Quasi-analytic directional wavelet packets: applications to image processing

Recently, a versatile library of quasi-analytic complex-valued wavelet packets (WPs) which originate from splines of arbitrary orders, was designed [1]. The real parts of the quasi-analytic WPs (qWPs) are the regular spline-based orthonormal WPs. The imaginary parts, which are slightly modified Hilbert transforms of the real parts, are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, are antisymmetric. Both regular and complementary WPs are well localized in time domain and their DFT spectra provide a variety of refined splits of the frequency domain. The waveforms can have arbitrary number of vanishing moments.Tensor products of 1D quasi-analytic WPs (qWPs) provide a diversity of 2D waveforms oriented in multiple directions. The designed computational scheme enables us to get fast and easy implementation of the qWP transforms. The shapes of real and imaginary parts of the qWPs can be regarded as directional cosine waves with different frequencies modulated by localized low-frequency signals. For example, the set of the fourth-level WPs comprises waveforms which are oriented in 314 different directions and are oscillating with 256 different frequencies. Various combinations of qWPs form multiple frames in the 2D signal space.The combination of the exceptional properties of the designed qWPs, such as unlimited directionality and oscillating structure of the waveforms, vanishing moments and refined frequency resolution, make them a powerful tool for image processing applications. The algorithms based on the qWPs proved to be competitive with the best existing methods in solving such classical image processing problems as denoising, inpainting and deblurring. The qWP algorithms are especially efficient for capturing edges and fine texture and oscillating patterns even in severely degraded images.Due to the above properties and next to unlimited diversity of testing waveforms, the qWPs have strong capabilities for extraction characteristic features from signals and images, which are utilized in the image classification algorithms in conjunction with Support Vector Machines and Convolutional Neural Networks.