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{\it Energy minimization problem in image processing}

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{\bf Yunho Kim}\\
 UCLA\\
{\tt yuno1123@math.ucla.edu}
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\noindent {\bf Abstract}:  We recover an unknown image from a noisy blurred
image $f$. We solve this problem by minimizing a regularized functional and
find solutions of the form $u+v$, where $u$ is cartoon modeled in a space of
bounded variation and $v$ is texture in a homogeneous Sobolev space of
negative differentiability.  With this model, unlike many other PDE models, we
can not only recover a clean unknown image but also decompose the image into
cartoon and texture parts capturing more details in natural images. We will
also prove that there is a minimizer of this problem satisfying highly
nonlinear Euler-Lagrange equations of the energy functional that we minimize
and investigate the characteristics of the minimizers. We implement this model
using gradient descent and finite difference scheme to see how good the
recovered images will be.

This is a joint work with my adviser Luminita A. Vese and will be submitted
for a paper.

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