Periodic spline-based frames for image restoration
Amir Z. Averbuch, Pekka Neittaanmäki, Valery A. Zheludev · Inverse Problems and Imaging · 2015
We present a design scheme that generates tight and semi-tightframes in discrete-time periodic signals space originated fromfour-channel perfect reconstruction periodic filter banks. Filter banksare derived from interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one linear phaselow-pass filter (in most cases interpolating) and one high-passfilter, whose magnitude's response mirrors that of a low-pass filter.These filter banks comprise two band-pass filters. We introduce local discrete vanishing moments (LDVM). When the frame is tight, analysis framelets coincidewith their synthesis counterparts. However, for semi-tight frames,we swap LDVM between synthesis and analysis framelets. The designscheme is generic and it enables us to design framelets with anynumber of LDVM. The computational complexity of the the framelet transforms,which consists of calculating the forward and the inverse FFTs, doesnot depend on the number of LDVM and does depend on the size of the the impulse response fi lters. The designed frames are used for image restorationtasks, which were degraded by blurring, random noise and missingpixels. The images were restored by the application of the SplitBregman Iterations method. The frames performances are evaluated. Apotential application of this methodology is the design of asnapshot hyperspectral imager that is based on a regular digitalcamera. All these imaging applications are described.