On B-Spline Framelets Derived from the Unitary Extension Principle

Abstract

The spline wavelet tight frames considered in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408--447] have been used widely in frame based image analysis and restorations (see, e.g., survey articles [B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS Lecture Notes Series, vol. 19, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute, 2010; Z. Shen, in Proceedings of the International Congress of Mathematicians, Vol. IV, Hindustan Book Agency, Hyderabad, India, 2010, pp. 2834--2863]). However, except for the properties of the tight frame and the approximation order of the truncated frame series (see [Ron and Shen, J. Funct. Anal., 148 (1997), pp. 408--447; I. Daubechies et al., Appl. Comput. Harmon. Anal., 14 (2003), pp. 1--46]), there are few other properties of this family of tight frames that are currently known. The aim of this paper is to present a few new properties of this family that will provide some reasons why it is efficient in image an...