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Re: Control points and DLT

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  • Re: Control points and DLT

    Herewith is my two bits (or bytes) worth regarding the discussion started
    by Rick
    Hinrichs, and so far commented upon by Brian Davis, Virgil Stokes, Bing Yu,
    and Scott Tashman

    1) The standard DLT has an inherent problem that has been pointed out by
    Hatze and others over a number of years. Namely, the technique solves for
    10 independent parameters using 11 equations, and the solution typically
    cannot provide an accurate representation of the physical camera system.
    This fact
    is relatively easy to demonstrate if one calculates the terms of the camera
    rotation matrix from the DLT parameters. The result is a rotation matrix
    which is not orthonormal. So how might such a distortion of the physical
    reality affect our ability to achieve accurate measurements ? All of the
    DLT parameters are
    compromised in order to satisfy the least-squares criterion used in solving
    the equations. The result is a measurement system whose model is
    inaccurate and in which the final measurements are sensitive to the number
    locations of the control points chosen for the calibration. It also
    explains why a number of studies have found that the DLT extrapolates
    poorly outside of the control point volume.

    2) Brian Davis proposes a technique which "corrects" points with the
    largest residuals in order to get a better solution. I agree with Virgil
    Stokes that this is a dangerous procedure. Too often calibration residuals
    are equated with system accurary. Under certain circumstances it is
    possible to have low residuals AND low accuracy. Any accurate measurement
    system must be based on an accurate standard, otherwise we are attempting
    to get something from nothing. In case of the DLT, even if we have highly
    accurate control points (which is rarely the case), we have an inaccurate
    model of the the camera system. I do believe that it is possible
    to achieve this "something from nothing (or at least very little)". If the
    number of measurements exceeds the number of independent parameters in the
    model, AND the model is accurate, then we can get an accurate system from
    less input. An example of this approach is the "wand" calibration
    technique which I developed, and which has become a part of several
    commercial 3D systems. [ In this technique two markers attached to the ends
    of a stick are moved around the calibration space while multiple frames of
    data are collected. The need for accurately measured control points is
    eliminated. Three approximately measured static points are used to define
    the location and orientation of the desired coordinate system.]

    3) The approach to 3D calibration that I advocate and which has been
    by a number of commercial 3D systems for quite some time, is a two step
    one. You first
    calibrate (linearize) each individual camera (Scott Tashman briefly
    described this technique
    in his comments), which provides all of the internal camera parameters,
    and then in the final 3D calibration only the six external camera
    parameters (3 of location, 3 of orientation) are solved for. There are a
    number of advantages to this technique which I will not
    detail here, but a principal one is that the model can accurately represent
    the physical measurement system.

    4) Regarding the question of the desirable number of control points, I have
    found that with the two step technique any more than eight accurately
    specified and measured control points provide very small gains in accuracy
    improvement. A paper ["Accurate Remote Measurement of Robot Trajectory
    Motion", A.Dainis M.Jubers. Proceedings of the IEEE International
    Conference on Robotics and Automation, St. Louis, Mo. 1985] concluded that
    the use of 62 and 108 control points only provide marginal improvements
    over eight control points. The control points were generated by a very high
    accuracy 3D coordinate measuring machine, and the measurements were
    performed by a modified Selspot system. I have also found in unpublished
    studies that this technique provides considerably more accurate
    extrapolations than the standard DLT method.

    5) On several occasions I have tried to compare the DLT and my two step
    methods in parallel. However, since the DLT parameters are corrupted for
    the reasons described in the first paragraph, there are no direct
    correspondences between parameters from the two techniques, and it is
    difficult to draw specific conclusions. However, one conclusion is that the
    standard DLT is capable of providing accurate results NEAR the control
    points, but is more limited at other spatial locations. The above detailed
    reasons also, I feel, explain why so many points are required for optimum
    accuracy when using the DLT method. Only six accurate control points should
    be necessary to compute the 11 parameters, a few more if you wish to
    include radial distortion corrections. But this obviously is not the case.

    Andy Dainis Ph.D.