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A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation. (English) Zbl 1278.68330
Summary: Variational image restoration models for both additive and multiplicative noise (MN) removal are rarely encountered in the literature. This paper proposes a new variational model and a fast algorithm for its numerical approximation to remove independent additive and MN from digital images. Two previous works by L. I. Rudin et al. [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] and Z. Jin and X. Yang [J. Math. Anal. Appl. 362, No. 2, 415–426 (2010; Zbl 1191.68788)] are used to develop the new model. As a result, developing a fast numerical algorithm is difficult because the associated Euler-Lagrange equation is highly nonlinear and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient nonlinear multigrid approach via a robust fixed-point smoother. Numerical tests using both synthetic and realistic images not only confirm that our new model delivers quality results but also that the proposed numerical algorithm allows a very fast numerical realization of the model.

MSC:
68U10 Computing methodologies for image processing
65F22 Ill-posedness and regularization problems in numerical linear algebra
65K10 Numerical optimization and variational techniques
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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