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

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

Homogeneous Penalizers and Constraints in Convex Image Restoration

Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form ${\rm argmin} \{ \Phi(x)$ subject to $\Psi(x) \le \tau \}$ and their penalized counterparts ${\rm argmin} \{\Phi(x) + \lambda \Psi(x)\}$. We recall general results on the topic by the help of an epigraphical projection. Then we deal with the special setting $\Psi := \| L \cdot\|$ with $L \in \mathbb{R}^{m,n}$ and $\Phi := \varphi(H \cdot)$, where $H \in \mathbb{R}^{n,n}$ and $\varphi: \mathbb R^n \rightarrow \mathbb{R} \cup \{+\infty\}$ meet certain requirements which are often fulfilled in image processing models. In this case we prove by incorporating the dual problems that there exists a bijective function such that the solutions of the constrained problem coincide with those of the penalized problem if and only if $\tau$ and $\lambda$ are in the graph of this function. We illustrate the relation between $\tau$ and $\lambda$ for various problems arising in image processing. In particular, we point out the relation to the Pareto frontier for joint sparsity problems. We demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise with the $I$-divergence as data fitting term $\varphi$ and in inpainting models with the constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level

Homogeneous Penalizers and Constraints in Convex Image Restoration

Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form $min\{\Phi(x)$ subject to $\Psi(x)\le\tau\}$ and their non-constrained, penalized counterparts $min\{\Phi(x)+\lambda\Psi(x)\}$. We start with general considerations of the topic and provide a novel proof which ensures that a solution of the constrained problem with given $\tau$ is also a solution of the on-constrained problem for a certain $\lambda$. Then we deal with the special setting that $\Psi$ is a semi-norm and $\Phi=\phi(Hx)$, where $H$ is a linear, not necessarily invertible operator and $\phi$ is essentially smooth and strictly convex. In this case we can prove via the dual problems that there exists a bijective function which maps $\tau$ from a certain interval to $\lambda$ such that the solutions of the constrained problem coincide with those of the non-constrained problem if and only if $\tau$ and $\lambda$ are in the graph of this function. We illustrate the relation between $\tau$ and $\lambda$ by various problems arising in image processing. In particular, we demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise and in inpainting models with constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level

Homogeneous Penalizers and Constraints in Convex Image Restoration

Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form $min\{\Phi(x)$ subject to $\Psi(x)\le\tau\}$ and their non-constrained, penalized counterparts $min\{\Phi(x)+\lambda\Psi(x)\}$. We start with general considerations of the topic and provide a novel proof which ensures that a solution of the constrained problem with given $\tau$ is also a solution of the on-constrained problem for a certain $\lambda$. Then we deal with the special setting that $\Psi$ is a semi-norm and $\Phi=\phi(Hx)$, where $H$ is a linear, not necessarily invertible operator and $\phi$ is essentially smooth and strictly convex. In this case we can prove via the dual problems that there exists a bijective function which maps $\tau$ from a certain interval to $\lambda$ such that the solutions of the constrained problem coincide with those of the non-constrained problem if and only if $\tau$ and $\lambda$ are in the graph of this function. We illustrate the relation between $\tau$ and $\lambda$ by various problems arising in image processing. In particular, we demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise and in inpainting models with constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level

Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors

We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain et al. (Another look at Sobolev spaces. In: Menaldi, Rofman, Sulem (eds) Optimal control and partial differential equations-innovations and applications: in honor of professor Alain Bensoussan’s 60th anniversary, IOS Press, Amsterdam, pp 439–455, 2001). For the proposed regularization functionals, we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.© The Author(s) 201