High-resolution image reconstruction with displacement errors: A framelet approach

High-resolution image reconstruction arises in many applications, such as remote sensing, surveillance, and medical imaging. The Bose and Boo (1998) model can be viewed as the passage of the high-resolution image through a blurring kernel built from the tensor product of a univariate low-pass filter of the form [1/2 + ε, 1,..., 1, 1/2 - ε], where e is the displacement error. When the number L of low-resolution sensors is even, tight-frame symmetric framlet filters were constructed (Chan et al., 2004b) from this low-pass filter using Ron and Shen's (1997) unitary extension principle. The framelet filters do not depend on e, and hence the resulting algorithm reduces to that of the case where ε = 0. Furthermore, the framelet method works for symmetric boundary conditions. This greatly simplifies the algorithm. However, both the design of the tight framelets and extension to symmetric boundary are only for even L and cannot, be applied to the case when L is odd. In this article, we design tight framelets and derive a tight-framelet algorithm with symmetric boundary conditions that work for both odd and even L An analysis of the convergence of the algorithms is also given. The details of the implementations of the algorithm are also given.