We present complex rotation-covariant multiresolution families aimed for image analysis. Since they are complex-valued functions, they provide the important phase information, which is missing in the discrete wavelet transform with real wavelets. Our basis elements have nice properties in Hilbert space such as smoothness of fractional order α ∈ $ R ^{ + } $ . The corresponding filters allow a FFT-based implementation and thus provide a fast algorithm for the wavelet transform.