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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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