Abstract
Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity—by measuring the lack of isometry, deal with similarity or scaling—by multiplying the Euclidean arc-length by the Gaussian curvature, and manage equi-affine transformations—by resorting to the special affine arc-length definition in classical equi-affine differential geometry. Here, we propose a computational framework that is invariant to the full affine group of transformations (similarity and equi-affine). Thus, by construction, it can handle non-rigid shapes. Technically, we add the similarity invariant property to an equi-affine invariant one and establish an affine invariant pseudo-metric. As an example, we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison.
Similar content being viewed by others
References
Aflalo, Y., Kimmel, R., & Raviv, D. (2013). Scale invariant geometry for nonrigid shapes. SIAM Journal on Imaging Sciences, 6(3), 1579–1597.
Alvarez, L., Guichard, F., Lions, P.-L., & Morel, J.-M. (1993). Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis, 123(3), 199–257.
Andrade, M., & Lewiner, T. (2012). Affine-invariant curvature estimators for implicit surfaces. Computer Aided Geometric Design, 29(2), 162–173.
Beg, M. F., & Miller, M. I. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision (IJCV), 61(2), 139–157.
Bérard, P., Besson, G., & Gallot, S. (1994). Embedding Riemannian manifolds by their heat kernel. Geometric and Functional Analysis, 4(4), 373–398.
Blaschke, W. (1923). Vorlesungen uber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitatstheorie, vol. 2. Berlin: Springer.
Bronstein, M. M., & Kokkinos, I. (2010). Scale-invariant heat kernel signatures for non-rigid shape recognition. In Proceedings of Computer Vision and Pattern Recognition (CVPR).
Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2006). Efficient computation of isometry-invariant distances between surfaces. SIAM Journal on Scientific Computing, 28(5), 1812–1836.
Bronstein, A. M., Bronstein, M. M., Kimmel, R., Mahmoudi, M., & Sapiro, G. (2010a). A Gromov–Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. International Journal of Computer Vision (IJCV), 89(2–3), 266–286.
Bronstein, A. M., Bronstein, M. M., Castellani, U., Falcidieno, B., Fusiello, A., Godil, A., Guibas, L. J., Kokkinos, I., Lian, Z., Ovsjanikov, M., Patané, G., Spagnuolo, M., & Toldo, R. (2010b) . SHREC 2010: Robust large-scale shape retrieval benchmark. In Proceedings of Workshop on 3D Object Retrieval (3DOR).
Brook, A., Bruckstein, A.M., & Kimmel, R. (2005). On similarity-invariant fairness measures. In Scale-space, LNCS 3459 (pp. 456–467). Hofgeismar, Germany: Springer, 7–9 April 2005.
Bruckstein, A. M., & Netravali, A. N. (1995). On differential invariants of planar curves and recognizing partially occluded planar shapes. Annals of Mathematics and Artificial Intelligence (AMAI), 13(3–4), 227–250.
Bruckstein, A. M., & Shaked, D. (1998). Skew symmetry detection via invariant signatures. Pattern Recognition, 31(2), 181–192.
Bruckstein, A. M., Rivlin, E., & Weiss, I. (1997). Scale-space local invariants. Image and Vision Computing, 15(5), 335–344.
Bruckstein, A. M., Katzir, N., Lindenbaum, M., & Porat, M. (1992). Similarity-invariant signatures for partially occluded planar shapes. International Journal of Computer Vision, 7(3), 271–285.
Bruckstein, A. M., Holt, R. J., Netravali, A. N., & Richardson, T. J. (1993). Invariant signatures for planar shape recognition under partial occlusion. Computer Vision, Graphics, and Image Processing: Image Understanding, 58, 49–65.
Calabi, E., Olver, P. J., Shakiban, C., Tannenbaum, A., & Haker, S. (1998). Differential and numerically invariant signature curves applied to object recognition. International Journal of Computer Vision, 26, 107–135.
Carlsson, S., Mohr, R., Moons, T., Morin, L., Rothwell, C. A., Van Diest, M., et al. (1996). Semi-local projective invariants for the recognition of smooth plane curves. International Journal of Computer Vision, 19(3), 211–236.
Chazal, F., Cohen-Steiner, D., Guibas, L. J., Mémoli, F., & Oudot, S. (2009). Gromov–Hausdorff stable signatures for shapes using persistence. Computer Graphics Forum, 28(5), 1393–1403.
Cohignac, T., Lopez, C., & Morel, J. M. (1994). Integral and local affine invariant parameter and application to shape recognition, vol. 1. In Proceedings of the 12th IAPR International Conference on Pattern Recognition (ICPR) (pp. 164–168), October 1994.
Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5–30.
Coifman, R. R., Lafon, S., Lee, A. B., Maggioni, M., Nadler, B., Warner, F., et al. (2005). Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. PNAS, 102(21), 7426–7431.
Davies, R. H., Twining, C. J., Cootes, T. F., Waterton, J. C., & Taylor, C. J. (2002). A minimum description length approach to a minimum description length approach to statistical shape modeling. IEEE Transactions on Medical Imaging, 21(5), 525–537.
Do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall.
Dziuk, G. (1988). Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations (pp. 142–155).
Elad, A., & Kimmel, R. (2001). Bending invariant representations for surfaces. In Proceedings of Computer Vision and Pattern Recognition (CVPR) (pp. 168–174).
Fletcher, P. T., Joshi, S., Lu, C., & Pizer, S. (2003). Gaussian distributions on Lie groups and their application to statistical shape analysis. In Proceedings of Information Processing in Medical Imaging (IPMI) (pp. 450–462).
Gray, A., Abbena, E., & Salamon, S. (2006). Modern differential geometry of curves and surfaces with mathematica (3rd ed.). Boca Raton, FL: CRC Press.
Hamza, A. B., & Krim, H. (2006). Geodesic matching of triangulated surfaces. IEEE Transactions on Image Processing, 15(8), 2249–2258.
Huang, H., Shen, L., Zhang, R., Makedon, F., Hettleman, B., & Pearlman, J. D. (2005). Surface alignment of 3D spherical harmonic models: Application to cardiac MRI analysis. In Proceedings of Medical Image Computing and Computer Assisted Intervention (MICCAI).
Kimmel, R. (1996). Affine differential signatures for gray level images of planar shapes, vol. 1. In IEEE Proceedings of the 13th International Conference on Pattern Recognition (pp. 45–49). Vienna, Austria: IEEE, 25–30 August 1996.
Kovnatsky, A., Bronstein, M. M., Raviv, D., Bronstein, A. M., & Kimmel, R. (2012). Affine-invariant photometric heat kernel signatures. In Proceedings of Eurographics workshop on 3D object retrieval (3DOR).
Ling, H., & Jacobs, D. W. (2005). Using the inner-distance for classification of articulated shapes, vol. 2. In Proceedings of Computer Vision and Pattern Recognition (CVPR) (pp. 719–726), San Diego, USA, 20–26 June 2005.
Lipman, Y., & Funkhouser, T. (2009). Möbius voting for surface correspondence, vol. 28. In Proceedings of ACM Transactions on Graphics (SIGGRAPH).
Lowe, D. (2004). Distinctive image features from scale-invariant keypoint. International Journal of Computer Vision (IJCV), 60(2), 91–110.
Mémoli, F., & Sapiro, G. (2005). A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics, 5(3), 313–347.
Meyer, M., Desbrun, M., Schroder, P., & Barr, A. H. (2003). Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and Mathematics, III, 35–57.
Moons, T., Pauwels, E., Van Gool, L. J., & Oosterlinck, A. (1995). Foundations of semi-differential invariants. International Journal of Computer Vision (IJCV), 14(1), 25–48.
Morel, J. M., & Yu, G. (2009). ASIFT: A new framework for fully affine invariant image comparison. SIAM Journal on Imaging Sciences, 2, 438–469.
Olver, P. J. (1999). Joint invariant signatures. Foundations of Computational Mathematics, 1, 3–67.
Olver, P. J. (2005). A survey of moving frames. In H. Li, P. J. Olver, & G. Sommer (Eds.), Computer algebra and geometric algebra with applications (pp. 105–138)., LNCS 3519 New York: Springer.
Ovsjanikov, M., Bronstein, A. M., Bronstein, M. M., & Guibas, L. J. (2009). Shape Google: A computer vision approach to invariant shape retrieval. In Proceedings of Non-Rigid Shape Analysis and Deformable Image Alignment (NORDIA).
Ovsjanikov, M., Mérigot, Q., Mémoli, F., & Guibas, L. J. (2010). One point isometric matching with the heat kernel, vol. 29. In Proceedings of Symposium on Geometry Processing (SGP) (pp. 1555–1564).
Pauwels, E., Moons, T., Van Gool, L. J., Kempenaers, P., & Oosterlinck, A. (1995). Recognition of planar shapes under affine distortion. International Journal of Computer Vision (IJCV), 14(1), 49–65.
Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision (JMIV), 25(1), 127–154.
Polthier, K., & Schmies, M. (1998). Straightest geodesics on polyhedral surfaces. In Mathematical visualization (pp. 135–150). Heidelberg: Springer.
Qiu, H., & Hancock, E. R. (2007). Clustering and embedding using commute times. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(11), 1873–1890.
Raviv, D., Bronstein, A. M., Bronstein, M. M., Kimmel, R., & Sochen, N. (2011a). Affine-invariant diffusion geometry of deformable 3D shapes. In Proceedings of Computer Vision and Pattern Recognition (CVPR).
Raviv, D., Bronstein, A. M., Bronstein, M. M., Kimmel, R., & Sochen, N. (2011b). Affine-invariant geodesic geometry of deformable 3D shapes. Computers & Graphics, 35(3), 692–697.
Raviv, D., Bronstein, A. M., Bronstein, M. M., Waisman, D., Sochen, N., & Kimmel, R. (2013). Equi-affine invariant geometry for shape analysis. Journal of Mathematical Imaging and Vision (JMIV).
Reuter, M., Rosas, H. D., & Fischl, B. (2010). Highly accurate inverse consistent registration: A robust approach. Neuroimage, 53(4), 1181–1196.
Rugis, J., & Klette, R. (2006). A scale invariant surface curvature estimator, vol. 4319. In Advances in Image and Video Technology, First Pacific Rim Symposium (PSIVT) (pp. 138–147).
Rustamov, R. (2007). Laplace–Beltrami eigenfunctions for deformation invariant shape representation. In Proceedings of Symposium on Geometry Processing (SGP) (pp. 225–233).
Sapiro, G. (1993). Affine Invariant Shape Evolutions. PhD thesis, Technion-IIT.
Su, B. (1983). Affine differential geometry. Beijing: Science Press.
Sun, J., Ovsjanikov, M., & Guibas, L. J. (2009). A concise and provably informative multi-scale signature based on heat diffusion. In Proceedings of Symposium on Geometry Processing (SGP).
Van Gool, L., Brill, M., Barrett, E., Moons, T., & Pauwels, E. J. (1992a). Semi-differential invariants for nonplanar curves. In J. Mundy & A. Zisserman (Eds.), Geometric invariance in computer vision, chap. 11 (pp. 293–309). Cambridge, MA: MIT Press.
Van Gool, L., Moons, T., Pauwels, E. J., & Oosterlinck, A. (1992b). Semi-differential invariants. In A. Zisserman & J. Mundy (Eds.), Geometric invariance in computer vision, Chap. 8. Cambridge, MA: MIT Press.
Wang, Y., Gupta, M., Zhang, S., Wang, S., Gu, X., Samaras, D., et al. (2008). High resolution tracking of non-rigid motion of densely sampled 3D data using harmonic maps. International Journal of Computer Vision (IJCV), 76(3), 283–300.
Weiss, I. (1988). Projective invariants of shapes. Technical Report CARTR-339, Center for Automation, University of Maryland, January 1988.
Acknowledgments
We thank the editor and the reviewers for their valuable comments that helped us improve the presentation and writeup of the paper. This research was supported by the Office of Naval Research (ONR) award number N00014-12-1-0517 and by Israel Science Foundation (ISF) Grant Number 1031/12.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Schnörr.
Rights and permissions
About this article
Cite this article
Raviv, D., Kimmel, R. Affine Invariant Geometry for Non-rigid Shapes. Int J Comput Vis 111, 1–11 (2015). https://doi.org/10.1007/s11263-014-0728-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-014-0728-2