Learning Affine Transformations of the Plane

for Model-based Object Recognition

Given a known and an unknown view of the same planar object, there is an affine transformation that can bring them into alignment. In this paper, we consider the problem of learning the mapping between the image coordinates of an unknown affine transformed view of an object and the parameters of the affine transformation that can align the known and the unknown views of the object. This is performed by first sampling the space of affine transformed views of the object and then by a training a Single Layer Neural Network (SL-NN) to learn the mapping between the "sample" affine transformed views and the parameters of the affine transformation which can align the known view of the object with the sample views. An important contribution of this work is the use of Singular Value Decomposition (SVD) and Interval Arithmetic (IA) for computing the range of values that the parameters of affine transformation can assume for a particular object. This is necessary for being able to sample the space of affine transformed views of the object. Also, we show that it is possible to decouple the x- and y-coordinates of the object's interest points and predict the parameters of the affine transformation in two steps, simplifying the proposed scheme considerably. Finally, we propose the use of Principal Components Analysis (PCA) as a front-end stage to the SL-NN. This stage has great practical benefits, since it eliminates the need of considering objects with exactly the same number of points, increases the noise tolerance of the method dramatically, and is can also guide us in deciding how many sample affine transformed views are necessary in order for the SL-NN to learn a good, noise tolerant, mapping. The proposed approach has been tested using real scenes.

George Bebis
January 19, 1996 at 7:02 PM