Frequency implementation of discrete wavelet transforms

The paper implements the discrete wavelet transform in the discrete Fourier domain. The need for such an approach arose out of our desire to find a convenient means of realizing a new class of non-separable orientation selective 2D wavelet filter banks that are designed directly in the DFT domain. The filter bank design process begins with a conventional separable 2D perfect reconstruction parallel filter bank that is not orientation selective. In the DFT domain, each non-low pass channel is decomposed into the sum of two orientation selective frequency responses that are each supported on only two quadrants of the 2D frequency plane. The resulting filter bank possesses the good joint localization properties of orthogonal wavelet transforms and offers both perfect reconstruction and orientation selectivity. However, the orientation selective channels are non-separable - they cannot be implemented as iterated 1D convolutions according to the usual separable 2D wavelet transform paradigm. To overcome this difficulty, we develop straightforward techniques for implementing the DWT directly in the DFT domain.