Image Interpolation
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1997-1998), Exposé no. 12, 15 p.

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.

@article{SEDP_1997-1998____A12_0,
     author = {Caselles, Vicent and Masnou, Simon and Morel, Jean-Michel and Sbert, Catalina},
     title = {Image {Interpolation}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:12},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1997-1998},
     mrnumber = {1660525},
     zbl = {1058.65020},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1997-1998____A12_0/}
}
TY  - JOUR
AU  - Caselles, Vicent
AU  - Masnou, Simon
AU  - Morel, Jean-Michel
AU  - Sbert, Catalina
TI  - Image Interpolation
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:12
PY  - 1997-1998
DA  - 1997-1998///
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_1997-1998____A12_0/
UR  - https://www.ams.org/mathscinet-getitem?mr=1660525
UR  - https://zbmath.org/?q=an%3A1058.65020
LA  - en
ID  - SEDP_1997-1998____A12_0
ER  - 
Caselles, Vicent; Masnou, Simon; Morel, Jean-Michel; Sbert, Catalina. Image Interpolation. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1997-1998), Exposé no. 12, 15 p. http://www.numdam.org/item/SEDP_1997-1998____A12_0/

[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. | MR 1225209 | Zbl 0788.68153

[2] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Math. 6, 551–561, 1967 | MR 217665 | Zbl 0158.05001

[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. | Numdam | MR 1233638 | Zbl 0797.49013

[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. | Zbl 0921.68116

[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. | MR 1799563

[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. | Zbl 0755.35015

[8] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press Inc., 1992. | MR 1158660 | Zbl 0804.28001

[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. | MR 1628359

[10] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the Sup Norm of the Gradient, Arch. Rat. Mech. Anal. 123 (1993), pp. 51-74. | MR 1218686 | Zbl 0789.35008

[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. | Zbl 0801.68171

[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. | MR 1628350 | Zbl 0881.65003