On The Continuous Steering of the Scale of Tight Wavelet Frames

TL;DR

This paper introduces a method for creating adaptable tight wavelet frames that can be scaled efficiently, enhancing frequency localization and drawing an analogy to steerable wavelets for improved signal analysis.

Contribution

It presents a novel construction of scalable tight wavelet frames using Fourier multipliers, enabling efficient frequency band scaling similar to steerable wavelets.

Findings

Wavelets can be scaled via Fourier multiplier scaling.

The method improves frequency localization.

Scaling is efficiently implemented with matrix multiplication.

Abstract

In analogy with steerable wavelets, we present a general construction of adaptable tight wavelet frames, with an emphasis on scaling operations. In particular, the derived wavelets can be "dilated" by a procedure comparable to the operation of steering steerable wavelets. The fundamental aspects of the construction are the same: an admissible collection of Fourier multipliers is used to extend a tight wavelet frame, and the "scale" of the wavelets is adapted by scaling the multipliers. As an application, the proposed wavelets can be used to improve the frequency localization. Importantly, the localized frequency bands specified by this construction can be scaled efficiently using matrix multiplication.