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\title[ROF Model and Split Bregman -- \insertframenumber/\inserttotalframenumber]
       {\large MA3113: Topics in Mathematical Image Processing I\\   
        ROF Model and Split Bregman Iterations}
\author[\copyright\ Suh-Yuh Yang (楊肅煜), Math. Dept., NCU, Taiwan]
       {
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       \small\textcolor{purple}{Suh-Yuh Yang (楊肅煜)}
       }
\institute[NCU, Taiwan]
       {\small 
        Department of Mathematics, National Central University \\
        Jhongli District, Taoyuan City 320317, Taiwan 
       }
%\date{\textcolor{blue}{\small February 14, 2025}} 
\date{\textcolor{blue}{\small \today}}  

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\indent\textcolor{blue}{\bf The bounded variation space $BV(\Omega)$} \\
\textcolor{orange}{\rule{110 mm}{1 mm}}
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\small 
Let $\Omega$ be an open subset of $\mathbb{R}^2$.  
The space of functions of bounded variation $BV(\Omega)$ is defined as 
the space of real-valued function $u\in L^1(\Omega)$ such that the total variation 
is finite, i.e., 
\textcolor{purple}{
\[
	BV(\Omega) = \{u\in L^1(\Omega): \|u\|_{TV(\Omega)} < \infty\}, 
\]}where  
\[ 
	 \|u\|_{TV(\Omega)}  
	 := \sup\Big\{\int_{\Omega} u (\nabla\cdot{\varphi})\,d{\mbf x}: 
	{\varphi}\in C_c^1(\Omega, \mathbb{R}^2), 
	\|{\varphi}\|_{(L^\infty(\Omega))^2} \leq 1\Big\},  
\]
$C_c^1(\Omega, \mathbb{R}^2)$ is the space of continuously 
differentiable vector functions
with compact support in $\Omega$,  
$L^1(\Omega)$ and $L^{\infty}(\Omega)$ are the usual $L^p(\Omega)$ space 
for $p=1$ and $p=\infty$, respectively. 
%equipped with the $\|\cdot\|_{L^p(\Omega)}$ norm. 
\vskip 0.5 cm

\textcolor{purple}{\it 
Then $BV(\Omega)$ is a Banach space with the norm,  
\[ 
	\|u\|_{BV(\Omega)} := \|u\|_{L^1(\Omega)} + \|u\|_{TV(\Omega)}.  
\]}  
%For a sufficiently smooth function $u$, we have $\int_{\Omega} |Du| = \int_{\Omega} |\nabla u|$. 
%Therefore, for simplicity, below we use the notation $\nabla u$ to denote $Du$. 
 
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\indent\textcolor{blue}{\bf The ROF total-variation regularization model} \\
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\small 
Let $f: \overline{\Omega}\subset\mathbb{R}^2\rightarrow\mathbb{R}$ be a given noisy image.   
Rudin, Osher, and Fatemi ({\it Physica D, 1992}) proposed the model for image denoising:    
\textcolor{purple}{
\[ 
	\min_{u\in BV(\Omega) \cap L^2(\Omega)} \Bigl(\underbrace{\|u\|_{TV(\Omega)}}_{regularizer}
	+ \frac{\lambda}{2}\underbrace{\int_{\Omega}\big(u({\mbf x}) - f({\mbf x})\big)^2\,d{\mbf x}}_{data\ fidelity}
	\Bigr), 
\]}where $\lambda > 0$ is a tuning parameter which controls the regularization strength.  
Notice that 
\begin{itemize} 
\item A smaller value of $\lambda$ will lead to a more regular solution. 

\item The space of functions with bounded variation 
         help remove spurious oscillations (noise) and preserve sharp signals (edges).  
         
\item The $TV$ term allows the solution to have discontinuities. 
\end{itemize}

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\indent\textcolor{blue}{\bf The existence, uniqueness and stability of solution} \\
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\small
{\bf Theorem:}\ 
\textcolor{purple}{\it If $u$ is smooth, then $\|u\|_{TV(\Omega)} = \int_{\Omega}|\nabla u|\,d{\mbf x}$.} 
\smallskip  

{\bf Theorem:}\ 
\textcolor{purple}{\it If $f\in L^2(\Omega)$, the minimizer exists and is unique and is stable 
	in $L^2$ with respect to perturbations in $f$. } 
\vskip 0.5 cm 

{\bf ROF model for image denoising:} Below we assume that $u$ is smooth, and we consider the model  
\textcolor{purple}{
\[ 
	\min_{u\in{\cal V}} \Bigl(\int_{\Omega}|\nabla u|\,d{\mbf x}  
	+ \frac{\lambda}{2}\int_{\Omega}\big(u({\mbf x}) - f({\mbf x})\big)^2\,d{\mbf x}\Bigr).  
\]}Let $E[\cdot]$ be the functional over the vector space ${\cal V}$, 
\[ 
	E[u] := \int_{\Omega}|\nabla u|\,d{\mbf x}  
	+ \frac{\lambda}{2}\int_{\Omega}\big(u({\mbf x}) - f({\mbf x})\big)^2\,d{\mbf x}. 
\] 	

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\indent\textcolor{blue}{\bf Numerical experiments (Cameraman)} \\
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\small 
\begin{center} 	
\resizebox{9.5 cm}{5.6 cm}{\includegraphics{ROF_cameraman.png}} 
\medskip 

\textcolor{orange}{\it $\lambda h = 1/5, 1/10, 1/15, 1/20$} 
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\indent\textcolor{blue}{\bf Numerical experiments (Lena)} \\
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\small 
\begin{center} 	
\includegraphics[scale=0.25]{ROF_Lenna.png}	
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\textcolor{orange}{\it $\lambda h = 1/5, 1/10, 1/15, 1/20$} 
\end{center} 

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\indent\textcolor{blue}{\bf References} \\
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\small 
\begin{enumerate}
\item L. I. Rudin, S. Osher, and E. Fatemi, 
	Nonlinear total variation based noise removal algorithms, 
	\textcolor{purple}{\it Physica D,} 60 (1992), pp. 259-268.  
	
\item T. Goldstein and S. Osher, 
	The split Bregman method for $L^1$ regularized problems,  
         \textcolor{purple}{\it SIAM Journal on Imaging Sciences,} 2 (2009), pp. 323-343. 

\item P. Getreuer, 
	Rudin-Osher-Fatemi total variation denoising using split Bregman,  
       \textcolor{purple}{\it Image Processing On Line,} 2 (2012), pp. 74-95. 
        
\end{enumerate}

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