Complex moments of the local power spectrum (CMP) are investigated in a multiscale context. The multiscale CMPs are shown to approximate well the 1D angular Fourier transform of the band in question. This observation is used to derive further properties of the power spectrum in terms of texture orientations or n-folded symmetry patterns. A method is presented to approximate the power spectrum using only separable filtering in the spatial domain. Interesting implications to the Gabor decomposition are shown. The number of orientations in the filter bank is related to the order of n-folded symmetry detectable. Furthermore, the multiscale CMPs can be estimated incrementally in the spatial domain, which is both fast and reliable. Experiments on power spectrum estimation, orientation estimation, and texture segmentation are presented.