Spline-based frames in the space of periodic signals

We present a design scheme to generate tight and semi-tight frames in the space of discrete-time periodic signals, which are originated from four-channel perfect reconstruction periodic filter banks. The filter banks are derived from interpolating and quasi-interpolating polynomial splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude’s response mirrors that of a low-pass filter. In addition, these filter banks comprise two bandpass filters. In the semi-tight frames case, all the filters have linear phase and (anti)symmetric impulse response, while in the tight frame case, some of band-pass filters are slightly asymmetric. We introduce the notion of local discrete vanishing moments (LDVM). In the tight frame case, analysis framelets coincide with their synthesis counterparts. However, in the semi-tight frames, we have the option to swap LDVM between synthesis and analysis framelets. The design scheme is generic and it enables us to design framelets with any number of LDVM. The computational complexity of the framelet transforms, which consists of calculation of the forward and the inverse fast Fourier transforms and simple arithmetic operations, practically does not depend on the number of LDVM and does depend on the size of the impulse response of filters . The designed frames are used for image restoration tasks, which are degraded by blurring, random noise and missing pixels. The images were restored by the application of the Split Bregman Iterations method. The frame performances are evaluated.