Directional wavelet packets originating from polynomial splines

The paper presents a versatile library of quasi-analytic complex-valued wavelet packets (qWPs) which originate from polynomial splines of arbitrary orders. Discrete Fourier transforms (DFT) of qWPs are located in either positive or negative half-band of the frequency domain. Consequently, the DFTs of 2D qWPs, which are derived by the tensor products of 1D qWPs, occupy one of quadrants of 2D frequency domain. Such a structure of the DFT spectra of the 2D qWPs results in the directionality of their real parts. Due to the fact that the spectra of qWPs are well localized in the frequency domain, the shapes of real parts of the qWPs are close to windowed cosine waves oscillating in a variety of different directions with a variety of frequencies. For example, a set of the fourth-level qWPs comprises 314 different directions and 256 different frequencies. The above properties combined with a fast transform implementation make directional qWPs a strong tool for the application to a variety of image processing tasks such as restoration of degraded images and extraction of characteristic features from images to use them in deep learning. A few illustrations of successful application of the designed qWPs to image denoising, inpainting, and classification are given in the paper.