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

• Many tasks in image processing can be tackled by modeling an appropriate data fidelity term $$\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$$ and then solve one of the regularized minimization problems \begin{align*} &{}(P_{1,\tau}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ \Phi(x) \;{\rm s.t.}\; \Psi(x) \leq \tau \big\} \\ &{}(P_{2,\lambda}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ \Phi(x) + \lambda \Psi(x) \}, \; \lambda > 0 \end{align*} with some function $$\Psi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$$ and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets $${\rm SOL}(P_{1,\tau})$$ and $${\rm SOL}(P_{2,\lambda})$$ of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} 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 \begin{align*} \tau \mapsto {}& {\rm SOL}(P_{1,\tau}), \; \tau \in (0,c) \\ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), \; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change $$g: (0,c) \rightarrow (0,d)$$. Moreover we show that the solver sets are changing all the time while $$\tau$$ runs from $$0$$ to $$c$$ and $$\lambda$$ 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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Verfasserangaben: René Ciak urn:nbn:de:hbz:386-kluedo-41000 Gabriele Steidl Dissertation Englisch 09.06.2015 2015 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 09.10.2014 09.06.2015 X, 155 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 00-XX GENERAL / 00Axx General and miscellaneous specific topics / 00A05 General mathematics Standard gemäß KLUEDO-Leitlinien vom 13.02.2015