Critical features of the minimizers of nonsmooth costfunctions;
application to image processing.
Mila Nikolova CNRS URA820 ENST Dpt. TSI, Paris, France
Friday, March 8, 2002 15:0016:00, Room 145, NIST North (820) Gaithersburg Friday, March 8, 2002 13:0014:00, Room 4550 Boulder
Abstract:
We address applications where a sought vector x
(an image, a signal) is recovered
from data y by minimizing a costfunction 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 nonsmooth, the minimizers of finvolve large regions where the differences between neighbors are null. This
explains in particular the staircasing effect observed in total variation
methods. This striking property cannot be
exhibited by the minimizers of smooth costfunctions.
The datafidelity term D comes from a statistical modelling of the
dataacquisition and is usually a smooth function (and often quadratic). We showthat if D is nonsmooth 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 datafidelity equations. This surprising property does notoccur if D is smooth.
An astute use of the properties explained above helps to conceive costfunctionswhose minimizers exhibit some special properties. E.g., we focus on the
reconstruction of binary images where the main difficulty comes from the
nonconvexity of the relevant costfunctions. We show how to construct
continuousvalued, convex costfunctions whose minimizers are quasibinary (theyinvolve only very few nonbinary pixels).
Next, we exhibit how to process outliers, or spiky images, by using nonsmooth
datafidelity terms. Numerical experiments will be shown for all these
applications.
Contact: A. S. CarassoNote: Visitors from outside NIST must contact
Robin Bickel; (301) 9753668;
at least 24 hours in advance.
