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with two finite rotations. The case of infinitesimal trans-
lations observed at a finite number of points is equivalent to
projective reconstruction in flatland. Five translations deter-
mine the points up to a 5th degree Cremona transformation,
six translations up to a two-fold plus projective ambiguity
and seven translations up to projective ambiguity.
The rest of the paper is organized as follows. In Section 2
we discuss related work. In Section 3 we derive our main
result. In Section 4 we summarize additional theoretical
results for various cases and assumptions. In Section 5 we
present some results on synthetic and real images. Section 6
concludes.2. Related Work
It is beyond the scope of this paper to give an account
of the history of calibration and self-calibration. The in-
terested reader is referred to [4, 5]. Classical calibration is
performed by imaging a known calibration object [12, 14].
Self-calibration alleviates the need for a known object. It
relies on observing the same a priori unknown scene in mul-
tiple images while keeping some internal parameters fixed.
The best constrained case is when all intrinsic parameters
are fixed [7], but it is also possible to obtain results while
changing the focal length [9], or in theory even all but one
intrinsic parameter [6]. Both classical calibration and self-
calibration rely on a camera model known up to a small
number of parameters. This constraint was dropped re-
cently, leading to very flexible calibration using a known
object undergoing known [3], or unknown [11] motion.
The goal of this paper is to investigate what can be done
with a non-parametric model of a single camera when the
scene is a priori unknown and not directly observable, lead-
ing to a theory of non-parametric self-calibration.
3. Main Theory
Assume that the camera produces images that are warped
versions of their Euclidean counterpart given by a calibrated
central perspective camera. Let the warp be described by
some function f x that indicates from which Euclidean
image point y the intensity of the distorted image point x is taken, i.e.y f x (2)
describes the correspondence between the distorted and the
Euclidean image.
In the infinitesimal motion case, we observe a flow field
v x in the distorted image. The constraint is that when un-
warped, this flow must be some underlying Euclidean flow
field m y , which even for general 3D motion is more lim-
ited than a general flow. Formally, the constraint is ∂f
∂x x is the matrix derivative of f at x.
We will find it convenient to think of points y f x
in the Euclidean image as points on the unit sphere and
the flow m y as vectors based at y and tangent to the unit
sphere. The flow induced at a point y in the Euclidean im-
age by an instantaneous rotation r is then m y r × y, (4)
which when inserted into Equation (3) giveswhere J ∂f
∂x x is a × matrix.
The information about the warping function f in the
neighborhood of a point x is contained in y f x and
J. Given the observed flow vectors v we wish to recover
the unwarped image locations y.
If we only observe the flow at a finite number of points,
it turns out that the rotations r can only be recovered up to
a common scale and multiplication by a × matrix, and
that the points y can only be determined up to a projective
transformation:
Theorem 2 If we observe the flow vectors at a finite num-
ber of points x of the distorted image, induced by instan-
taneous rotations r, we can at best recover the undistorted
points y f x corresponding to x up to a projective trans-
formation, and the rotations r up to a common scale and