Periodic Spline-Based Frames for Image Restoration

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 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 low-pass filter. In addition, these filter banks comprise two band-pass filters. 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 semitight 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 calculating the forward and the inverse fast Fourier transforms, practically 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.