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**Application of ADI iterative methods to the restoration of noisy images.**
*(English)*
Zbl 0849.65101

The restoration of two-dimensional images in the presence of noise is studied. Here an application of the alternating direction implicit (ADI) iteration method and a generalization thereof to the computation of the minimum mean square error estimate of a two-dimensional image in the presence of Gaussian white noise are described.

It is shown that when the noise is white and Gaussian, and under suitable assumptions on the image, the linear system of equations arising after application of the minimum mean square method can be written as a Sylvester’s equation for the matrix representing the restored image. The authors show that the alternating direction implicit iteration method is well suited for the solution of Sylvester’s equations. This is illustrated with computed examples for the case when the image is described by a separated first-order Markov process. The authors consider generalizations of the alternating direction implicit iteration method for the generation of iteration parameters. The competitiveness of the new numerical schemes is illustrated.

It is shown that when the noise is white and Gaussian, and under suitable assumptions on the image, the linear system of equations arising after application of the minimum mean square method can be written as a Sylvester’s equation for the matrix representing the restored image. The authors show that the alternating direction implicit iteration method is well suited for the solution of Sylvester’s equations. This is illustrated with computed examples for the case when the image is described by a separated first-order Markov process. The authors consider generalizations of the alternating direction implicit iteration method for the generation of iteration parameters. The competitiveness of the new numerical schemes is illustrated.

Reviewer: I.Dimov (Sofia)

### MSC:

65C99 | Probabilistic methods, stochastic differential equations |

65F10 | Iterative numerical methods for linear systems |

68U10 | Computing methodologies for image processing |

15A24 | Matrix equations and identities |

60J05 | Discrete-time Markov processes on general state spaces |

60G35 | Signal detection and filtering (aspects of stochastic processes) |