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

Calvetti, D.; Reichel, L.

SIAM J. Matrix Anal. Appl. 17, No. 1, 165-186 (1996).

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.

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)

Keywords:

restoration of noisy images; noise reduction; Wiener filter; alternating direction implicit iteration method; minimum mean square method; Sylvester’s equation; first-order Markov process