# Phase-Shifting Separable Haar Wavelets and Applications

## TL;DR

This paper introduces a novel phase-shifting method for Haar wavelets that preserves key properties and enables accurate, efficient shift and rotation operations in the wavelet domain, with applications to image processing.

## Contribution

It derives closed-form expressions for phase shifting in Haar wavelets, including non-integer shifts, and demonstrates their application to image rotation and interpolation.

## Key findings

- Accurate phase shifting formulas for Haar wavelets
- Effective image rotation and interpolation using the new method
- Preservation of Haar wavelet properties during shift operations

## Abstract

This paper presents a new approach for tackling the shift-invariance problem in the discrete Haar domain, without trading off any of its desirable properties, such as compression, separability, orthogonality, and symmetry. The paper presents several key theoretical contributions. First, we derive closed form expressions for phase shifting in the Haar domain both in partially decimated and fully decimated transforms. Second, it is shown that the wavelet coefficients of the shifted signal can be computed solely by using the coefficients of the original transformed signal. Third, we derive closed-form expressions for non-integer shifts, which have not been previously reported in the literature. Fourth, we establish the complexity of the proposed phase shifting approach using the derived analytic expressions. As an application example of these results, we apply the new formulae to image rotation and interpolation, and evaluate its performance against standard methods.