Tel Aviv University
  School of Mathematical Sciences
    Applied Mathematics Seminar
Date:          Tuesday March 29, 2011, 15:10
Place:             Schreiber Bldg, Room 309
Speaker:     Leah Bar
         Tel Aviv University
Title: Generalized Newton-Type Methods for Energy Formulations in Image
Many problems in image processing are addressed via the minimization
of a cost functional. The most prominently used optimization technique
is gradient-descent, often used due to its simplicity and applicability
where other techniques, e.g., those coming from discrete optimization,
cannot be applied. Yet, gradient-descent suffers from slow convergence,
and often to just local minima which highly depend on the initialization
and the condition number of the functional Hessian. Newton-type methods,
on the other hand, are known to have a faster, quadratic convergence. In
its classical form, the Newton method relies on the L2-type norm to
define the descent direction. In this work, we generalize and
reformulate this very important optimization method by introducing
Newton-type methods based on more general norms. Such norms are
introduced both in the descent computation (Newton step) and in the
corresponding stabilizing trust-region. This generalization opens up new
possibilities in the extraction of the Newton step, including benefits
such as mathematical stability and the incorporation of smoothness
constraints. We first present the derivation of the modified Newton step
in the calculus of variation framework needed for image processing.
Then, we demonstrate the method with two common objective functionals:
variational image deblurring and geometric active contours for image
segmentation. We show that in addition to the fast convergence, norms
adapted to the problem at hand yield different and superior results.
Joint work with Prof. Guillermo Sapiro, university of Minnesota.
For information about future seminars, see
Dr. Adi Ditkowski                                                     |
Department of Applied Mathematics                                     |
School of Mathematical Sciences          phone:  972-3-640-5987       |
Tel Aviv University,                     fax  :  972-3-640-9357       |
Tel Aviv, 69978 Israel                   email:   <>  |
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <> 
Announcement from: Adi Ditkowski   <>