On B-spline framelets derived from the unitary extension principle

TL;DR

This paper explores new properties of spline wavelet tight frames derived from the unitary extension principle, including recurrence formulas and their relation to derivatives of Gaussian functions, enhancing understanding of their efficiency in image analysis.

Contribution

It introduces a recurrence formula for higher order spline wavelet tight frames and links their generators to derivatives of Gaussian functions, providing deeper insight into their structure.

Findings

Generators of high order spline wavelet tight frames approximate derivatives of Gaussian functions.

Wavelet systems from derivatives of Gaussian functions form nearly tight frames.

New properties help explain the effectiveness of spline wavelet frames in image processing.

Abstract

Spline wavelet tight frames of Ron-Shen have been used widely in frame based image analysis and restorations. However, except for the tight frame property and the approximation order of the truncated series, there are few other properties of this family of spline wavelet tight frames to be known. This paper is to present a few new properties of this family that will provide further understanding of it and, hopefully, give some indications why it is efficient in image analysis and restorations. In particular, we present a recurrence formula of computing generators of higher order spline wavelet tight frames from the lower order ones. We also represent each generator of spline wavelet tight frames as certain order of derivative of some univariate box spline. With this, we further show that each generator of sufficiently high order spline wavelet tight frames is close to a right order of derivative of a properly scaled Gaussian function. This leads to the result that the wavelet system generated by a finitely many consecutive derivatives of a properly scaled Gaussian function forms a frame whose frame bounds can be almost tight.