloid, d
some constant . Therefore, for the hyperboloid the factor is:
which increases as d p increases and d v increases. So, for both ellipsoids and hyperboloids,
the factor in Equation (39) increases with r. Hence, both hyperboloidal and ellipsoidal
catadioptric sensors constructed with a uniform resolution conventional camera will have
their highest resolution around the periphery, a useful property for certain applications
such as teleconferencing.
3.1 The Orthographic Case
The orthographic case is slightly simpler than the projective case and is illustrated in

Figure

10. Again, we assume that the image plane is frontal; i.e. perpendicular to the
direction of orthographic projection. Then, the resolution of the conventional orthographic
camera is:
dA
where the constant M is the linear magnification of the camera. If the solid angle d!
images the area dS of the mirror and OE is the angle between the mirror normal and the
direction of orthographic projection, we have:
Combining Equations (35), (42), and (43) yields:
dA
d-
d!
For the paraboloid
, the multiplicative factor r 2 simplifies to:
Hence, as for both the ellipsoid and the hyperboloid, the resolution of paraboloid based
catadioptric sensors increases with r, the distance from the center of the mirror.
viewpoint, v=(0,0)
r
image plane
camera boundary
z -
direction of
orthographic projection
mirror area, dS
normal
f
solid angle, du
solid angle, dw
mirror point, (r,z)
world point
pixel area, dA image of world point

Figure

10: The geometry used to derive the spatial resolution of a catadioptric sensor in the orthographic
case. Again, assuming that the image plane is frontal and the conventional orthographic
camera has a linear magnification M , its spatial resolution is dA
. The solid angle d! equals
cos OE \Delta dS, where dS is the area of the mirror imaged and OE is the angle between the mirror normal
and the direction of orthographic projection. Combining this information with Equation (35)
yields the spatial resolution of the orthographic catadioptric sensor as dA
d! .
4 Defocus Blur of a Catadioptric Sensor
In addition to the normal causes present in conventional dioptric systems, such as diffraction
and lens aberrations, two factors combine to cause defocus blur in catadioptric sensors.
They are: (1) the finite size of the lens aperture, and (2) the curvature of the mirror. To
analyze how these two factors cause defocus blur, we first consider a fixed point in the world
and a fixed point in the lens aperture. We then find the point on the mirror which reflects
a ray of light from the world point through that lens point. Next, we compute where on
the image plane this mirror point is imaged. By considering the locus of imaged mirror
points as the lens point varies, we can compute the area of the image plane onto which a
fixed world point is imaged. In Section 4.1, we derive the constraints on the mirror point
at which the light is reflected, and show how it can be projected onto the image plane. In
Section 4.2, we extend the analysis to the orthographic case. Finally, in Section 4.3, we
present numerical results for hyperboloid, ellipsoid, and paraboloid mirrors.
4.1 Analysis of Defocus Blur
To analyze defocus blur, we need to work in 3-D. We use the 3-D cartesian frame (v; -
where v is the location of the effective viewpoint, p is the location of the effective pinhole, - z
is a unit vector in the direction ~
vp, the effective pinhole is located at a distance c from the
effective viewpoint, and the vectors -
x and -
y are orthogonal unit vectors in the plane
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 11 for an illustration of this configuration.
viewpoint, v=(0,0,0)
pinhole, p=(0,0,c)
c
image plane, z=c+u
focal plane, z=c
l=(d-cosl,d-sinl,c)
mirror
lens aperture
z
focused plane, z=c-v
world point, w= (x,y,z)
normal, n
blur region
plane, z=0
principal ray
l

Figure

11: The geometry used to analyze the defocus blur. We work in the 3-D cartesian
frame (v; - x; -
x and -
y are orthogonal unit vectors in the plane z = 0. In addition to
the assumptions of Section 3, we also assume that the effective pinhole is located at the center
of the lens and that the lens has a circular aperture. If a ray of light from the world point
kmk
is reflected at the mirror point through the
lens point sin -; c), there are three constraints on must lie on the mirror,
(2) the angle of incidence must equal the angle of reflection, and (3) the normal n to the mirror
at and the two vectors must be coplanar.
Consider a point on the mirror and a point
kmk
in the
world, where l ? kmk. Then, since the hyperboloid mirror satisfies the fixed viewpoint
constraint, a ray of light from w which is reflected by the mirror at m 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 w
is reflected at the point on the mirror and then passes through the lens
aperture 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 l and the image plane as l
varies over the lens aperture 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 m 1 and l, we can find the point on the image plane where the
ray of light through these points is imaged. First, the line through m 1 in the direction ~
is extended to intersect the focused plane. By the thin lens law [Hecht and Zajac, 1974]
the focused plane is:
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 l and the image plane
are the x and y coordinates of:
and the z coordinate is the z coordinate of the image plane c
Given the lens point c) and the world point
kmk
there are three constraints on the point must lie on the mirror
and so (for the hyperboloid) we have:
Secondly, the incident ray (w reflected ray (m and the normal to the mirror
at must lie in the same plane. The normal to the mirror at m 1 lies in the direction:
for the hyperboloid. Hence, the second constraint is:
Finally, the angle of incidence must equal the angle of reflection and so the third constraint
on the point m 1 is:
These three constraints on m 1 are all multivariate polynomials in x 1 , y 1 , and z 1 : Equation
(48) and Equation (50) are both of order 2, and Equation (51) is of order 5. We were
unable to find a closed form solution to these three equations (Equation (51) has 25 terms
in general and so it is probable that none exists) but we did investigate numericals solution.
Before we present the results, we briefly describe the orthographic case.
4.2 Defocus Blur in the Orthographic Case
The orthographic case is slightly different, as is illustrated in Figure 12. One way to convert
a thin lens to produce orthographic projection is to place an aperture at the focal point
behind the lens [Nayar, 1997b] . Then, the only rays of light that reach the image plane are
those that are (approximately) parallel to the optical axis. For the orthographic case, there
viewpoint, v=(0,0,0)
pinhole, p=(0,0,c)
c
image plane, z=c+u
focal plane, z=c
l=(d-cosl,d-sinl,c)
mirror
lens aperture
focused plane, z=c-v
normal, n
blur region
plane, z=0
principal ray
world point, w= (x,y,z)
l
focus, f=(0,0,c+f)
focal aperture

Figure

12: The geometry used to analyze defocus blur in the orthographic case. One way to
create orthographic projection is to add a (circular) aperture at the rear focal point (the one
behind the lens) [ Nayar, 1997b ] . Then, the only rays of light that reach the image plane are
those which are (approximately) parallel to the optical axis. The analysis of defocus blur is then
essentially the same as in the perspective case except that we need to check whether each ray of
light passes through this aperture when computing the blur region.
is therefore only one difference to the analysis. When estimating the blur region, we need
to check that the ray of light actually passes through the (circular) aperture at the rear
focal point. This task is straightforward. The intersection of the ray of light with the rear
focal plane is computed using linear interpolation of the lens point and the point where the
mirror point is imaged on the image plane. It is then checked whether this point lies close
enough to the optical axis.
4.3 Numerical Results
In our numerical experiments we set the distance between the effective viewpoint and the
pinhole to be meter, and the distance from the viewpoint to the world point w to be
meters. For the hyperboloidal and ellipsoidal mirrors, we set the radius of the lens
aperture to be 10 mm. For the paraboloidal mirror, the limiting aperture is the one at the
focal point. We chose the size of this aperture so that it lets through exactly the same rays
of light that the front 10 mm one would for a point 1 meter away on the optical axis. We
assumed the focal length to be 10 cm and therefore set the aperture to be 1 mm. With
these settings, the F-stop for the paraboloidal mirror is 2 \Theta 1=5. The results for
the other two mirrors are independent of the focal length, and hence the F-stop.
To allow the three mirror shapes to be compared on an equal basis, we used values
for k and h that correspond to the same mirror radii. The radius of the mirror is taken
to be the radius of the mirror cut off by the plane z = 0; i.e. the mirrors are all taken to
image the entire upper hemisphere. Some values of k and h are plotted in Table 1 against
the corresponding mirror radius, for

Table

1: The mirror radius as a function of the mirror parameters (k and h) for
Mirror Radius Hyperboloid (k) Ellipsoid (k) Paraboloid (h)
4.3.1 Area of the Blur Region
In

Figures

13-15, we plot the area of the blur region (on the ordinate) against the distance
to the focused plane v (on the abscissa) for the hyperboloidal, ellipsoidal, and paraboloidal
mirrors. In each figure, we plot separate curves for different world point directions. The
angles are measures in degrees from the plane z = 0, and so the curve at 90 ffi corresponds
to the (impossible) world point directly upwards in the direction of the z-axis. For the
hyperboloid we set for the ellipsoid 0:11, and for the paraboloid
0:1. As can be seen in Table 1, these settings correspond to a mirror with radius 10 cm.
Qualitatively similar results were obtained for the other radii. Section 4.3.3 contains related
results for the other radii.
The smaller the area of the blur region, the better focused the image will be. We
see from the figures that the area never reaches exactly zero, and so an image formed using
these catadioptric sensors can never be perfectly focused. However, the minimum area is
very small, and in practice there is no problem focusing the image for a single world point.
Moreover, it is possible to use additional corrective lenses to compensate for most of this
effect [Hecht and Zajac, 1974].
Note that the distance at which the image of the world point will be best focused (i.e.
somewhere in the range 0.9-1.15 meters) is much less than the distance from the pinhole
to the world point (approximately 1 meter from the pinhole to the mirror plus 5 meters
from the mirror to the world point). The reason for this effect is that the mirror is curved.
For the hyperboloidal and paraboloidal mirrors which are convex, the curvature tends to
increase the divergence of rays coming from the world point. For these rays to be converged
and the image focused, a larger distance to the image plane u is needed. A larger value of
u corresponds to a smaller value of v, the distance to the focused plane. For the concave
ellipsoidal mirror, the mirror converges the rays to the extent that a virtual image is formed
between the mirror and the lens. The lens must be focused on this virtual image.
4.3.2 Shape of the Blur Region
Next, we provide an explanation of the fact that the area of the blur region never exactly
reaches zero. For a conventional lens, the blur region is a circle. In this case, as the focus
setting is adjusted to focus the lens, all points on the blur circle move towards the center
of the blur circle at a rate which is proportional to their distance from the center of the
blur circle. Hence, the blur circle steadily shrinks until the blur region has area 0 and the
lens is perfectly focused. If the focus setting is moved further in the same direction, the
blur circle grows again as all the points on it move away from the center.
For a catadioptric sensor using a curved mirror, the blur region is only approximately
a circle for all three of the mirror shapes. Moreover, as the image is focused, the speed
with which points move towards the center of this circle is dependent on their position in a
much more complex way than for a single lens. The behavior is qualitatively the same for
all of the mirrors and is illustrated in Figure 16. From Figure 16(a) to Figure 16(e), the
for the hyperboloidal mirror with In this example, we have meter, the radius of
the lens aperture 10 millimeters, and the distance from the viewpoint to the world point
meters. We plot curves for 7 different world points, at 7 different angles from the plane
The area of the blur region never becomes exactly zero and so the image can never be perfectly
focused. However, the area does become very small and so focusing on a single point is not a
problem in practice. Note that the distance at which the image will be best focused (around 1.0-
1.15 meters) is much less than the distance from the pinhole to the world point (approximately
1 meter from the pinhole to the mirror plus 5 meters from the mirror to the world point.) The
reason is that the mirror is convex and so tends to increase the divergence of rays of light.
for the ellipsoidal mirror with 0:11. The other settings are the same as for the hyperboloidal
mirror in Figure 13. Again, the distance to the focused plane is less than the distance to the point
in the world, however the reason is different. For the concave ellipsoidal mirror, a virtual image
is formed between the mirror and the lens. The lens needs to focus on this virtual image.
for the paraboloidal mirror with 0:1. The settings are the same as for the hyperboloidal
mirror, except the size of the apertures. The limiting aperture is the one at the focal point. It
is chosen so that it lets through exactly the same rays of light that the 10 mm one does for the
hyperboloidal mirror for a point 1 meter away on the optical axis. The results are qualitatively
very similar to the hyperboloidal mirror.
-0.006
-0.0020.0020.006
(a) Hyperboloid 1082 mm (b) Hyperboloid 1083.25 mm
(c) Ellipsoid 1003.75 mm (d) Ellipsoid 1004 mm
Paraboloid 1068.63 mm (f) Paraboloid 1069 mm

Figure

16: The variation in the shape of the blur region as the focus setting is varied. Note that
all of the blur regions in this figure are relatively well focused. Also, note that the scale of the 6
figures are all different.
blur region gets steadily smaller, and the image becomes more focused. In Figure 16(f), the
focus is beginning to get worse again. In Figure 16(a) the blur region is roughly a circle,
however as the focus gets better, the circle begins to overlap itself, as shown in Figure 16(b).
The degree of overlap increases in Figures 16(c) and(d). (These 2 figures are for the ellipse
and are shown to illustrate how similar the blur regions are for the 3 mirror shapes. The
only difference is that the region has been reflected about a vertical axis since the ellipse
is a concave mirror.) In Figure 16(e), the image is as well focused as possible and the blur
region completely overlaps itself. In Figure 16(f), the overlapping has begun to unwind.
Finally, in Figure 17, we illustrate how the blur regions vary with the angle of the
point in the world, for a fixed focal setting. In this figure, which displays results for the
hyperboloid with 0:11, the focal setting is chosen so that the point at 45 ffi is in focus.
As can be seen, for points in the other directions the blur region can be quite large and so
points in those directions are not focused. This effect, known as field curvature [Hecht and
Zajac, 1974], is studied in more detail in the following section.
4.3.3 Focal Settings
Finally, we investigated how the focus setting that minimizes the area of the blur region
(see

Figures

changes with the angle ' which the world point w makes with the plane
The results are presented in Figures 18-20. As before, we set assumed
the radius of the lens aperture to be 10 millimeters (1 millimeter for the paraboloid), and
fixed the world point to be l = 5 meters from the effective viewpoint. We see that the best
focus setting varies considerably across the mirror for all of the mirror shapes. Moreover,
the variation is roughly comparable for all three mirrors (of equal
In practice, these results, often referred to as "field curvature" [Hecht and Zajac,
1974], mean that it can sometimes be difficult to focus the entire scene at the same time.
-0.3
-0.2
-0.10.20.4
-0.4
-0.3
-0.2
(a) 1018.8 mm,
-0.4
-0.3
-0.2
-0.10.20.4
-0.4
-0.3
-0.2
(a) 1018.8 mm,
-0.4
-0.3
-0.2
-0.10.20.4
-0.4
-0.3
-0.2
(a) 1018.8 mm,

Figure

17: An example of the variation in the blur region as a function of the angle of the point
in the world. In this example for the hyperboloid with the point at 45 ffi is in focus, but
the points in the other directions are not.
Hyperboloid, k=6.10
Hyperboloid, k=11.0
Hyperboloid, k=21.0
Hyperboloid, k=51.0

Figure

18: The focus setting which minimizes the area of the blur region in Figure 13 plotted
against the angle ' which the world point w makes with the plane z = 0. Four separate curves
are plotted for different values of the parameter k. See Table 1 for the corresponding radii of
the mirrors. We see that the best focus setting for w varies considerably across the mirror. In
practice, these results mean that it can sometimes be difficult to focus the entire scene at the
same time, unless additional compensating lenses are used to compensate for the field curvature
[ Hecht and Zajac, 1974 ] . Also, note that this effect becomes less important as k increases and the
mirror gets smaller.
Ellipsoid, k=0.24
Ellipsoid, k=0.11
Ellipsoid, k=0.02

Figure

19: The focus setting which minimizes the area of the blur region in Figure 14 plotted
against the angle ' which the world point w makes with the plane z = 0. Four separate curves
are plotted for different values of the parameter k. See Table 1 for the corresponding radii of
the mirrors. The field curvature for the ellipsoidal mirror is roughly comparable to that for the
Paraboloid, k=0.20
Paraboloid, k=0.10
Paraboloid, k=0.05
Paraboloid, k=0.02

Figure

20: The focus setting which minimizes the area of the blur region in Figure 15 plotted
against the angle ' which the world point w makes with the plane z = 0. Four separate curves
are plotted for different values of the parameter h. See Table 1 for the corresponding radii of the
mirrors. The field curvature for the paraboloidal mirror is roughly comparable to that for the
Either the center of the mirror is well focused or the points around the periphery are
focused, but not both. Fortunately, it is possible to introduce additional lenses which
compensate for the field curvature [Hecht and Zajac, 1974]. (See the discussion at the end
of this paper for more details.) Also note that as the mirrors become smaller in size (k
increases for the hyperboloid, k decreases for ellipsoid, and h decreases for the paraboloid)
the effect becomes significantly less pronounced.
In this paper, we have studied three design criteria for catadioptric sensors: (1) the shape
of the mirrors, (2) the resolution of the cameras, and (3) the focus settings of the cameras.
In particular, we have derived the complete class of mirrors that can be used with a single
camera to give a single viewpoint, found an expression for the resolution of a catadioptric
sensor in terms of the resolution of the conventional camera(s) used to construct it, and
presented detailed analysis of the defocus blur caused by the use of a curved mirror.
There are a number of possible uses for the (largely theoretical) results presented
in this paper. Throughout the paper we have touched on many of their uses by a sensor
designer. The results are also of interest to a user of a catadioptric sensor. We now briefly
mention a few of the possible uses, both for sensor designers and users:
ffl For applications where a fixed viewpoint is not a requirement, we have derived the
locus of the viewpoint for several mirror shapes. The shape and size of these loci may
be useful for the user of such a sensor requiring the exact details of the geometry. For
example, if the sensor is being used in an stereo rig, the epipolar geometry needs to
be derived precisely.
ffl The expression for the resolution of the sensor could be used by someone applying
image processing techniques to the output of the sensor. For example, many image
enhancement algorithms require knowledge of the solid angles of the world integrated
over by each pixel in sensor.
ffl Knowing the resolution function also allows a sensor designer to design a CCD with
non-uniform resolution to get an imaging system with a known (for example uniform)
resolution.
ffl The defocus analysis could be important to the user of a catadioptric sensor who
wishes to apply various image processing techniques, from deblurring to restoration
and super-resolution.
ffl Knowing the defocus function also allows a sensor designer to compensate for the
field curvature introduced by the use of a curved mirror. One method consists of
introducing optical elements behind the imaging lens. For instance, a plano-concave
lens placed flush with the CCD permits a good deal of field curvature correction.
(Light rays at the periphery of the image travel through a greater distance within the
plano-concave lens). Another method is to use a thick meniscus lens right next to
the imaging lens (away from the CCD). The same effect is achieved. In both cases,
the exact materials and curvatures of the lens surfaces are optimized using numerical
simulations. Optical design is almost always done this way as analytical methods are
far too cumbersome. See [Born and Wolf, 1965] for more details.
We have described a large number of mirror shapes in this paper, including cones,
spheres, planes, hyperboloids, ellipsoids, and paraboloids. Practical catadioptric sensors
have been constructed using most of these mirror shapes. See, for example, [Rees, 1970],
[Charles et al., 1987] , [Nayar, 1988], [Yagi and Kawato, 1990], [Hong, 1991], [Goshtasby and
Gruver, 1993], [Yamazawa et al., 1993], [Bogner, 1995], [Nalwa, 1996], and [Nayar, 1997a].
As described in [Chahl and Srinivassan, 1997], even more mirror shapes are possible if we
relax the single-viewpoint constraint. Which then is the "best" mirror shape to use?
Unfortunately, there is no simple answer to this question. If the application requires
exact perspective projection, there are three alternatives: (1) the ellipsoid, (2) the
hyperboloid, and (3) the paraboloid. The major limitation of the ellipsoid is that only a
hemisphere can be imaged. As far as the choice between the paraboloid and the hyperboloid
goes, using an orthographic imaging system does require extra effort on behalf of the
optical designer, but may also make construction and calibration of the entire catadioptric
system easier, as discussed in Section 2.4.
If the application at hand does not require a single viewpoint, many other practical
issues may become more important, such as the size of the sensor, its resolution variation
across the field of view, and the ease of mapping between coordinate systems. In this paper
we have restricted attention to single-viewpoint systems. The reader is referred to other
papers proposing catadioptric sensors, such as [Yagi and Kawato, 1990] , [Yagi and Yachida,
1991], [Hong, 1991], [Bogner, 1995], [Murphy, 1995], and [Chahl and Srinivassan, 1997], for
discussion of the practical merits of catadioptric systems with extended viewpoints.

Acknowledgements

The research described in this paper was conducted while the first author was a Ph.D.
student in the Department of Computer Science at Columbia University in the City of
New York. This work was supported in parts by the VSAM effort of DARPA's Image
Understanding Program and a MURI grant under ONR contract No. N00014-97-1-0553.
The authors would also like to thank the anonymous reviewers for their comments which
have greatly improved the paper.



--R

The plenoptic function and elements of early vision.
A theory of catadioptric image forma- tion
Introduction to panoramic imaging.
Principles of Optics.
Reflective surfaces for panoramic imaging.
How to build and use an all-sky camera
A natural classification of curves and surfaces with reflection properties.
The lumi- graph
Design of a single-lens stereo camera system

Image based homing.
A stereo viewer based on a single camera with view-control mechanism
Application of panoramic imaging to a teleoperated lunar rover.
A true omnidirectional viewer.
Catadioptric image formation.
Recovering depth using a single camera and two specular spheres.
Catadioptric omnidirectional camera.
Omnidirectional video camera.
Stereo with mirrors.
Generation of perspective and panoramic video from omnidirectional video.
Panoramic television viewing system.
Panoramic scene analysis with conic projection.


Omnidirectional imaging with hyperboloidal projection.
Obstacle avoidance with omnidirectional image sensor HyperOmni Vision.
--TR
The lumigraph
Catadioptric Omnidirectional Camera
A Theory of Catadioptric Image Formation
Stereo with Mirrors

--CTR
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panoramic imaging;defocus blur;sensor resolution;omnidirectional imaging;sensor design;image formation