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\noindent285J, L. Vese  

\noindent{\bf Assignment 3:} Due on Monday, November 2 

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\noindent{\bf [1]} Implement a numerical scheme for the Euler-Lagrange equation of the ROF model in the presence of known blur: 
$$\inf_{u}\int_{\Omega}\sqrt{\epsilon^2+|\nabla u|^2}dxdy+\frac{\lambda}{2}\int_{\Omega}|k*u-f|^2dxdy,$$
where you can choose a Gaussian blur or a uniform blur for $k$ (of mean 1), with small support (between $3\times 3$ and $9\times 9$). If you compute the convolution in the spatial domain, then you can use the commands ``conv2'' or ``imfilter'' in matlab. You can also evaluate the convolutions in the frequency domain. Define a blurry data $f$ (with a small amount of noise). Output the root-mean-square-error between $u$ and $u_{orig}$, and plot the numerical energy versus iterations. Choose a value $\lambda$ that gives better results. 

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\noindent{\bf [2]} Assume that $\phi(z)$ is even and differentiable, increasing on $[0,\infty)$ and positive. We know that the time-dependent Euler-Lagrange equation obtained by minimizing
$$\inf_{u\in W^{1,1}(\Omega)}F(u)=\int_{\Omega}\phi(|\nabla u|)dxdy+\frac{\lambda}{2}\int_{\Omega}|f-u|^2dxdy$$ is formally given by 

$$(1) \ \ \ \ \ \ \ \ \ \ \frac{\partial u}{\partial t}=-\lambda(u-f)+\mbox{div}\Big(\frac{\phi'(|\nabla u|)}{|\nabla u|}\nabla u\Big).$$ 

\noindent(i) Express the above differential operator in (1) as $()u_{\vec N\vec N}+()u_{\vec T\vec T}$, where $\vec N=\frac{\nabla u}{|\nabla u|}$, and $\vec T$ is normalized and orthogonal to $\vec N$. 

\noindent(ii) Assume $\lambda=0$ in (1). Under what conditions is the obtained PDE (weakly) parabolic ? (in other words, under what conditions on $\phi$, does the quasi-linear 2nd order operator $\mbox{div}\Big(\frac{\phi'(|\nabla u|)}{|\nabla u|}\nabla u\Big)=\sum_{i,j=1}^2a_{ij}u_{x_i,x_j}$ satisfy the weakly elliptic property $\sum_{i,j=1}^2a_{ij}\xi_i\xi_j\geq0$ for all $\xi_1,\xi_2\in R$ ?)

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\noindent{\bf [3]} Let $u$ be sufficiently smooth and satisfy 
$$\frac{\partial u}{\partial t}=|\nabla u|G(\mbox{curv}u),$$ 
where $\mbox{curv}u=\mbox{div}\Big(\frac{\nabla u}{|\nabla u|}\Big)$ is the curvature operator, 
and $G$ is a function such that $kG(k)\geq0$. Show that this flow decreases
the total variation of $u$ in time. 

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\noindent{\bf [4]} Let $g\in C^1(R)$ be a function, with $g'>0$. Let $v=g(u)$. If $u$ 
satisfies 
$$\frac{\partial u}{\partial t}=|\nabla u|G(\mbox{curv}(u)),$$
so does $v$ (this is called contrast invariance or geometric invariance).


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