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    Abstract In this paper we develop a theory of non-parametric self-calibration. Recently, schemes have been devised for non-parametric laboratory calibration, but not for self-calibration.
    We allow an arbitrary warp to model the intrinsic map-ping, with the only restriction that the camera is central and that the intrinsic mapping has a well-defined non-singular matrix derivative at a finite number of points under study.We give a number of theoretical results, both for in-finitesimal motion and finite motion, for a finite number of observations and when observing motion over a dense im-age, for rotation and translation. 22815
    Our main result is that through observing the flow in-
    duced by three instantaneous rotations at a finite number
    of points of the distorted image, we can perform projective
    reconstruction of those image points on the undistorted im-
    age. We present some results with synthetic and real data.
    1. Introduction
    Classical calibration of the intrinsic parameters of a cam-
    era is done by identifying the image points corresponding
    to known points on a calibration object. Self-calibration on
    the other hand is performed by identifying correspondences
    while observing an unknown static scene undergoing an un-
    known but rigid motion. In both classical calibration and
    self-calibration, the internals of the camera are assumed to
    follow some model with a fairly limited number of param-
    eters. Recently, methods have been developed that perform
    calibration without assuming a parametric model for the
    camera intrinsics. This is done by calibrating each image
    ray separately while observing a known object undergoing
    known or unknown motion.
    The topic of this paper is to develop a theory of self-
    calibration without requiring a parametric form for the in-
    trinsics of the camera. We focus on the case of flow induced by three infinitesimal rotations and observed at a finite num-
    ber of locations in the distorted image. Our main result is
    in short that it is possible to perform projective reconstruc-
    tion of the observed locations. The main result is relatively
    easy to state precisely, so we state it here as a preview of the
    theory. The following is essentially a restatement of Theo-
    rem 4:
    Theorem 1 If we observe the flow vectors a x , b x and
    c x at the points x of the distorted image, induced by three
    instantaneous rotations, we obtain a projective reconstruc-
    tion of the undistorted image points y by setting If the scene is distant enough relative to the camera ge-
    ometry and motion, the camera can always be modeled as
    central and the motion approximated by a rotation. Some
    insects rotate their heads intensively, and although we do
    not claim that they perform self-calibration in the manner
    suggested by our theory, we find self-calibration with few
    or no assumptions on the camera an exciting prospect. In
    principle, one could then over time cope with any aberra-
    tions present, and in theory solve for ’everything’ including
    an unknown scene and unknown camera.
    To provide some context to our main result, we also sum-
    marize theory arising from various assumptions, including
    infinitesimal and finite rotations and translations, and mo-
    tion observed in a finite number of locations as well as
    over the complete dense image sphere. With two flow-fields
    from a purely rotating camera, it is possible in theory to re-
    cover the intrinsics of the camera without a priori assuming
    a parametric form. Moreover, starting from a smooth but
    otherwise arbitrary intrinsic mapping, two flow-fields from
    a purely translating camera determine the intrinsic mapping
    up to a projective transformation of the image plane. Fur-
    ther, non-parametric self-calibration is in theory possible
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