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{\bf Math 285J}

{\bf Assignment 4:} Due on Wednesday, November 25 

\bigbreak 

\noindent{\bf [1]} {\it Affine invariance:} Let $A=\left(
\begin{array}{ll}
a & b\\
c & d
\end{array}
\right)
$
be an arbitrary matrix, such that $ad-bc>0$, and let $X=(x,y)$. Check that, if 
$u$ satisfies 
$$\frac{\partial u}{\partial t}=|\nabla u|\mbox{curv}(u)^{1/3},$$
then $v(X)=u(AX)$ satisfies 
$$\frac{\partial v}{\partial t}=c(A)|\nabla v|\mbox{curv}(v)^{1/3}.$$
What is $c(A)$ ?

\bigbreak 

\noindent{\bf [2]} {\it Computational exercise:} Segmentation via the Ambrosio-Tortorelli elliptic approximations to the Mumford and Shah functional

\medbreak

Consider given image data $f\in L^{\infty}(\Omega)\subset L^2(\Omega)$. We wish to solve in practice the following minimization problem  (where $\varepsilon\rightarrow 0^+$ is a small parameter) 
\begin{equation}
\inf_{u,v\in H^1(\Omega)}G^{AT}_{\varepsilon}(u,v)=\int_{\Omega}\Big[\varepsilon|\nabla v|^2+\alpha 
(v^2+o_{\varepsilon})|\nabla u|^2+\frac{(v-1)^2}{4\varepsilon}+\beta|u-f|^2\Big]dxdy,
\end{equation}
where $o_{\varepsilon}$ is any non negative infinitesimal faster than 
$\varepsilon$, and $\alpha,\beta$ are positive parameters. The unknown $u=u_{\varepsilon}$ will be an optimal piecewise-smooth approximation of the data $f$, while the unknown $v=v_{\varepsilon}$ will be an edge function: $0\leq v \leq 1$, $v\approx 0$ near edges, and $v\approx 1$ inside homogenous regions. 

(i) Give the Euler-Lagrange equations in $u$ and $v$, associated with the minimization (using alternating minimization), together with the corresponding boundary conditions. 

(ii) Discretize the obtained system and implement it for an image $f$. Visualize the energy decrease versus iterations, and the final $u$ and $v$ at steady state. 



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