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harmful - Projective Geometry Considered Harmful Berthold...

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Projective Geometry Considered Harmful Berthold K.P. Horn Copright © 1999 Introduction Methods based on projective geometry have become popular in machine vision because they lead to elegant mathematics, and easy-to-solve linear equations [Longuett-Higgins 81, Hartley 97a, Quan & Lan 99]. It is often not realized that one pays a heavy price for this. Such method do not correctly model the physics of image formation and as a result require more correspondences and are considerably more sensitive to measurement error than methods based on true perspective projection. Projective geometry based methods are rarely used in photogrammetry [Wolf 74, Slama 80]. There the cost of acquiring the data is high and every effort is made to extract information about the scene and about the image taking geometry that is as accurate as possible. Since linear equations are easier to solve, there may appear to be an advan- tage in computational cost, but this advantage if any has been eroded by the reduced cost and increased speed of computation. Given the noisy nature of image measurements, one simply cannot afford to throw away accuracy. The issue of the limitations of projective geometry when applied to pho- togrammetric problems has been raised before, particuarly in the context of the relative orientation problem that arises in binocular stereo [Hartley 97b]. But rel- ative orientation is a relatively complex problem where it is hard to gain insight from simple geometric arguments or numerical experiments. As a result, not all researchers have been persuaded that methods based on projective geometry are in fact inferior. Still, it is hard to see the attraction of linear methods for relative orientation, since good methods for solving the least squares problem of relative orientation do exist [Horn 90, 91]. We revisit this topic here in the context of a simpler problem, that of exterior orientation with respect to a planar object. We examine the difference between the mapping from the object plane to the image plane defined by true perspective projection and that defined by projective geometry. We show that virtually none of the transformations allowed by projective geometry correspond to real camera image-taking situations.
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2 We then compare the algorithms and study the sensitivity to noise using Monte Carlo methods and show that the error sensitivity of projective geometry based methods is much higher. Projective Geometry versus Perspective Projection. The true mapping from coordinates in the object coordinate system to image coordinates consists of two steps: (i) Rigid body transformation of the object coordinate system into the camera coordinate system. Rigid body transformations are combinations of rota- tion and translation. If, for now, we use orthonormal matrices to represent rotation, we can write x c x t x o y c y t y o = R + (1) z c z t z o where R is orthonormal and represents rotation, while t = (x o , y o , z o ) T is the translation (position of the object coordinate
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