February  2013, 7(1): 107-122. doi: 10.3934/ipi.2013.7.107

Local uniqueness of the circular integral invariant

1. 

Department of Mathematics, University of Vienna, Nordbergstr. 15, A-1090 Wien, Austria

2. 

Computational Science Center, University of Vienna, Nordbergstr. 15, A-1090 Wien, Austria, Austria

Received  July 2011 Revised  December 2012 Published  February 2013

This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant most common in use is the circular integral invariant. A major drawback of this curve descriptor, however, is the absence of any uniqueness result for this representation. This article serves as a contribution towards closing this gap by showing that the circular integral invariant is injective in a neighbourhood of the circle. In addition, we provide a stability estimate valid on this neighbourhood. The proof is an application of Riesz--Schauder theory and the implicit function theorem in a Banach space setting.
Citation: Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems and Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107
References:
[1]

É. Cartan, La méthode du repère mobile, la théorie des groupes continus et les espaces généralisées, Actual. Scient. et Industr., 194 (1935), 65 pp.

[2]

B. E. J. Dahlberg, The converse of the four vertex theorem, Proc. Amer. Math. Soc., 133 (2005), 2131-2135 (electronic). doi: 10.1090/S0002-9939-05-07788-9.

[3]

A. Duci, A. J. Yezzi, Jr., S. K. Mitter and S. Soatto, Shape representation via harmonic embedding, in "Computer Vision, 2003. Proceedings, Ninth IEEE International Conference on," 1, IEEE, (2003), 656-662.

[4]

A. Duci, A. J. Yezzi, Jr., S. Soatto and K. Rocha, Harmonic embeddings for linear shape analysis, J. Math. Imaging Vision, 25 (2006), 341-352. doi: 10.1007/s10851-006-7249-8.

[5]

T. Fidler, M. Grasmair and O. Scherzer, Identifiability and reconstruction of shapes from integral invariants, Inverse Probl. Imaging, 2 (2008), 341-354. doi: 10.3934/ipi.2008.2.341.

[6]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[7]

Q.-X. Huang, S. Flöry, N. Gelfand, M. Hofer and H. Pottmann, Reassembling fractured objects by geometric matching, in "ACM SIGGRAPH 2006 Papers," SIGGRAPH '06, ACM, New York, NY, (2006), 569-578.

[8]

E. Klassen, A. Srivastava, W. Mio and S. H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), 372-383.

[9]

S. Lang, "Differential and Riemannian manifolds," Third edition, Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.

[10]

S. Lie, "Über differentialinvarianten," Math. Ann., 24 (1884), 537-578.

[11]

S. Manay, B.-W. Hong, A. J. Yezzi, Jr. and S. Soatto, Integral invariant signatures, in "Computer Vision - ECCV 2004" (eds. T. Pajdla and J. Matas), Lecture Notes in Computer Science, 3024, Springer Berlin/Heidelberg, (2004), 87-99.

[12]

P. W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, in "Phase Space Analysis of Partial Differential Equations" (eds. A. Bove, F. Colombini, D. Del Santo), Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, (2006), 133-215. doi: 10.1007/978-0-8176-4521-2_11.

[13]

P. J. Olver, "Equivalence, Invariants, and Symmetry," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.

[14]

S. C. Preston, The geometry of whips, Ann. Global Anal. Geom., 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z.

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E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, Int. J. Comput. Vision, 70 (2006), 55-75.

[16]

Y.-L. Yang, Y.-K. Lai, S.-M. Hu and H. Pottmann, Robust principal curvatures on multiple scales, in "Proceedings of the Fourth Eurographics Symposium on Geometry Processing" (eds. K. Polthier and A. Sheffer), Aire-la-Ville, Eurographics Association, Switzerland, (2006), 223-226.

[17]

K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

show all references

References:
[1]

É. Cartan, La méthode du repère mobile, la théorie des groupes continus et les espaces généralisées, Actual. Scient. et Industr., 194 (1935), 65 pp.

[2]

B. E. J. Dahlberg, The converse of the four vertex theorem, Proc. Amer. Math. Soc., 133 (2005), 2131-2135 (electronic). doi: 10.1090/S0002-9939-05-07788-9.

[3]

A. Duci, A. J. Yezzi, Jr., S. K. Mitter and S. Soatto, Shape representation via harmonic embedding, in "Computer Vision, 2003. Proceedings, Ninth IEEE International Conference on," 1, IEEE, (2003), 656-662.

[4]

A. Duci, A. J. Yezzi, Jr., S. Soatto and K. Rocha, Harmonic embeddings for linear shape analysis, J. Math. Imaging Vision, 25 (2006), 341-352. doi: 10.1007/s10851-006-7249-8.

[5]

T. Fidler, M. Grasmair and O. Scherzer, Identifiability and reconstruction of shapes from integral invariants, Inverse Probl. Imaging, 2 (2008), 341-354. doi: 10.3934/ipi.2008.2.341.

[6]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[7]

Q.-X. Huang, S. Flöry, N. Gelfand, M. Hofer and H. Pottmann, Reassembling fractured objects by geometric matching, in "ACM SIGGRAPH 2006 Papers," SIGGRAPH '06, ACM, New York, NY, (2006), 569-578.

[8]

E. Klassen, A. Srivastava, W. Mio and S. H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), 372-383.

[9]

S. Lang, "Differential and Riemannian manifolds," Third edition, Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.

[10]

S. Lie, "Über differentialinvarianten," Math. Ann., 24 (1884), 537-578.

[11]

S. Manay, B.-W. Hong, A. J. Yezzi, Jr. and S. Soatto, Integral invariant signatures, in "Computer Vision - ECCV 2004" (eds. T. Pajdla and J. Matas), Lecture Notes in Computer Science, 3024, Springer Berlin/Heidelberg, (2004), 87-99.

[12]

P. W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, in "Phase Space Analysis of Partial Differential Equations" (eds. A. Bove, F. Colombini, D. Del Santo), Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, (2006), 133-215. doi: 10.1007/978-0-8176-4521-2_11.

[13]

P. J. Olver, "Equivalence, Invariants, and Symmetry," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.

[14]

S. C. Preston, The geometry of whips, Ann. Global Anal. Geom., 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z.

[15]

E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, Int. J. Comput. Vision, 70 (2006), 55-75.

[16]

Y.-L. Yang, Y.-K. Lai, S.-M. Hu and H. Pottmann, Robust principal curvatures on multiple scales, in "Proceedings of the Fourth Eurographics Symposium on Geometry Processing" (eds. K. Polthier and A. Sheffer), Aire-la-Ville, Eurographics Association, Switzerland, (2006), 223-226.

[17]

K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

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