Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T23:08:33.897Z Has data issue: false hasContentIssue false
  • English
  • Français

Equi-affine differential invariants for invariant feature point detection

Part of: Classical differential geometry

Published online by Cambridge University Press:  06 March 2019

STANLEY L. TUZNIK ,
PETER J. OLVER Open the ORCID record for PETER J. OLVER [Opens in a new window]  and
ALLEN TANNENBAUM
STANLEY L. TUZNIK
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA e-mail: stanley.tuznik@stonybrook.edu
PETER J. OLVER*
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA e-mail: olver@umn.edu
ALLEN TANNENBAUM
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA e-mail: stanley.tuznik@stonybrook.edu Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA e-mail: arobertan@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.

Keywords

Differential invariantfeature detectionequi-affine groupmoving fram

MSC classification

Primary: 53A15: Affine differential geometry
Secondary: 53A55: Differential invariants (local theory), geometric objects 68U10: Image processing
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 2 , April 2020 , pp. 277 - 296
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported by AFOSR [grant number FA9550-17-1-0435] and the National Institutes of Health [grant number R01-AG048769].

References

Akivis, M. & Rosenfeld, B. (1993) Élie Cartan (1869–1951). Translations Math. Monographs, Vol. 123, American Math. Soc., Providence, R.I.Google Scholar
Alcantarilla, P. F., Bartoli, A. & Davison, A. J. (2012) Kaze features. In: European Conference on Computer Vision, Springer, pp. 214–227.Google Scholar
Astrom, K. (1995) Fundamental limitations on projective invariants of planar curves. IEEE Trans. Pattern Anal. Mach. Intell. 17 (1), 77–81.CrossRefGoogle Scholar
Baumberg, A. (2000) Reliable feature matching across widely separated views. In: Proceedings IEEE Conference on Computer Vision and Pattern Recognition, 2000, Vol. 1. IEEE, pp. 774–781.Google Scholar
Bay, H., Tuytelaars, T. & Gool, L. V. (2006) Surf: Speeded up Robust Features, ECCV 2006, Springer, pp. 404–417.Google Scholar
Burdis, J. M., Kogan, I. A. & Hong, H. (2013) Object-image correspondence for algebraic curves under projections. SIGMA: Symmetry Integr. Geom. Methods Appl. 9 (023), 1–31.Google Scholar
Calabi, E., Olver, P. J. & Tannenbaum, A. (1996) Affine geometry, curve flows, and invariant numerical approximations. Adv. Math. 124 (1), 154–196.CrossRefGoogle Scholar
COLLEU, T., SHEN, J.-K., MATUSZEWSKI, B., SHARK, L.-K. & CARIOU, C. (2006) Feature-based deformable image registration with ransac based search correspondence. In: AECRIS’06-Atlantic Europe Conference on Remote Imaging and Spectroscopy, pp. 57–64.Google Scholar
Fels, M. & Olver, P. J. (1999) Moving coframes: II. regularization and theoretical foundations. Acta Appl. Math. 55(2), 127–208.CrossRefGoogle Scholar
Fischler, M. A. & Bolles, R. C. (1981) Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM. 24 (6), 381–395.CrossRefGoogle Scholar
Guggenheimer, H. (1963) Differential Geometry, McGraw–Hill, New York.Google Scholar
Hann, C. E. (2001) Recognising Two Planar Objects Under a Projective Transformation. PhD dissertation, University of Canterbury, Mathematics and Statistics.Google Scholar
Hann, C. E. & Hickman, M. (2002) Projective curvature and integral invariants. Acta Appl. Math. 74(2), 177–193.CrossRefGoogle Scholar
Harris, C. & Stephens, M. (1988) A combined corner and edge detector. In: Alvey Vision Conference, Vol. 15, no. 50, Manchester, UK, pp. 10–5244.Google Scholar
Kogan, I. A. & Olver, P. J. (2015) Invariants of objects and their images under surjective maps. Lobachevskii J. Math. 36(3), 260–285.CrossRefGoogle Scholar
Lowe, D. G. (2004) Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vision. 60(2), 91–110.CrossRefGoogle Scholar
Mikolajczyk, K. & SCHMID, C. (2005) A performance evaluation of local descriptors. IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1615–1630.CrossRefGoogle ScholarPubMed
Mikolajczyk, K., Tuytelaars, T., Schmid, C., Matas, A. Z. J., Schaffalitzky, F., Kadir, T., & Gool, L. V. (2005) A comparison of affine region detectors. Int. J. Comput. Vision. 65 (1–2), 43–72.CrossRefGoogle Scholar
Olver, P. J. (1995) Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Olver, P. J. (2001) Joint invariant signatures. Found. Comput. Math. 1(1), 3–67.CrossRefGoogle Scholar
Olver, P. J. (2003) Moving frames. J. Symb. Comput. 36(3), 501–512.CrossRefGoogle Scholar
Olver, P. J. (2015) Modern developments in the theory and applications of moving frames. In: Impact 150 Stories, Vol. 1, London Mathematical Society, London, UK, pp. 14–50.Google Scholar
Olver, P. J. (2015) Moving Frame Derivation of the Fundamental Equi-Affine Differential Invariants for Level Set Functions. preprint, University of Minnesota, Minneapolis, MN, USA.Google Scholar
Olver, P. J., Sapiro, G. & Tannenbaum, A. (1994) Differential invariant signatures and flows in computer vision: a symmetry group approach. In: Ter Haar Romeny, B. M. (editor), Geometry-Driven Diffusion in Computer Vision, Kluwer Acad. Publ., Dordrecht, The Netherlands, pp. 255–306.CrossRefGoogle Scholar
Olver, P. J., Sapiro, G. & Tannenbaum, A. (1997) Invariant geometric evolutions of surfaces and volumetric smoothing. Siam. J. Appl. Math. 57 (1), 176–194.Google Scholar
Olver, P. J., Sapiro, G. & Tannenbaum, A. (1999) Affine invariant detection: edge maps, anisotropic diffusion, and active contours. Acta Appl. Math. 59 (1), 45–77.CrossRefGoogle Scholar
Perona, P. & Malik, J. (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12 (7), 629–639.CrossRefGoogle Scholar
Sapiro, G. & Tannenbaum, A. (1993) Affine invariant scale-space. Int. J. Comput. Vision. 11 (1), 25–44.CrossRefGoogle Scholar
Sethian, J. A. (1999) Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK.Google Scholar
Ter Haar Romeny, B. M. (1994) Geometry-Driven Diffusion in Computer Vision. Kluwer Acad. Publ., Dordrecht, The Netherlands.CrossRefGoogle Scholar
Welk, M., Kim, P. & Olver, P. J. (2007) Numerical invariantization for morphological PDE schemes. In: International Conference on Scale Space and Variational Methods in Computer Vision. Springer, pp. 508–519.CrossRefGoogle Scholar
Witkin, A. (1984) Scale-space filtering: a new approach to multi-scale description. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP’84, Vol. 9. IEEE, pp. 150–153.CrossRefGoogle Scholar
Yu, G. & Morel, J.-M. (2011) ASIFT: an algorithm for fully affine invariant comparison. Image Process. On Line. 1, 11–38.CrossRefGoogle Scholar