# Analytic and directional wavelet packets in the space of periodic signals

## TL;DR

This paper introduces a flexible library of complex wavelet packets derived from discrete splines, enabling efficient 2D directional analysis and demonstrating effectiveness in image restoration tasks.

## Contribution

It develops a new set of analytic and quasi-analytic wavelet packets from discrete splines, with a fast implementation scheme for enhanced signal and image processing.

## Key findings

- Effective in restoring images with high levels of noise or missing pixels
- Provides a diverse set of oriented 2D wavelet packets for directional analysis
- Enables fast and easy implementation of wavelet packet transforms

## Abstract

The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [2]. The imaginary parts are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, they 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 designed computational scheme in the paper enables us to get fast and easy implementation of the WP transforms. The presented WPs proved to be efficient in signal/image processing applications such as restoration of images degraded by either additive noise or missing of up to 90% of their pixels.