Translation and scale invariants of Legendre moments

By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or scaled images using complex or geometric moments. However, the derivation of these functions is not based on Legendre polynomials. This is mainly due to the fact that it is difficult to extract a common displacement or scale factor from Legendre polynomials. In this paper, we introduce a new set of translation and scale invariants of Legendre moments based on Legendre polynomials. The descriptors remain unchanged for translated, elongated, contracted and reflected non-symmetrical as well as symmetrical images. The problems associated with the vanishing of odd-order Legendre moments of symmetrical images are resolved. The performance of the proposed descriptors is experimentally confirmed using a set of binary English, Chinese and Latin characters. In addition to this, a comparison of computational speed between the proposed descriptors and the aforesaid geometric moments-based method is also presented. (C) 2003 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.