Directional wavelet packets originating from polynomial splines

TL;DR

This paper introduces a comprehensive library of complex wavelet packets derived from polynomial splines, offering enhanced directionality and frequency resolution for advanced image processing tasks.

Contribution

It presents a new set of quasi-analytic wavelet packets with arbitrary directional orientations, derived from polynomial splines, expanding the tools for image analysis.

Findings

Provides wavelet packets with 62 directions at fourth level

Offers refined frequency resolution and strong directionality

Waveforms exhibit symmetry and oscillating structures

Abstract

The paper presents a versatile library of quasi-analytic complex-valued wavelet packets (WPs) which originate from polynomial splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [1]. The imaginary parts are the so-called complementary orthonormal WPs that are derived from the Hilbert transforms of the regular WPs and, unlike the symmetric regular WPs, are antisymmetric. Tensor products of 1D quasi-analytic WPs provide a diversity of 2D WPs oriented in multiple directions. For example, a set of the fourth-level WPs comprises 62 different directions. The properties of the presented WPs are refined frequency resolution, directionality of waveforms with unlimited number of orientations, (anti-)symmetry of waveforms and windowed oscillating structure of waveforms with a variety of frequencies. Directional WPs have a strong potential to be used in various image processing applications such as restoration of degraded images and extraction of characteristic features from the images.