Phase-Shifting Separable Haar Wavelets and Applications
This addresses a technical bottleneck in signal processing for applications like image manipulation, though it appears incremental as it builds on existing Haar wavelet theory.
The paper tackles the shift-invariance problem in discrete Haar wavelets by deriving closed-form expressions for phase shifting, including non-integer shifts, and applies this to image rotation and interpolation with performance evaluation against standard methods.
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.