Geodesic
The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row $k$ of this triangle correspond to the Fourier series coefficients of the moment of order $k$ of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The motivating application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries.
Keywords: moments; symmetry detection; moving frame; shape recognition. 
@article{SIGMA_2013_9_a30,
     author = {M. Boutin and Sh. Huang},
     title = {The {Pascal} {Triangle} of {a~Discrete} {Image:} {Definition,} {Properties} and {Application} to {Shape}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a30/}
}
TY  - JOUR
AU  - M. Boutin
AU  - Sh. Huang
TI  - The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a30/
LA  - en
ID  - SIGMA_2013_9_a30
ER  - 
%0 Journal Article
%A M. Boutin
%A Sh. Huang
%T The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a30/
%G en
%F SIGMA_2013_9_a30
M. Boutin; Sh. Huang. The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a30/

[1] Fels M., Olver P. J., “Moving coframes. I: A practical algorithm”, Acta Appl. Math., 51 (1998), 161–213 | DOI | MR

[2] Fels M., Olver P. J., “Moving coframes. II: Regularization and theoretical foundations”, Acta Appl. Math., 55 (1999), 127–208 | DOI | MR | Zbl

[3] Flusser J., Zitova B., Suk T., Moments and moment invariants in pattern recognition, John Wiley Sons Ltd., Chichester, 2009 | Zbl

[4] Gustafsson B., He C., Milanfar P., Putinar M., “Reconstructing planar domains from their moments”, Inverse Problems, 16 (2000), 1053–1070 | DOI | MR | Zbl

[5] Haddad A. W., Huang S., Boutin M., Delp E. J., “Detection of symmetric shapes on a mobile device with applications to automatic sign interpretation”, Proc. SPIE, 8304, 2012, 83040G, 13 pp. | DOI

[6] Milanfar P., Verghese G. C., Karl W., Willsky A. S., “Reconstruction polygons from moments with connections to array processing”, IEEE Trans. Signal Process., 43 (1995), 432–443 | DOI

[7] Olver P. J., Classical invariant theory, London Mathematical Society Student Texts, 44, Cambridge University Press, Cambridge, 1999 | DOI | MR

[8] Rostampour A. R., Madhvapathy P. R., “Shape recognition using simple measures of projections”, Proceedings of Seventh Annual International Phoenix Conference on Computers and Communications (Scottsdale, AZ, 1988), IEEE, Arizona State University, 1988, 474–479 | DOI