Image Interpolation

Vicent Caselles[1]; Simon Masnou[2]; Jean-Michel Morel[3]; Catalina Sbert[4]

  • [1] Dept. de Matemàtiques, Univ. de les Illes Balears, 07071 Palma de Mallorca, Spain,
  • [2] CEREMADE, Université de Paris-Dauphine, 75775 Paris Cedex 16, France, masnou@ceremade.dauphine.fr
  • [3] CMLA, Ecole Normale Supérieure de Cachan, 94235 Cedex, France
  • [4] Dept. de Matemàtiques, Univ. de les Illes Balears, 07071 Palma de Mallorca, Spain

Séminaire Équations aux dérivées partielles (1997-1998)

  • Volume: 1997-1998, page 1-15

Abstract

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We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The Absolute Minimal Lipschitz Extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range. We also analyse the problem of unsmooth interpolation and show how it permits a subsidiary variational method.

How to cite

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Caselles, Vicent, et al. "Image Interpolation." Séminaire Équations aux dérivées partielles 1997-1998 (1997-1998): 1-15. <http://eudml.org/doc/10938>.

@article{Caselles1997-1998,
abstract = {We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The Absolute Minimal Lipschitz Extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range. We also analyse the problem of unsmooth interpolation and show how it permits a subsidiary variational method.},
affiliation = {Dept. de Matemàtiques, Univ. de les Illes Balears, 07071 Palma de Mallorca, Spain,; CEREMADE, Université de Paris-Dauphine, 75775 Paris Cedex 16, France, masnou@ceremade.dauphine.fr; CMLA, Ecole Normale Supérieure de Cachan, 94235 Cedex, France; Dept. de Matemàtiques, Univ. de les Illes Balears, 07071 Palma de Mallorca, Spain},
author = {Caselles, Vicent, Masnou, Simon, Morel, Jean-Michel, Sbert, Catalina},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {numerical examples; image processing; image interpolation; algorithms; absolute minimal Lipschitz extension model; variational method},
language = {eng},
pages = {1-15},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Image Interpolation},
url = {http://eudml.org/doc/10938},
volume = {1997-1998},
year = {1997-1998},
}

TY - JOUR
AU - Caselles, Vicent
AU - Masnou, Simon
AU - Morel, Jean-Michel
AU - Sbert, Catalina
TI - Image Interpolation
JO - Séminaire Équations aux dérivées partielles
PY - 1997-1998
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1997-1998
SP - 1
EP - 15
AB - We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The Absolute Minimal Lipschitz Extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range. We also analyse the problem of unsmooth interpolation and show how it permits a subsidiary variational method.
LA - eng
KW - numerical examples; image processing; image interpolation; algorithms; absolute minimal Lipschitz extension model; variational method
UR - http://eudml.org/doc/10938
ER -

References

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  1. L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mechanics and Anal. , 16, IX (1993), pp. 200-257. Zbl0788.68153MR1225209
  2. G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Math. 6, 551–561, 1967 Zbl0158.05001MR217665
  3. G. Bellettini, G. Dal Maso and M. Paolini, Semicontinuity and relaxation properties of a curvature depending functional in 2D, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20, 247–297, 1993. Zbl0797.49013MR1233638
  4. J.R. Casas and L.Torres, Strong edge features for image coding, In R.W.Schafer P.Maragos and M.A. Butt, editors, Mathematical Morphology and its Applications to Image and Signal Processing, pp 443–450. Kluwer Academic Publishers, Atlanta, GA, May 1996. Zbl0921.68116
  5. V. Caselles, T. Coll and J.M. Morel, A Kanizsa programme, TR 9539, CEREMADE, Université Paris-Dauphine, France, 1995. 
  6. V. Caselles, J.M. Morel and C. Sbert, An Axiomatic Approach to Image Interpolation, TR 9712, CEREMADE, Université Paris-Dauphine, France, 1995. Text containing all mathematical proofs. Zbl0980.68124MR1799563
  7. M. G. Crandall, H. Ishii and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. 27 (1992) pp. 1-67. Zbl0755.35015
  8. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press Inc., 1992. Zbl0804.28001MR1158660
  9. F. Guichard and J.M. Morel, Introduction to Partial Differential Equations on image processing, Tutorial, ICIP-95, Washington. Extended version to appear as book in Cambridge University Press. MR1628359
  10. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the Sup Norm of the Gradient, Arch. Rat. Mech. Anal. 123 (1993), pp. 51-74. Zbl0789.35008MR1218686
  11. G. Kanizsa, Grammaire du Voir, Diderot, 1996. 
  12. S. Masnou and J.M. Morel, Level lines based disocclusion, in Proc. ICIP’98, IEEE, 1998. 
  13. M. Nitzberg, D. Mumford and T. Shiota, “Filtering, Segmentation and Depth”, Lecture Notes in Computer Science, Vol. 662, Springer-Verlag, Berlin, 1993. Zbl0801.68171
  14. M.J.D. Powell, A review of methods for multivariable interpolation at scattered data points, Numerical Analysis Reports, NA11, DAMTP, University of Cambridge, 1996. To appear in State of the Art in Numerical Analysis, Cambridge University Press. Zbl0881.65003MR1628350

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