Wavelet Frames Generated by Spline Based p-Filter Banks
This chapter presents a design scheme to generate tight and so-called semi-tight frames in the space of discrete-time periodic signals. The frames originate from three- and four-channel perfect reconstruction periodic filter banks. The filter banks are derived from interpolating and quasi-interpolating polynomial splines and from 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 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. The design scheme enables to design framelets with any number of LDVMs.
- 3.J. Cai, S. Osher, and Z. Shen. Split Bregman methods and frame based image restoration. Multiscale Model. Simul., 8(2):337–369, 2009/10.Google Scholar
- 11.D. Gabor, Theory of communications. J. Inst. Electr. Eng. 93, 429–457 (1946)Google Scholar
- 18.Z. Shen. Wavelet frames and image restorations. In R. Bhatia, editor, Proceedings of the International Congress of Mathematicians, Vol. IV, pages 2834–2863, New Delhi, 2010. Hindustan Book Agency.Google Scholar
- 20.V. Zheludev, V. N. Malozemov, and A. B. Pevnyi. Filter banks and frames in the discrete periodic case. In N. N. Uraltseva, editor, Proceedings of the St. Petersburg Mathematical Society, Vol. XIV, volume 228 of Amer. Math. Soc. Transl., Ser. 2, pages 1–11, 2009.Google Scholar