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Tight frame: an efficient way for high-resolution image reconstruction. (English) Zbl 1046.42026

Summary: High-resolution image reconstruction arise in many applications, such as remote sensing, surveillance, and medical imaging. The model proposed by N. Bose and K. Boo [Int. J. Imaging Syst. Technol. 9, 294–304 (1998), per bibl.] 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 \([1/2+\varepsilon,1,\dots,1,1/2-\varepsilon]\), where \(\varepsilon\) is the displacement error. Using a wavelet approach, bi-orthogonal wavelet systems from this low-pass filter were constructed in [R. H. Chan, T. F. Chan, L. Shu and Z. Shen, SIAM J. Sci. Comput. 24, No. 4, 1408–1432 (2003; Zbl 1031.68127); Linear Algebra Appl. 366, 139–155 (2003; Zbl 1025.65064)] to build an algorithm. The algorithm is very efficient for the case without displacement errors, i.e., when all \(\varepsilon=0\). However, there are several drawbacks when some \(\varepsilon \neq 0\). 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 \(\varepsilon\) change. In this paper, we design tight-frame symmetric wavelet filters by using the unitary extension principle of [A. Ron and Z. Shen, J. Funct. Anal. 148, No. 2, 408–447 (1997; Zbl 0891.42018)]. The wavelet filters do not depend on \(\varepsilon\), and hence our algorithm essentially reduces to that of the case where \(\varepsilon=0\). This greatly simplifies the algorithm and resolves the drawbacks of the bi-orthogonal approach.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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References:

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