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A fast algorithm for deblurring models with Neumann boundary conditions. (English) Zbl 0951.65038

The authors deal with the blur removal which is a fundamental issue in signal and image processing. They consider the use of the Neumann boundary conditions which are corresponding to a reflection of the original scene at the boundary. In this case the resulting matrices are (block) Toeplitz-plus-Hankel matrices. It is shown that for symmetric blurring functions these matrices can always be diagonalized by a discrete cosine transform. Thus the cost of inversion is significantly lower than if zero or periodic boundary conditions are used, and the generalized cross-validation estimate of the regularization parameter can be done in a straightforward way. In the case of nonsymmetric blurring functions the authors show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. The results from the numerical tests reported in the paper illustrate the efficiency of using the Neumann boundary conditions which provide an effective model for image restoration problems in terms of both the computational cost and minimizing the ringing effects near the boundary.
The paper is interesting for numerical mathematicians dealing with efficient algorithms in signal and image processing.
Reviewer: K.Georgiev (Sofia)

MSC:

65F22 Ill-posedness and regularization problems in numerical linear algebra
68U10 Computing methodologies for image processing
65T50 Numerical methods for discrete and fast Fourier transforms
65F10 Iterative numerical methods for linear systems
65Y20 Complexity and performance of numerical algorithms
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