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Many tasks in image processing can be tackled by modeling an appropriate datafidelity term $\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$ andthen 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 tasksarise naturally here: \begin{align*} {}& \text{1. Study the solver sets ${\rmSOL}(P_{1,\tau})$ and  ${\rm SOL}(P_{2,\lambda})$ of the minimization problems.} \\ {}& \text{2.Ensure that the minimization problems have solutions.} \end{align*} This thesisprovides contributions to both tasks: Regarding the first task for a morespecial setting we prove that there are intervals $(0,c)$ and $(0,d)$ such thatthe 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 thesame, 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 haveadditionally coercivity. We regard lower semicontinuity and coercivity from atopological point of view and develop a new technique for proving lowersemicontinuity plus coercivity.  Dropping any lower semicontinuity assumption we also prove a theorem on thecoercivity of a sum of functions.

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