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 analysis and restorations. In particular, we first present a recurrence formula for computing the generators of higher order spline wavelet tight frames from lower order ones. This simplifies the computations of the exact values of the functions in applications. Second, we represent each generator of spline wavelet tight frames as a certain order of derivative of some univariate box spline that satisfies a few additional properties. This is a crucial property used in [J.-F. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], where connections between total variational and wavelet frame based approaches for image restorations are established. Finally, we further show that each generator of sufficiently high order spline wavelet tight frames is close to a derivative of a properly scaled Gaussian function. This leads to the result that the wavelet system generated by finitely many consecutive derivatives of a properly scaled Gaussian function forms a frame whose frame bounds can be almost tight. © 2013 Society for Industrial and Applied Mathematics.