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<p><b>Abstract</b>—Similarity measurements between 3<it>D</it> objects and 2<it>D</it> images are useful for the tasks of object recognition and classification. We distinguish between two types of similarity metrics: metrics computed in image-space (<it>image metrics</it>) and metrics computed in transformation-space (<it>transformation metrics</it>). Existing methods typically use image metrics; namely, metrics that measure the difference in the image between the observed image and the nearest view of the object. Example for such a measure is the Euclidean distance between feature points in the image and their corresponding points in the nearest view. (This measure can be computed by solving the <it>exterior orientation calibration problem</it>.) In this paper we introduce a different type of metrics: <it>transformation metrics</it>. These metrics penalize for the deformations applied to the object to produce the observed image.</p><p>In particular, we define a <it>transformation metric</it> that optimally penalizes for "affine deformations" under weak-perspective. A closed-form solution, together with the nearest view according to this metric, are derived. The metric is shown to be equivalent to the Euclidean image metric, in the sense that they bound each other from both above and below. It therefore provides an easy-to-use closed-form approximation for the commonly-used least-squares distance between models and images. We demonstrate an image understanding application, where the true dimensions of a photographed battery charger are estimated by minimizing the transformation metric.</p>
Affine deformations, 3D-to-2D metric, object recognition, exterior orientation calibration.

D. Weinshall and R. Basri, "Distance Metric Between 3D Models and 2D Images for Recognition and Classification," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 18, no. , pp. 465-470, 1996.
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