A PDE based method for image restoration is described by the heat equation with Neumann and initial conditions, where diffusivity is chosen as a rapidly decreasing function of the gradient magnitude of the solution and the initial condition represents the observed image. A widely used method for the numerical solution of this parabolic equation is the theta-method, that leads, at each time level, to a solution of a system of nonlinear equations of the form (I- tau A(u)) u= w, where the matrix A(u) satisfies irreducibility, vanishing column sum, negativity of diagonal entries and nonnegativity of the off-diagonal entries, Lipschitz continuity and symmetry. In this work we consider the application of splitting methods to the semi-implicit theta-method and to the implicit theta-method. In the case of semi-implicit theta-method, the Arithmetic Mean method is considered for the solution of the linear system that occurs at each time level; this method has within its overall mathematical structure certain well defined substructures that can be executed simultaneously in order to increase the degree of multiprogramming. For the solution of the nonlinear system that occurs at each time level of the implicit theta-method, multiplicative and additive operator splitting methods are examined. Because of the noncommutativity of the matrices of the splitting, the order of applying these matrices in the multiplicative operator splitting methods, as the AlternatingDirection Implicit or the Fractional Step methods, can affect the final result in image denoising. Indeed, the filtered two-dimensional image will not be the same after a rotation of 90 degrees. A symmetric strategy does not suffer from this deficiency; this motivates the advantage of using additive operator splitting methods in image restoration problems.
GALLIGANI, Emanuele, RUGGIERO, V. e ZANGHIRATI, G.. "Splitting methods for nonlinear diffusion filtering" Working paper, Dipartimento di Matematica Giuseppe Vitali - Università di Modena e Reggio Emilia, 2005. https://doi.org/10.25431/11380_593962
|Titolo:||Splitting methods for nonlinear diffusion filtering|
|Autore/i:||GALLIGANI, Emanuele; V., RUGGIERO; G., ZANGHIRATI|
|Data di pubblicazione:||2005|
|Mese di pubblicazione:||Novembre|
|Digital Object Identifier (DOI):||10.25431/11380_593962|
|Serie:||Quaderni del Dipartimento, n. 73|
|Citazione:||GALLIGANI, Emanuele, RUGGIERO, V. e ZANGHIRATI, G.. "Splitting methods for nonlinear diffusion filtering" Working paper, Dipartimento di Matematica Giuseppe Vitali - Università di Modena e Reggio Emilia, 2005. https://doi.org/10.25431/11380_593962|
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