### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 080, 29 pages      arXiv:1603.09335      https://doi.org/10.3842/SIGMA.2016.080

### Möbius Invariants of Shapes and Images

Stephen Marsland a and Robert I. McLachlan b
a) School of Engineering and Advanced Technology, Massey University, Palmerston North, New Zealand
b) Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

Received April 01, 2016, in final form August 08, 2016; Published online August 11, 2016

Abstract
Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the Möbius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known Möbius invariants, and then develop an algorithm by which shapes can be recognised that is Möbius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a Möbius-invariant signature of grey-scale images.

Key words: invariant; invariant signature; Möbius group; shape; image.

pdf (1957 kb)   tex (1548 kb)

References

1. Abu-Mostafa Y.S., Psaltis D., Recognitive aspects of moment invariants, IEEE Trans. Pattern Anal. Machine Intell. 6 (1984), 698-706.
2. Aghayan R., Ellis T., Dehmeshki J., Planar numerical signature theory applied to object recognition, J. Math. Imaging Vision 48 (2014), 583-605.
3. Ahlfors L.V., Cross-ratios and Schwarzian derivatives in ${\bf R}^n$, in Complex Analysis, Editors J. Hersch, A. Huber, Birkhäuser, Basel, 1988, 1-15.
4. Ames A.D., Jalkio J.A., Shakiban C., Three-dimensional object recognition using invariant Euclidean signature curves, in Analysis, Combinatorics and Computing, Nova Sci. Publ., Hauppauge, NY, 2002, 13-23.
5. Åström K., Fundamental difficulties with projective normalization of planar curves, in Applications of Invariance in Computer Vision, Lecture Notes in Computer Science, Vol. 825, Springer, Berlin - Heidelberg, 1994, 199-214.
6. Bandeira A.S., Cahill J., Mixon D.G., Nelson A.A., Saving phase: injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal. 37 (2014), 106-125, arXiv:1302.4618.
7. Barrett D.E., Bolt M., Cauchy integrals and Möbius geometry of curves, Asian J. Math. 11 (2007), 47-53.
8. Bauer M., Bruveris M., Michor P.W., Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision 50 (2014), 60-97, arXiv:1305.1150.
9. Calabi E., Olver P.J., Shakiban C., Tannenbaum A., Haker S., Differential and numerically invariant signature curves applied to object recognition, Int. J. Comput. Vis. 26 (1998), 107-135.
10. Edelsbrunner H., Harer J., Persistent homology - a survey, in Surveys on Discrete and Computational Geometry, Contemp. Math., Vol. 453, Amer. Math. Soc., Providence, RI, 2008, 257-282.
11. Feng S., Kogan I., Krim H., Classification of curves in 2D and 3D via affine integral signatures, Acta Appl. Math. 109 (2010), 903-937, arXiv:0806.1984.
12. Fridman B., Kuchment P., Lancaster K., Lissianoi S., Mogilevsky M., Ma D., Ponomarev I., Papanicolaou V., Numerical harmonic analysis on the hyperbolic plane, Appl. Anal. 76 (2000), 351-362.
13. Gauthier J.P., Smach F., Lemaître C., Miteran J., Finding invariants of group actions on function spaces, a general methodology from non-abelian harmonic analysis, in Mathematical Control Theory and Finance, Springer, Berlin, 2008, 161-186.
14. Ghorbel F., A complete invariant description for gray-level images by the harmonic analysis approach, Pattern Recognition Lett. 15 (1994), 1043-1051.
15. Glaunès J., Qiu A., Miller M.I., Younes L., Large deformation diffeomorphic metric curve mapping, Int. J. Comput. Vis. 80 (2008), 317-336.
16. Hann C.E., Hickman M.S., Projective curvature and integral invariants, Acta Appl. Math. 74 (2002), 177-193.
17. Hickman M.S., Euclidean signature curves, J. Math. Imaging Vision 43 (2012), 206-213.
18. Hoff D.J., Olver P.J., Extensions of invariant signatures for object recognition, J. Math. Imaging Vision 45 (2013), 176-185.
19. Kakarala R., The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach, J. Math. Imaging Vision 44 (2012), 341-353, arXiv:0902.0196.
20. Lenz R., Group theoretical methods in image processing, Lecture Notes in Computer Science, Vol. 413, Springer-Verlag, Berlin, 1990.
21. Manay S., Cremers D., Hong B., Yezzi A.J., Soatto S., Integral invariants for shape matching, IEEE Trans. Pattern Anal. Machine Intell. 28 (2006), 1602-1618.
22. Marsland S., McLachlan R.I., Modin K., Perlmutter M., Geodesic warps by conformal mappings, Int. J. Comput. Vis. 105 (2013), 144-154, arXiv:1203.3982.
23. Michor P.W., Manifolds of differentiable mappings, Shiva Mathematics Series, Vol. 3, Shiva Publishing Ltd., Nantwich, 1980.
24. Michor P.W., Mumford D., An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal. 23 (2007), 74-113, math.DG/0605009.
25. Milnor J.W., The geometry of growth and form, Talk given at the IAS, Princeton, 2010, available at http://www.math.sunysb.edu/~jack/gfp-print.pdf.
26. Mumford D., Pattern theory and vision, in Questions Mathématiques En Traitement Du Signal et de L'Image, Chapter 3, Institute Henri Poincaré, Paris, 1998, 7-13.
27. O'Hara J., Solanes G., Möbius invariant energies and average linking with circles, Tohoku Math. J. 67 (2015), 51-82, arXiv:1010.3764.
28. Olver P.J., Moving frames and singularities of prolonged group actions, Selecta Math. (N.S.) 6 (2000), 41-77.
29. Olver P.J., Joint invariant signatures, Found. Comput. Math. 1 (2001), 3-67.
30. Olver P.J., Moving frames - in geometry, algebra, computer vision, and numerical analysis, in Foundations of Computational Mathematics (Oxford, 1999), London Math. Soc. Lecture Note Ser., Vol. 284, Cambridge University Press, Cambridge, 2001, 267-297.
31. Olver P.J., A survey of moving frames, in Computer Algebra and Geometric Algebra with Applications, Lecture Notes in Computer Science, Vol. 3519, Springer, Berlin - Heidelberg, 2005, 105-138.
32. Olver P.J., The symmetry groupoid and weighted signature of a geometric object, J. Lie Theory 26 (2016), 235-267.
33. Patterson B.C., The differential invariants of inversive geometry, Amer. J. Math. 50 (1928), 553-568.
34. Petukhov S.V., Non-Euclidean geometries and algorithms of living bodies, Comput. Math. Appl. 17 (1989), 505-534.
35. Shakiban C., Lloyd P., Signature curves statistics of DNA supercoils, in Geometry, Integrability and Quantization, Softex, Sofia, 2004, 203-210.
36. Shakiban C., Lloyd P., Classification of signature curves using latent semantic analysis, in Computer Algebra and Geometric Algebra with Applications, Lecture Notes in Computer Science, Vol. 3519, Springer, Berlin - Heidelberg, 2005, 152-162.
37. Taylor M.E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, Vol. 22, Amer. Math. Soc., Providence, RI, 1986.
38. Thompson D.W., On growth and form, Cambridge University Press, Cambridge, England, 1942.
39. Turski J., Geometric Fourier analysis of the conformal camera for active vision, SIAM Rev. 46 (2004), 230-255.
40. Turski J., Geometric Fourier analysis for computational vision, J. Fourier Anal. Appl. 11 (2005), 1-23.
41. Turski J., Computational harmonic analysis for human and robotic vision systems, Neurocomputing 69 (2006), 1277-1280.
42. Van Gool L., Moons T., Pauwels E., Oosterlinck A., Vision and Lie's approach to invariance, Image Vision Comput. 13 (1995), 259-277.
43. Wallace A., D'Arcy Thompson and the theory of transformations, Nature Rev. Genet. 7 (2006), 401-406.