Chan, Raymond H.; Chan, Tony F.; Shen, Lixin; Shen, Zuowei Wavelet deblurring algorithms for spatially varying blur from high-resolution image reconstruction. (English) Zbl 1025.65064 Linear Algebra Appl. 366, 139-155 (2003). Summary: High-resolution image reconstruction refers to reconstructing a higher resolution image from multiple low-resolution samples of a true image. The authors [Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong (2000)] considered the case where there are no displacement errors in the low-resolution samples, i.e., the samples are aligned properly, and hence the blurring operator is spatially invariant.In this paper, we consider the case where there are displacement errors in the low-resolution samples. The resulting blurring operator is spatially varying and is formed by sampling and summing different spatially invariant blurring operators. We represent each of these spatially invariant blurring operators by a tensor product of a lowpass filter which associates the corresponding blurring operator with a multiresolution analysis of \(L^2(\mathbb R^2)\). Using these filters and their duals, we derive an iterative algorithm to solve the problem based on the algorithmic framework of our research report [loc. cit.]. Our algorithm requires a nontrivial modification to the algorithms in Chan et al. [loc. cit.], which apply only to spatially invariant blurring operators. Our numerical examples show that our algorithm gives higher peak signal-to-noise ratios and lower relative errors than those from the Tikhonov least squares approach. Cited in 1 ReviewCited in 7 Documents MSC: 65T60 Numerical methods for wavelets 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 68U10 Computing methodologies for image processing Keywords:wavelet; high-resolution image reconstruction; Tikhonov least squares method; deblurring algorithms; blurring operators; lowpass filter; multiresolution analysis; iterative algorithms; numerical examples Software:Stony Brook PDFBibTeX XMLCite \textit{R. H. Chan} et al., Linear Algebra Appl. 366, 139--155 (2003; Zbl 1025.65064) Full Text: DOI References: [1] Bose, N.; Boo, K., High-resolution image reconstruction with multisensors, Int. J. Imag. Syst. Technol., 9, 294-304 (1998) [3] Cohen, A.; Daubechies, I.; Feauveau, J., Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, 485-500 (1992) · Zbl 0776.42020 [4] Daubechies, I., Ten lectures on wavelets, (CBMS Conference Series in Applied Mathematics, vol. 61 (1992), SIAM: SIAM Philadelphia) · Zbl 0776.42018 [5] Donoho, D.; De-noising by soft-thresholding, IEEE Trans. Inform. Theory, 41, 613-627 (1995) · Zbl 0820.62002 [6] Donoho, D.; Johnstone, I., Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81, 425-455 (1994) · Zbl 0815.62019 [7] Gonzalez, R.; Woods, R., Digital Image Processing (1993), Addison-Wesley [8] Lawton, W.; Lee, S.; Shen, Z., Stability and orthonormality of multivariate refinable functions, SIAM J. Matrix Anal. Applicat., 28, 999-1014 (1997) · Zbl 0872.41003 [9] Ng, M.; Chan, R.; Chan, T.; Yip, A., Cosine transform preconditioners for high resolution image reconstruction, Linear Algebra Applicat., 316, 89-104 (2000) · Zbl 0993.65056 [10] Ng, M.; Chan, R.; Tang, W., A Fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21, 851-866 (2000) · Zbl 0951.65038 [11] Shen, Z., Refinable function vectors, SIAM J. Math. Appl., 29, 235-250 (1998) · Zbl 0913.42028 [12] Skiena, S., The Algorithm Design Manual (1997), Springer-Verlag: Springer-Verlag Telos · Zbl 0885.68002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.