## EECS CSPL SEMINAR SERIES

WINTER TERM 1997

Mila Nikolova
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Mila Nikolova

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UFR Mathématiques et Informatique

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Université Paris V, France

Tuesday, April 8, 1997

4:30-5:30 pm

1003 EECS

### Regularization Functions and Estimators

Abstract -

We are concerned with the class of estimators defined as the minimizer of
a coupled criterion, joining a quadratic data-fidelity term and a
regularization term. The latter term is added to favour the recovery of
features known a priori, and it sums the contributions of the differences
between neighbouring pixels, weighted using general potential functions
(PFs). Such estimators arise in both the regularization framework and in
Bayesian MAP estimation, the regularization term being a Markovian energy.
Quite a variety of PFs, convex or bounded, non-smooth or discontinuous at
some points, are used in practice which correspond to different prior
models. Here we explore the mutual relation between the shape of the PF
and the main characteristics of the estimate such as recovery of
homogeneous regions, stability, edge recovery, bias over edges, and
resolution.

First we give necessary conditions for local minima of a piecewise smooth
multimodal criterion. Then, we show that the global minimum is regularized
while strict local minima evolve continuously with the data. Estimators
incorporating non-convex PFs are piecewise continuous, so they are locally
stable. Furthermore, a solution presenting strictly homogeneous regions
can be obtained if and only if the PF is non-smooth at zero. These
strictly homogeneous regions remain unchanged under small variations of
the data. The presence of local concavities in the PF non-convex PFs
affect edge detection performance. Moreover, concave
non-differentiability implies a classification of the differences w.r.t. a
fixed threshold, thus refining the resolution. In particular, we show that
edges may be recovered without bias only if the PF is constant beyond a
threshold.

Biosketch -

Mila Nikolova received the Ph.D. degree in Signal Processing from
the Universite de Paris-Sud in 1995. Currently, she is teaching and
research assistant at the Universite Paris V. Her research interests
are in inverse problems and image reconstruction.

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