hen we can derive the moment invariants by the following algebraic method. If we subject both u, 0 and u’, v’ to the following transformation:Рthen the orthogonal transformation is converted into the following simple relations,РBy substituting (36) and (37) into (34), we have the following identities:Рwhere Ipo, . . . , I,, and Igo, * . . , I&, are the corresponding coefficients after the substitutions. From the identity in U and V, the coefficients of the various monomials Up-rVr on the two sides must be the same. Therefore,РThese are (p + 1) linearly independent moment invariants under proper orthogonal transformations, and △=eiθ which is not the determinant of the transformation.РFrom the identity of first two expressions in (38), it can be seen that Ir,p-r is plex conjugate of Ip-r,r ,
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