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Unformatted text preview: Metric Rectification for Perspective Images of Planes David Liebowitz and Andrew Zisserman Robotics Research Group Department of Engineering Science University of Oxford Oxford OX1 3PJ, UK. We describe the geometry, constraints and algorithmic implementation for metric rectification of planes. The recti- fication allows metric properties, such as angles and length ratios, to be measured on the world plane from a perspect- ive image. The novel contributions are: first, that in a stratified context the various forms of providing metric information, which include a known angle, two equal though unknown angles, and a known length ratio; can all be represented as circular constraints on the parameters of an affine trans- formation of the plane — this provides a simple and uniform framework for integrating constraints; second, direct recti- fication from right angles in the plane; third, it is shown that metric rectification enables calibration of the internal camera parameters; fourth, vanishing points are estimated using a Maximum Likelihood estimator; fifth, an algorithm for automatic rectification. Examples are given for a num- ber of images, and applications demonstrated for texture map acquisition and metric measurements. 1 Introduction It is well known that under perspective imaging a plane is mapped to the image by a plane projective transformation (a homography) . This transformation is used in many areas of computer vision including planar object recogni- tion , mosaicing , and photogrammetry . The projective transformation is determined uniquely if the Eu- clidean world coordinates of four or more image points are known. Once the transformation is determined, Euclidean measurements, such as lengths and angles, can be made on the world plane directly from image measurements. Fur- thermore, the image can be rectified by a projective warping to one that would have been obtained from a fronto-parallel view of the plane (i.e. parallel to the image plane). In this paper we show that it is not necessary to provide the Euclidean coordinates of four points; instead metric properties on the world plane, such as a length ratio and an angle, can be used directly to partially determine the pro- jective transformation up to a particular (metric) ambigu- ity. This partial determination requires far less information about the world plane to be known, but is neverless suffi- cient to enable metric measurements of entities on the world plane to be made from their images. Collins and Beveridge  made a significant step in this direction by showing that once the vanishing line of the plane is identified, the transformation from world to image plane can be reduced to an affinity. They used this result to reduce the dimension of the search, from eight to six, in re- gistering satellite images. We improve on this result in four ways....
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- Spring '08
- Euclidean geometry, length ratio, metric rectiﬁcation, O. D. Faugeras, world plane