## Abstract

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 L^{2}(R^{2}). 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.

Original language | English (US) |
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Pages (from-to) | 139-155 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 366 |

Issue number | SPEC. ISS. |

DOIs | |

State | Published - Jun 1 2003 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗Corresponding author. E-mail address: rechan@math.cuhk.edu.hk (R.H. Chan). 1 Research supported in part by HKRGC Grant CUHK4212/99P and CUHK Grant DAG 2060183. 2Research supported in part by the ONR under contract N00014-96-1-0277 and by the NSF under grant DMS-9973341. 3 Research supported in part by the Center of Wavelets, Approximation and Information Processing (CWAIP) funded by the National Science and Technology Board, the Ministry of Education under Grant RP960 601/A.

## Keywords

- High-resolution image reconstruction
- Tikhonov least squares method
- Wavelet

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics