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Restoring images degraded by spatially variant blur. (English) Zbl 0919.65091
A mathematical model of the restoration of blurred camera images is an integral equation of the first kind, \(\int k(s,t)f^*(dt)= g(s)+ \eta(s)= g^*(s)\), where \(s,t\in R^1\), \(g(s)\) is the measured image, known only for certain values of \(s_1,\dots, s_p\), and \(\eta(s)\) is an additive noise. In the paper it is assumed that the kernel \(k\) is spatially invariant, \(k(s,t)= k(s- t)\), and that the integration domain is partitioned into \(p\) nonoverlapping regions, \(k(s,t)= \sum^p_{i= 1}\delta_i(s) k_i(s- t+ s_i)\), \(\delta_i(s)\) being the Heaviside 0-1 function. In matrix form it corresponds to a block Toeplitz matrix \(K= \sum^p_{i=1} D_iK_i\) where \(D\) is a diagonal matrix. Fast matrix-vector products are introduced, a preconditioner for this matrix is devised, which differs from the identity by a matrix of small rank plus a matrix of small norm. Numerical examples are from astronomy and from checkerboard.
Reviewer: R.Lepp (Tallinn)

MSC:
65R20 Numerical methods for integral equations
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65R30 Numerical methods for ill-posed problems for integral equations
65F30 Other matrix algorithms (MSC2010)
45B05 Fredholm integral equations
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