Learning a$ne transformations

Under the assumption of weak perspective, two views of the same planar object are related through an a$ne transformation. In this paper, we consider the problem of training a simple neural network to learn to predict the parameters of the a$ne transformation. Although the proposed scheme has similarities with other neural network schemes, its practical advantages are more profound. First of all, the views used to train the neural network are not obtained by taking pictures of the object from di!erent viewpoints. Instead, the training views are obtained by sampling the space of a$ne transformed views of the object. This space is constructed using a single view of the object. Fundamental to this procedure is a methodology, based on singular-value decomposition (SVD) and interval arithmetic (IA), for estimating the ranges of values that the parameters of a$ne transformation can assume. Second, the accuracy of the proposed scheme is very close to that of a traditional least squares approach with slightly better space and time requirements. A front-end stage to the neural network, based on principal components analysis (PCA), shows to increase its noise tolerance dramatically and also to guides us in deciding how many training views are necessary in order for the network to learn a good, noise tolerant, mapping. The proposed approach has been tested using both arti"cial and real data. ( 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.