  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.
