Lie Group Transformations of Objects in Images

Suppose we take a set of greyscale images obtained from videophotography of a moving object in three dimensions, such as an aeroplane, and suppose we compute moments or fourier descriptors, or some other set of smooth features of the resulting image to get a point in a feature space Fn. Then provided we compute enough moments or other descriptors, i.e. provided n is big enough, and provided the object is not symmetric, the result of this is to give a sample of points from a smooth, one-one map of the group of transformations into the feature space. The value of n which is `big enough' is, almost always, 2d+1, where d is the dimension of the group. This result has applications in, for example, interpolating between di erent views of an object. In this paper we state and prove the above result formally, and give some simple applications.