Structural Properties of Solutions to Total Variation Regularization Problems

Wolfgang Ring

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 4, page 799-810
  • ISSN: 0764-583X

Abstract

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In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.

How to cite

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Ring, Wolfgang. "Structural Properties of Solutions to Total Variation Regularization Problems." ESAIM: Mathematical Modelling and Numerical Analysis 34.4 (2010): 799-810. <http://eudml.org/doc/197509>.

@article{Ring2010,
abstract = { In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two. },
author = {Ring, Wolfgang},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Total variation regularization; piecewise constant function; convex optimization; Lebesgue decomposition; singular measures.; optimality system; total variation regularization; Lebesgue decomposition; singular measures; ill-posed problem; least-squares problem},
language = {eng},
month = {3},
number = {4},
pages = {799-810},
publisher = {EDP Sciences},
title = {Structural Properties of Solutions to Total Variation Regularization Problems},
url = {http://eudml.org/doc/197509},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Ring, Wolfgang
TI - Structural Properties of Solutions to Total Variation Regularization Problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 4
SP - 799
EP - 810
AB - In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.
LA - eng
KW - Total variation regularization; piecewise constant function; convex optimization; Lebesgue decomposition; singular measures.; optimality system; total variation regularization; Lebesgue decomposition; singular measures; ill-posed problem; least-squares problem
UR - http://eudml.org/doc/197509
ER -

References

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  14. M. Nikolova, Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. (to appear).  
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