TY - JOUR
T1 - Cosine transform preconditioners for high resolution image reconstruction
AU - NG, Kwok Po
AU - Chan, Raymond H.
AU - Chan, Tony F.
AU - Yip, Andy M.
N1 - Funding Information:
∗ Corresponding author. Tel.: +852-28592252; fax: +852-25592225. E-mail addresses: [email protected] (M.K. Ng), [email protected] (R.H. Chan), [email protected] (T.F. Chan), [email protected] (A.M. Yip). 1 Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and HKU CRCG Grant No. 10202720. 2 Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. 3 Research supported by ONR under grant N00014-96-1-0277 and NSF under grant DMS 96-26755.
PY - 2000/9/1
Y1 - 2000/9/1
N2 - This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L2 or H1 norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.
AB - This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L2 or H1 norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.
KW - Discrete cosine transform
KW - Image reconstruction
KW - Neumann boundary condition
KW - Toeplitz matrix
UR - http://www.scopus.com/inward/record.url?scp=0034415728&partnerID=8YFLogxK
U2 - 10.1016/S0024-3795(99)00274-8
DO - 10.1016/S0024-3795(99)00274-8
M3 - Journal article
AN - SCOPUS:0034415728
SN - 0024-3795
VL - 316
SP - 89
EP - 104
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 1-3
ER -