Periodic discrete-time frames: Design and applications for image restoration

We present diverse families of tight and semi-tight frames in the space of discrete-time periodic signals, which are generated by threeand four-channel perfect reconstruction periodic filter banks. The filter banks are derived from the interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one interpolating linear phase low-pass filter and one high-pass filter, whose magnitude’s response mirrors that of the low-pass filter. Depending on the channels number, these filter banks comprise one or two band-pass 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. Since the common definition of the vanishing moments is invalid in periodic setting, 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 options to swap LDVM between the synthesis and the analysis framelets or to add smoothness to the synthesis framelets. The design scheme is generic and makes it possible to design framelets with any number of LDVM. The computational complexity of the framelet transforms, which consists of calculation of direct and inverse fast Fourier transforms and simple arithmetical operations, practically does not depend on the number of LDVM. The designed frames are tested for image restoration, which are degraded by blurring, affected by random noise and parts of pixels are missing. The images were restored by the application of the Split Bregman Iterations method. The frame performances are compared.