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.
Categories: math.OC cs.CV math.CA math.FA