Periodic spline-based frames for image restoration

We present a design scheme that generates tight and semi-tight frames in discrete-time periodic signals space originated from four-channel perfect reconstruction periodic filter banks. Filter banks are derived from interpolating and quasi-interpolating polynomial and discrete 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 lowpass filter. These filter banks comprise two band-pass filters. We introduce local discrete vanishing moments (LDVM). When the frame is tight, analysis framelets coincide with their synthesis counterparts. However, for semi-tight frames, we 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 calculating the forward and the inverse FFTs, does not depend on the number of LDVM and does depend on the size of the impulse response filters. The designed frames are used for image restoration tasks, which were degraded by blurring, random noise and missing pixels. The images were restored by the application of the Split Bregman Iterations method. The frames performances are evaluated. A potential application of this methodology is the design of a snapshot hyperspectral imager that is based on a regular digital camera. All these imaging applications are described.