Tight frame: an efficient way for high-resolution image reconstruction

High-resolution image reconstruction arise in many applications, such as remote sensing, surveillance, and medical imaging. The model proposed by Bose and Boo [Int. J. Imaging Syst. Technol. 9 (1998) 294–304] can be viewed as passing the high-resolution image through a blurring kernel, which is the tensor product of a univariate low-pass filter of the form [1/2+ε,1, . . . ,1,1/2−ε], where ε is the displacement error. Using a wavelet approach, bi-orthogonal wavelet systems from this low-pass filter were constructed in [R. Chan et al., SIAM J. Sci. Comput. 24 (4) (2003) 1408–1432; R. Chan et al., Linear Algebra Appl. 366 (2003) 139–155] to build an algorithm. The algorithm is very efficient for the case without displacement errors, i.e., when all ε = 0. However, there are several drawbacks when some ε = 0. First, the scaling function associated with the dual low-pass filter has low regularity. Second, only periodic boundary conditions can be imposed, and third, the wavelet filters so constructed change when some ε change. In this paper, we design tight-frame symmetric wavelet filters by using the unitary extension principle of [A. Ron, Z. Shen, J. Funct. Anal. 148 (1997) 408–447]. The wavelet filters do not depend on ε, and hence our algorithm essentially reduces to that of the case where ε = 0. This greatly simplifies the algorithm and resolves the drawbacks of the bi-orthogonal approach. * Corresponding author. E-mail addresses: [email protected] (S.D. Riemenschneider), [email protected] (L. Shen). 1 Research supported in part by HKRGC Grants CUHK4243/01P and CUHK DAG 2060220. 2 This work was supported by Grant NSF-EPSCoR-0132740. The work was partially done while author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2003. The visit was partially supported by the institute. 3 This work was supported by Grant NSF-EPSCoR-0132740. 4 Research supported in part by several grants at the National University of Singapore. 1063-5203/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.acha.2004.02.003 92 R.H. Chan et al. / Appl. Comput. Harmon. Anal. 17 (2004) 91–115  2004 Elsevier Inc. All rights reserved.