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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
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\input{tcilatex}

\begin{document}


\begin{center}
Diffusion Equation
\end{center}

\bigskip

\begin{align*}
\frac{\partial u}{\partial t} &= k\frac{\partial ^{2}u}{\partial x^{2}}%
\qquad -\infty <x<\infty \\
u(x,0) &= f(x)
\end{align*}

\bigskip

\begin{center}
Properties of Diffusion equation
\end{center}

\begin{itemize}
\item Maximum Principle \noindent For \quad $0\leq x\leq l\quad 0\leq t\leq
T $ \newline
\noindent the maximum of $u(x,t)$ occurs either at \newline
\noindent $t=0$ or $x=0$ or $x=l$

\noindent (proof at maximum $\frac{\partial u}{\partial x}=0$ and $\frac{%
\partial ^{2}u}{\partial x^{2}}\leq 0$ )

\item The solution exists for all $f$ and is unique

\item If $u(x,0)=f(x)$ and $v(x,0)=g(x)$

\noindent Then

\noindent $\dint (u(x,t)-v(x,t))^{2}dx\leq \dint (f(x)-g(x))^{2}dx$

\item Fourier integral in x

\begin{equation*}
\widehat{u}(\xi ,t)=\dint\limits_{-\infty }^{\infty }e^{-ix\xi }u(x,t)dx
\end{equation*}

\noindent Then%
\begin{equation*}
\widehat{u}_{t}=-k\xi ^{2}\widehat{u}
\end{equation*}

\noindent So%
\begin{equation*}
\widehat{u}(\xi ,t)=\widehat{u}(\xi ,0)e^{-k\xi ^{2}t}
\end{equation*}

\noindent So

\begin{description}
\item[(a)] $k>0$ \qquad gives exponential decay

\item[(b)] $k<0$ \qquad gives exponential growth \qquad not well posed !!
\end{description}

\item explicit solution

For $-\infty <x<\infty $%
\begin{equation*}
u(x,t)=\frac{1}{\sqrt{4\pi kt}}\dint\limits_{-\infty }^{\infty }e^{-\frac{%
(x-y)^{2}}{4kt}}f(y)dy
\end{equation*}

\item So

\begin{description}
\item[(a)] $k>0$ \qquad diffusion operator is smoothing

\item[b] $k<0$ \qquad (-diffusion operator) is sharpening
\end{description}
\end{itemize}

\bigskip

\begin{definition}
\begin{equation*}
K_{\sigma }=\frac{1}{2\pi \sigma }e^{-\frac{x^{2}}{2\sigma ^{2}}}
\end{equation*}
\end{definition}

Then the solution to the diffusion equation is%
\begin{equation*}
u(x,t)=%
\begin{cases}
f(x) & t=0 \\ 
(K_{\sqrt{2t}}\ast f)(x) & t>0%
\end{cases}%
\end{equation*}

and%
\begin{equation*}
|u(x,t)|\leq Me^{ax^{2}}
\end{equation*}

i.e. 
\begin{eqnarray*}
\sigma &=&\sqrt{2t} \\
T &=&\frac{1}{2}\sigma ^{2}
\end{eqnarray*}

Note:%
\begin{equation*}
\mathcal{F}(K_{\sigma })=e^{-\frac{\omega ^{2}}{\frac{2}{\sigma ^{2}}}}
\end{equation*}

\bigskip \newpage

\begin{center}
Scale Space
\end{center}

\bigskip

\noindent Image representation at a continuum of scales.

\noindent Consider a sequence $T_{t}f$ with%
\begin{align*}
T_{0}f &= f \\
T_{t+s} &= T_{t}(T_{s}f)\qquad \text{semi-group}
\end{align*}

We want the following properties

\begin{itemize}
\item no new level curves

\item non-enhancement of local extrema

\item decreasing number of extrema

\item maximum principle
\end{itemize}

\begin{definition}
Gaussian Scale Space: Convolution of the picture with a Gaussian with
increasing $\sigma^2$.
\end{definition}

\noindent Note that this is equivalent to the diffusion equation with $t
\rightarrow 0$.

\begin{theorem}
\noindent Let 
\begin{equation*}
\Phi (f,x,\sigma )=\dint\limits_{-\infty }^{\infty } \phi
(f(x^{\prime}),x,x^{\prime },\sigma )dx^{\prime}
\end{equation*}%
\noindent Assume

\begin{itemize}
\item $\Phi (Af,x,\sigma )=A\Phi (f,x,\sigma )$

\item $\Phi (f(x^{^{\prime }}-a),x,\sigma )=\Phi (f,x-a,\sigma )$

\item $\Phi (f(\frac{{x^\prime }}{\lambda },x,\sigma )= \Phi (f,\frac{x}{%
\lambda },\sigma )$

\item $\Phi \left[ \Phi (f(x^{\prime \prime}),x^{\prime},
\sigma_{1}),x,\sigma _{2}\right] = \Phi (f(x^{\prime \prime}),x,\sigma _{3})$

\item $\Phi (f,x,\sigma )>0\qquad $when $f>0$ \ \ and $\sigma >0$

\noindent Then%
\begin{equation*}
\Phi (f,x,\sigma )=\dint\limits_{-\infty }^{\infty }f(x^{\prime })e^{-\frac{%
(x-x^{\prime })}{4\sigma ^{2}}}dx^{\prime }=K_{\frac{\sigma }{\sqrt{2}}}\ast
f
\end{equation*}
\end{itemize}
\end{theorem}

\newpage

\begin{center}
Perona-Malik
\end{center}

\bigskip

\noindent Gaussian (linear) diffusion smooths noise but also smooths edges

\noindent We want a process that reduces diffusion at the edges (or even
sharpens them)

\noindent So we have piecewise smoothing between the edges.

Consider%
\begin{equation*}
\frac{\partial u}{\partial t}=div\left[ g(|\nabla u|^{2})\,\;grad(u)\right]
\end{equation*}

\noindent with%
\begin{equation*}
g(s^{2})=\frac{1}{1+\frac{s^{2}}{\lambda ^{2}}}\qquad \lambda >0\text{ \ a
parameter}
\end{equation*}

\noindent In one dimension%
\begin{align*}
\frac{\partial u}{\partial t} &= \frac{\partial }{\partial x}g\left( \left( 
\frac{\partial u}{\partial x}\right) ^{2}\right) \;\frac{\partial u}{%
\partial x} \\
&= g\left( \left( \frac{\partial u}{\partial x}\right) ^{2}\right) \frac{%
\partial ^{2}u}{\partial x^{2}}+g^{\prime }2\frac{\partial u}{\partial x}%
\frac{\partial ^{2}u}{\partial x^{2}}\frac{\partial u}{\partial x} \\
&= \left[ g+2\left( \frac{\partial u}{\partial x}\right) ^{2}g^{\prime }%
\right] \frac{\partial ^{2}u}{\partial x^{2}}
\end{align*}

\noindent Define%
\begin{equation*}
\Phi (s)=sg(s^{2})
\end{equation*}

\noindent then%
\begin{equation*}
\frac{\partial u}{\partial t}=\Phi ^{\prime }(|\nabla u|^{2})\Delta u
\end{equation*}

\noindent For Perona-Malik%
\begin{align*}
\Phi (s)& =sg(s^{2})=\frac{s}{1+\frac{s^{2}}{\lambda ^{2}}} \\
\Phi ^{\prime }(s)& =\frac{1-\frac{s^{2}}{\lambda ^{2}}}{\left( 1+\frac{s^{2}%
}{\lambda ^{2}}\right) ^{2}} \\
& =%
\begin{cases}
\text{positive} & s<\lambda \\ 
\text{negative} & s>\lambda%
\end{cases}%
\end{align*}

\bigskip

\noindent To see maximum principle we consider simpler equation%
\begin{align*}
\frac{1}{2}\frac{\partial }{\partial t}\left( \frac{\partial u}{\partial x}%
\right) ^{2} &= \frac{\partial u}{\partial x}\frac{\partial }{\partial x}%
\left( \frac{\partial u}{\partial t}\right) \\
&= \Phi ^{\prime \prime }\frac{\partial u}{\partial x}\frac{\partial ^{2}u}{%
\partial x^{2}}+\Phi ^{\prime }\frac{\partial u}{\partial x}\frac{\partial
^{3}u}{\partial x^{3}}
\end{align*}

\noindent Now assume that $\left( \frac{\partial u}{\partial x}\right) ^{2}$
has a maximum at $(x_{0},t).$Then%
\begin{align*}
0 &= \frac{1}{2}\frac{\partial }{\partial x}\left( \frac{\partial u}{%
\partial x}\right) ^{2}=\frac{\partial u}{\partial x}\frac{\partial ^{2}u}{%
\partial x^{2}}\qquad \text{and\qquad }\frac{\partial u}{\partial x}\frac{%
\partial ^{3}u}{\partial x^{3}}<0 \\
\text{Also }\frac{\partial }{\partial x}\left( \frac{\partial u}{\partial x}%
\frac{\partial ^{2}u}{\partial x^{2}}\right) &= \frac{\partial u}{\partial x}%
\frac{\partial ^{3}u}{\partial x^{3}}+\left( \frac{\partial ^{2}u}{\partial
x^{2}}\right) ^{2}
\end{align*}

\noindent So%
\begin{equation*}
\frac{1}{2}\frac{\partial }{\partial t}\left( \frac{\partial u}{\partial x}%
\right) ^{2}=\Phi ^{\prime }\frac{\partial u}{\partial x}\frac{\partial ^{3}u%
} {\partial x^{3}}=\Phi ^{\prime }\cdot \text{negative}
\end{equation*}

\noindent So

\begin{equation*}
\frac{1}{2}\frac{\partial }{\partial t}\left( \frac{\partial u}{\partial x}%
\right) ^2 {\LARGE \vert_{x=x_0}=%
\begin{cases}
\text{negative} & s<\lambda \\ 
\text{positive} & s>\lambda%
\end{cases}%
}
\end{equation*}

\newpage

\begin{center}
Two Dimensions
\end{center}

\bigskip 
\begin{equation*}
\frac{\partial u}{\partial t}=\Phi ^{\prime }(|\nabla u|^{2})\frac{\partial
^{2}u}{\partial \eta ^{2}}+g(|\nabla u|^{2})\frac{\partial ^{2}u}{\partial
\xi ^{2}}
\end{equation*}

\noindent where $\xi $ direction is perpendicular to $\nabla u$ (direction
of greatest change in $u$)

\noindent and $\eta $ is parallel to $\nabla u$

\bigskip

\noindent So

\noindent $\xi $ direction is lines of constant gray level

\noindent $\eta $ direction is lines of maximal change in gray levels

\bigskip

\noindent So we have forward diffusion in the $\xi $ direction and

\noindent backward diffusion (sharpening) in the $\eta $ direction

\begin{center}
Well-posedness
\end{center}

\bigskip

\begin{itemize}
\item Existence of a weak solution in $c^{1}$ for finite time. \newline
\qquad Maximum principle

\item Unstable with respect to perturbations

\item staircasing - constant states divided by edges \newline
\qquad finer discretization gives more edges
\end{itemize}

\newpage

\begin{center}
Total Variation (TV) schemes
\end{center}

\bigskip

\begin{definition}
\begin{equation*}
\text{TV}(u)=\int_{\Omega }|\nabla u|dx=\int_{\Omega }\sqrt{\left( \frac{%
\partial u}{\partial x}\right) ^{2}+\left( \frac{\partial u}{\partial y}%
\right) ^{2}}dxdy
\end{equation*}
\end{definition}

\bigskip

\begin{center}
Osher-Rudin
\end{center}

\bigskip 
\begin{align*}
\frac{\partial u}{\partial t}& =-|\nabla u|F(\mathcal{L}(u)) \\
y(x,o)& =f(x)\qquad \text{present image}
\end{align*}%
$\mathcal{L}(u)$ is second order elliptic PDE whose zerocrossings correspond
to edges

\noindent For example $\mathcal{L}(u)=\Delta u$

\noindent or else $\mathcal{L}(u)=\frac{\partial ^{2}u}{\partial \eta ^{2}}$
with $\eta $ normal to the edge i.e. $\eta \parallel \nabla u\parallel $

\bigskip

\begin{center}
Minimization
\end{center}

\bigskip

\noindent Consider%
\begin{align*}
&\min_{u}\int |\nabla u| &  \\
\text{subject to \ }\int (u-f)^{2} &=\sigma ^{2}=\text{given variance} &  \\
\text{and }\qquad \qquad \qquad \int udx &= & \int fdx
\end{align*}

\noindent This implies%
\begin{equation*}
div\left( \frac{\nabla u}{|\nabla u|}\right) -\mu -\lambda (u-f)=0
\end{equation*}

\noindent $\lambda ,\mu $ are Lagrange multipliers

\bigskip

\noindent To improve the well-posedness of the solutions we regularize the
equation. So now we consider%
\begin{align*}
\frac{\partial u}{\partial t} &= div\left( g(|\nabla u_{\sigma }|)\nabla
u\right) \\
u_{\sigma } &= K_{\sigma }\ast u
\end{align*}

\noindent $\lambda $ and $\sigma $ are constants that can be played with.

\noindent The filter is insensitive to noise at scales $<\lambda $

\noindent Strong oscillations due to noise are amplified by the original
Perona-Malik equation

\bigskip

\noindent Alternatively one can use a time-delay regularization%
\begin{align*}
\frac{\partial u}{\partial t} &= \partial _{x}(g(v)\frac{\partial u}{%
\partial x}) \\
\frac{\partial v}{\partial t} &= \frac{1}{\tau } \left( \left\vert\frac{%
\partial u}{\partial x}\right\vert-v\right)
\end{align*}

\end{document}
