TY - JOUR
T1 - Wavelet deblurring algorithms for spatially varying blur from high-resolution image reconstruction
AU - Chan, Raymond H.
AU - Chan, Tony F.
AU - Shen, Lixin
AU - Shen, Zuowei
PY - 2003/6/1
Y1 - 2003/6/1
N2 - High-resolution image reconstruction refers to reconstructing a higher resolution image from multiple low-resolution samples of a true image. In Chan et al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000), we 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 L2(ℝ2). Using these filters and their duals, we derive an iterative algorithm to solve the problem based on the algorithmic framework of Chanet al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000). Our algorithm requires a nontrivial modification to the algorithms in Chan et al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000), 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.
AB - High-resolution image reconstruction refers to reconstructing a higher resolution image from multiple low-resolution samples of a true image. In Chan et al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000), we 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 L2(ℝ2). Using these filters and their duals, we derive an iterative algorithm to solve the problem based on the algorithmic framework of Chanet al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000). Our algorithm requires a nontrivial modification to the algorithms in Chan et al. (Wavelet algorithms for high-resolution image reconstruction, Research Report #CUHK-2000-20, Department of Mathematics, The Chinese University of Hong Kong, 2000), 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.
KW - High-resolution image reconstruction
KW - Tikhonov least squares method
KW - Wavelet
UR - http://www.scopus.com/inward/record.url?scp=0037409762&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-0037409762&origin=recordpage
U2 - 10.1016/S0024-3795(02)00497-4
DO - 10.1016/S0024-3795(02)00497-4
M3 - RGC 21 - Publication in refereed journal
VL - 366
SP - 139
EP - 155
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -