Several Studies on Framelets Derived From Compactly Supported Refinable Vector Functions

• Author / Creator

  Lu, Ran

• Generalizing wavelets by adding desired redundancy and flexibility, framelets (a.k.a. wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multi-scale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. The theory and applications of scalar framelets have been extensively studied in the literature. However, vector framelets, or equivalently multiframelets, are far from being well understood. This thesis provides a theoretical investigation of multiframelets.

  Framelets are often derived from refinable vector functions via the popular oblique extension principle (OEP), and such framelets are called OEP-based framelets. Constructing OEP-based tight multiframelets with several desired features is a well known challenging problem. We circumvent this issue by considering quasi-tight multiframelets, which are special dual multiframelets but behave almost identical as tight multiframelets. We will show in Chapter 2 that from any compactly supported univariate refinable vector function with at least two entries, one can always obtain a quasi-tight multiframelet such that: (1) its associated discrete framelet transform is compact and has the highest possible balancing order; (2) all compactly supported framelet generators have the highest possible order of vanishing moments. Several illustrative examples will be provided.

  In Chapter 3, we extend the theory of univariate quasi-tight multiframelets in Chapter 2 to arbitrary dimensions. The generalization is not straight forward. Several new challenges and elements are involved.

  In Chapter 4, we will discuss the more general question on how to construct multivariate dual multiframelets satisfying all desired properties from any pair of compactly supported refinable vector functions. Our study on constructing OEP-based multiframelets relies on a newly developed normal form of matrix-valued filters, which is of independent interest and importance for greatly reducing the difficulty of studying refinable vector functions and multiframelets.

  In Chapter 5, we introduce framelets with mixed dilation factors. Unlike a traditional framelet system which only involves a single dilation factor, we consider framelet systems involves different dilation factors. Recent advances on constructing tight framelets with low redundancy and good directionality give us a taste of framelets with mixed dilation factors, and demonstrate the interest and importance of establishing the corresponding theory. In this thesis, we develop the basic theory of framelets with mixed dilation factors.

• Subjects / Keywords

  • Multiframelets; Refinable vector functions; Normal form of a matrix filter; Oblique extension principle

• Graduation date

  Spring 2021

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-stfw-9m77

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.