Signal Representation and Processing Using Operator Groups

This thesis presents a signal representation in terms of operators The signal is as sumed to be an element of a vector space and subject to transformations of operators The operators form continuous groups so called Lie groups The representation can be used for signals in general in particular if spatial relations are unde ned and it does not require a basis of the signal space to be useful Special attention is given to orthogonal operator groups which are generated by anti Hermitian operators by means of the exponential mapping It is shown that the eigen system of the group generator is strongly related to properties of the corresponding operator group For one parameter orthogonal operator groups a phase concept is introduced This phase can for instance be used to distinguish between spatially even and odd signals and therefore corresponds to the usual phase for multi dimensional signals Given one operator group that represents the variation of the signal and one operator group that represents the variation of a corresponding feature descriptor an equiv ariant mapping maps the signal to the descriptor such that the two operator groups correspond Su cient conditions are derived for a general mapping to be equivariant with respect to a pair of operator groups These conditions are expressed in terms of the generators of the two operator groups As a special case second order homo geneous mappings are considered and examples of how second order mappings can be used to obtain di erent types of feature descriptors are presented in particular for operator groups that are homomorphic to rotations in two and three dimensions respectively A generalization of directed quadrature lters is made All feature extraction algorithms that are presented are discussed in terms of phase invariance Simple procedures that estimate group generators which correspond to one parameter groups are derived and tested on an example The resulting generator is evaluated by using its eigensystem in implementations of two feature extraction algorithms It is shown that the resulting feature descriptor has good accuracy with respect to the corresponding feature value even in the presence of signal noise