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Generalized newton-type methods for energy formulations in image processing

Publication ,  Journal Article
Bar, L; Sapiro, G
Published in: SIAM Journal on Imaging Sciences
January 1, 2009

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 paper, 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.

Duke Scholars

Published In

SIAM Journal on Imaging Sciences

DOI

EISSN

1936-4954

Publication Date

January 1, 2009

Volume

2

Issue

2

Start / End Page

508 / 531

Related Subject Headings

  • Artificial Intelligence & Image Processing
  • 4901 Applied mathematics
  • 4603 Computer vision and multimedia computation
 

Citation

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Bar, L., & Sapiro, G. (2009). Generalized newton-type methods for energy formulations in image processing. SIAM Journal on Imaging Sciences, 2(2), 508–531. https://doi.org/10.1137/080722436
Bar, L., and G. Sapiro. “Generalized newton-type methods for energy formulations in image processing.” SIAM Journal on Imaging Sciences 2, no. 2 (January 1, 2009): 508–31. https://doi.org/10.1137/080722436.
Bar L, Sapiro G. Generalized newton-type methods for energy formulations in image processing. SIAM Journal on Imaging Sciences. 2009 Jan 1;2(2):508–31.
Bar, L., and G. Sapiro. “Generalized newton-type methods for energy formulations in image processing.” SIAM Journal on Imaging Sciences, vol. 2, no. 2, Jan. 2009, pp. 508–31. Scopus, doi:10.1137/080722436.
Bar L, Sapiro G. Generalized newton-type methods for energy formulations in image processing. SIAM Journal on Imaging Sciences. 2009 Jan 1;2(2):508–531.

Published In

SIAM Journal on Imaging Sciences

DOI

EISSN

1936-4954

Publication Date

January 1, 2009

Volume

2

Issue

2

Start / End Page

508 / 531

Related Subject Headings

  • Artificial Intelligence & Image Processing
  • 4901 Applied mathematics
  • 4603 Computer vision and multimedia computation