By Fuensanta Andreu-Vaillo

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2003.

This ebook incorporates a precise mathematical research of the variational method of photograph recovery in response to the minimization of the whole edition submitted to the restrictions given by means of the picture acquisition version. This version, at the start brought by means of Rudin, Osher, and Fatemi, had a robust impression within the improvement of variational tools for photo denoising and recovery, and pioneered using the BV version in picture processing. After a whole research of the version, the minimizing overall edition stream is studied lower than assorted boundary stipulations, and its major qualitative houses are exhibited. specifically, numerous specific strategies of the denoising challenge are computed.

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5. Review of Numerical Methods where u E W I,2(S1) and v E L2(S1), E :::; v :::; ~. Starting from any u l and instance vI = 1), we construct iteratively u n+I v vI (for . E( u,v n) , = argmmuEW1,2(0) n+I -_ argmmc

Let (S(t))t~O be the semigroup generated by A. Then, we have con- servation of mass, that is, L S(t)uo dx = L Uo dx, for all t 2 o. Proof. Given Uo E Ll(O), let u(t) = S(t)uo. Then, (u(t), -u'(t)) E A. 18), we obtain that Jrl u'(t) = o. Consequently, the function t f----t Jrl u(t) is constant, and the proof concludes. 20. Let (S(t))t>o be the semigroup generated by A. Then IIS(t)uo - (uo)rlill ----+ where (uo)rl = Moreover, ifuo E £N1(0) as t 0 ----+ 00, In uo(x) dx. LOO(O) there exists a constant C, independent ofuo, such that for all t Proof.

1). e. on [0, T]. 17). 1) in (0, T) x O. 5. e. on [0, T]. e. on [0, T], Ut(t) = div(z(t)). e. e. e. e. on [0, T]. 17). 17). 4) follow from the complete accretivity of A. 5 An LN_Loo Regularizing Effect Let us first remark that there is no L1_LCXJ or L1_L2 regularizing effect. 1) in (0,1) x B1 (0) with initial datum lfhr vo(x) = IIxll~/2' Observe that v(t) E L 1(B 1(0)) \L 2(B1(0)), O:S t < 1. 1) which is in L1(Q) \ L2(Q). Indeed, given A ~ 1, let u(t, x) the solution of the Neumann problem in (0,00) x B1(0) with u(O,x) = AN' Since VA(t,X) = A is a solution IIxll 2 of the Neumann problem in (0,00) x B 1 (0), by comparison, we have A = VA(t,X):S u(t,x) in (0,00) x B 1(0).