Math @ Duke

Publications [#264865] of Guillermo Sapiro
Papers Published
 Bar, L; Sapiro, G, Generalized newtontype methods for energy formulations in image processing,
Siam Journal on Imaging Sciences, vol. 2 no. 2
(January, 2009),
pp. 508531, Society for Industrial & Applied Mathematics (SIAM) [doi]
(last updated on 2019/06/20)
Abstract: © 2009 Society for Industrial and Applied Mathematics. Many problems in image processing are addressed via the minimization of a cost functional. The most prominently used optimization technique is gradientdescent, often used due to its simplicity and applicability where other techniques, e.g., those coming from discrete optimization, cannot be applied. Yet, gradientdescent suffers from slow convergence, and often to just local minima which highly depend on the initialization and the condition number of the functional Hessian. Newtontype methods, on the other hand, are known to have a faster, quadratic convergence. In its classical form, the Newton method relies on the L2type norm to define the descent direction. In this paper, we generalize and reformulate this very important optimization method by introducing Newtontype methods based on more general norms. Such norms are introduced both in the descent computation (Newton step) and in the corresponding stabilizing trustregion. 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.


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