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

Abstract

High-resolution image reconstruction arises in many applications, such as remote sensing, surveillance, and medical imaging. The model of Bose and Boo [2] 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 + , 1, · · · , 1, 1 2 − , where is the displacement error. When the number L of low-resolution sensors is even, tight frame symmetric framelet filters were constructed in [8] from this low-pass filter using the unitary extension principle of [43]. The framelet filters do not depend on , and hence the resulting algorithm reduces to that of the case where = 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 paper, 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.

Tight Frame Based Method for High-Resolution Image Reconstruction

“…The algorithms of second type are modified versions of the algorithm of the first type by incorporating various denoising techniques. As we will see in subsection 4.2, some of algorithms are already appeared in our previous papers [9,10,11,12,13], but, theoretical results on the convergence of those algorithms given here are new.…”

“…This viewpoint suggests that a framework of multiresolution analysis can be naturally adopted to produce an HR image from a set of low-resolution images of the same scene with sub-pixel shifts. In this fashion, a series of work has been done recently, see, e.g., [9,10,11,12,13]. Extension of these work will be discussed in the paper.…”

“…This algorithm was proposed and studied in [5,8,10,11,12,13]. However, a complete analysis for this algorithm is not available.…”

“…It was proposed and studied in [7,8,11], and can be seen as an extension of the algorithm in [16] from orthornormal system to tight frame systems; another possible extension can be found in [18]. The following convergence theorem for (31) is proved in [7,8].…”

Wavelet Frames Generated by Perfect Reconstruction Filter Banks

“…Thus, frames may serve as a tool for error correction of signals that are transmitted through lossy/noisy channels. Recently, over-complete representations of signals were applied to image reconstruction [6,7]. Combination of wavelet frames with the Bregman iterations techniques [3,21] impacted the image processing applications such as deconvolution, inpainting, denoising, to name a few [4,5,10,19,20].…”