Coercive functions from a topological viewpoint and properties of minimizing sets of convex functions appearing in image restoration

06/23/2015 ∙ by René Ciak, et al. ∙ 0

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term Φ: R^n →R∪{+∞} and then solve one of the regularized minimization problems &(P_1,τ) argmin_x ∈ R^n{Φ(x) s.t. Ψ(x) ≤τ} &(P_2,λ) argmin_x ∈ R^n{Φ(x) + λΨ(x) }, λ > 0 with some function Ψ: R^n →R∪{+∞} and a good choice of the parameter(s). Two tasks arise naturally here: & 1. Study the solver sets SOL(P_1,τ) and SOL(P_2,λ) of the minimization problems. & 2. Ensure that the minimization problems have solutions. This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals (0,c) and (0,d) such that the setvalued curves τ& SOL(P_1,τ), τ∈ (0,c) λ& SOL(P_2,λ), λ∈ (0,d) are the same, besides an order reversing parameter change g: (0,c) → (0,d). Moreover we show that the solver sets are changing all the time while τ runs from 0 to c and λ runs from d to 0. In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity. Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.



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