Similarity and Aane Distance between Point Sets

We develop the expressions for measuring the distance between 2D point sets, which are invariant to either 2D aane transformations or 2D similarity transformations of the sets, and assuming a known correspondence between the point sets. We discuss the image normalization to be applied to the images before their comparison so that the computed distance is symmetric with respect to the two images. We then give a general (metric) deenition of the distance between images, which leads to the same expressions for the similarity and aane cases. This deenition avoids ad-hoc decisions about normalization. Moreover, it makes it possible to compute the distance between images under diierent conditions, including cases where the images are treated asymmetrically. We demonstrate these results with real and simulated images. 1 Background When comparing images to other images or models, one would like to somehow cancel camera transformations. In general there is no way to normalize images of 3D objects so that all projections are equivalent (other than a normalization that makes all images equivalent). However, under the weak perspective (scaled orthographic) projection model assumed here, it is possible to remove the eeects of certain camera transformations, such as rotations about the optical axis and translations. More speciically, there exist standard methods of image normalization with respect to the following image transformations 3]: Translations: the image is shifted so that its centroid is at the origin. Rotations: the image is rotated so that its principal axis has some standard orientation. Normalization with respect to rotations can be replaced by normalization with respect to linear transformations, or normalization by moments, where an image is transformed with a linear transformation so that its second moments have given values. This method is related to the Whitening transformation, a linear transformation of data which transforms its covariance matrix into the unit matrix 1, 4]. This transformation does not preserve Euclidean distances.