Critical features of the minimizers of non-smooth cost-functions;
application to image processing.
Mila Nikolova
CNRS URA820 ENST Dpt. TSI, Paris, France
Friday, March 8, 2002 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Friday, March 8, 2002 13:00-14:00,
Room 4550
Boulder
Abstract:
We address applications where a sought vector x
(an image, a signal) is recovered
from data y by minimizing a cost-function f(x,y)=D(x,y)+R(x),
where D is a data fidelity term and R is a regularization term.
Our goal is to exhibit how the shapes of the functions D and R determine
the essential features exhibited by the minimizers of f(.,y).
Typically, the regularization term R is defined over the differences between
neighboring samples of x. We show that when R is non-smooth, the minimizers of finvolve large regions where the differences between neighbors are null. This
explains in particular the stair-casing effect observed in total variation
methods. This striking property cannot be
exhibited by the minimizers of smooth cost-functions.
The data-fidelity term D comes from a statistical modelling of the
data-acquisition and is usually a smooth function (and often quadratic). We showthat if D is non-smooth at the origin, typical data y lead to minimizers x of
f(.,y) which fit exactly part of the data entries, i.e. they satisfy exactly a
certain number of the data-fidelity equations. This surprising property does notoccur if D is smooth.
An astute use of the properties explained above helps to conceive cost-functionswhose minimizers exhibit some special properties. E.g., we focus on the
reconstruction of binary images where the main difficulty comes from the
non-convexity of the relevant cost-functions. We show how to construct
continuous-valued, convex cost-functions whose minimizers are quasi-binary (theyinvolve only very few non-binary pixels).
Next, we exhibit how to process outliers, or spiky images, by using non-smooth
data-fidelity terms. Numerical experiments will be shown for all these
applications.
Contact: A. S. CarassoNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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