On B-Spline Framelets Derived from the Unitary Extension Principle

Spline wavelet tight frames of [20] have been used widely in frame based image analysis and restorations (see, e.g. survey articles [15, 22]). However, except for the tight frame property and the approximation order of the truncated series (see [13,20]), 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.