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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
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