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.