[Paper Review] A wavelet Plancherel theory with application to sparse continuous wavelet transform

Ron Levie, Nir Sochen|arXiv (Cornell University)|Dec 7, 2017

Image and Signal Denoising Methods30 references3 citations

TL;DR

This paper introduces a wavelet-Plancherel theory that embeds signals into a window-signal space, enabling an isometric isomorphism between the signal space and phase space via the wavelet-Plancherel transform. By leveraging this isomorphism, the method enables efficient sparse approximation through a coefficient search in the window-signal space, reducing computational complexity compared to standard matching pursuit algorithms.

ABSTRACT

We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger abstract signal space, which we call the window-signal space. There is a canonical extension of the wavelet transform on the window-signal space, which is an isometric isomorphism from the window-signal space to a space of functions on phase space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation on phase space can be pulled-back to an operation in the window-signal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the window-signal space. We can thus avoid inefficient calculations on phase space, performing all calculations entirely in the window-signal space. We consider in this paper a matching pursuit algorithm based on this coefficient search approach. Our method has lower computational complexity than matching pursuit algorithms based on a naive coefficient search.

Motivation & Objective

• To develop a theoretical framework for sparse signal approximation using continuous wavelet systems.
• To address the high computational cost of naive coefficient searches in phase space during matching pursuit.
• To embed the signal space into a larger window-signal space to enable isometric isomorphism with phase space.
• To enable efficient computation by pulling back operations from phase space to the window-signal space.
• To reduce the computational complexity of sparse approximation algorithms based on matching pursuit.

Proposed method

• The signal space is embedded into a larger abstract space called the window-signal space.
• An extended wavelet transform, the wavelet-Plancherel transform, is defined as an isometric isomorphism from the window-signal space to a function space on phase space.
• Operations in phase space, such as coefficient search, are pulled back to the window-signal space for computation.
• The pull-back property allows all calculations to be performed entirely within the window-signal space, avoiding direct phase space operations.
• A matching pursuit algorithm is constructed based on this coefficient search approach in the window-signal space.
• The method avoids inefficient phase space sampling by operating in the isometrically equivalent window-signal space.

Experimental results

Research questions

• RQ1 How can continuous wavelet systems be extended to enable efficient sparse approximation in signal processing?
• RQ2 What mathematical framework supports an isometric isomorphism between signal space and phase space for wavelet transforms?
• RQ3 Can coefficient search for matching pursuit be optimized by transforming the problem into a larger signal space?
• RQ4 What is the computational advantage of performing operations in the window-signal space over direct phase space computation?
• RQ5 How does the wavelet-Plancherel transform enable reduced complexity in sparse approximation algorithms?

Key findings

• The wavelet-Plancherel transform establishes an isometric isomorphism between the window-signal space and functions on phase space.
• The isomorphism allows all phase space operations to be equivalently performed in the window-signal space.
• The method enables a more efficient coefficient search for matching pursuit by operating in the window-signal space.
• The computational complexity of the resulting sparse approximation algorithm is lower than that of naive phase space-based matching pursuit.
• The framework provides a theoretical foundation for sparse signal representation using continuous wavelet systems with reduced computational overhead.