\documentclass[12pt]{article} 

\usepackage{makeidx}
\usepackage[]{graphicx}


\usepackage[]{amsmath,epsfig,latexsym, amsfonts, amssymb, amstext, enumerate,amsthm}
\newcommand{\R}{{\mathbb R}}

\begin{document}

{\bf HW \#3: due on Monday, November 16} 

\bigbreak 

{\bf [1]} Given $f\in L^2(\Omega)$, consider the minimization problem 
$$\inf_{u\in H^1(\Omega)}F(u)=\int_{\Omega}\lambda|f-Ku|^2dxdy +\sqrt{\int_{\Omega}|\nabla u|^2dxdy},$$ 
where $\lambda >0$ and $K$ is linear and continuous operator from $L^2(\Omega)$ to $L^2(\Omega)$, with adjoint $K^*$, such that $K1=1$. 

Formulate and show a similar characterization of minimizers (as done in class for the BV ROF model). Define the dual star norm $\|\cdot\|_{*}$ necessary in this formulation and mention to what known norm this corresponds. 

\bigbreak 

{\bf [2]} Let $u:\R^2\rightarrow \R$. The upper level set of an image function $u$ at level $\lambda\in \R$ is the set 
$\chi_{\lambda}(u)=\{x\in \R^2:\ u(x)\geq \lambda\}$. 
Show that $u$ can be retrieved by the reconstruction formula 
$$u(x)=sup\{\lambda:\ x\in \chi_{\lambda}(u)\}.$$

\bigbreak 

{\bf [3]} Let $u,v:\R^2\rightarrow \R$. Assume that two image functions $u$ and $v$ have the same 
level sets, that is for all $\lambda\in \R$, there is $\mu\in \R$ such that 
$\chi_{\lambda}(u)=\chi_{\mu}(v)$. Let us define $g$ by $g(\lambda)=\sup\Big\{
\mu:\ \chi_{\lambda}(u)=\chi_{\mu}(v)\Big\}$. Then $g$ is nondecreasing and 
$v=g\circ u$. (show first that $g$ is nondecreasing, then show that $v\geq g\circ u$ and 
that $g\circ u\geq v$). 


\bigbreak

{\bf [4]} Implement the projection algorithm by A. Chambolle, introduced in the paper posted on the class web-page, for ROF total variation minimization (implement equation (9) from the paper, and obtain the denoised output image $u$ using (7)). Compare with your implementation from the previous assignment. 

\end{document}
