Abstract
We define the space of affine shapes of k points in R n to be the topological quotient of (R n)k modulo the natural action of the affine group of R n. These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ k n in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh k n in detail. Each Sh k n is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for k>n, expressing the metric purely in terms of “pre-shape” data, i.e. configurations of k points in R n.
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References
Berthilsson, R., Astrom, K., Heyden, A.: Reconstruction of curves in R3, using factorization and bundle adjustment. In: The Proceedings of the Seventh IEEE International Conference on Computer Vision, pp. 674–679 (1999)
Bala, E., Cetin, A.E.: Computationally efficient wavelet affine invariant functions for shape recognition. IEEE Trans. Pattern Anal. Mach. Intell. 26(8), 1095–1099 (2004)
Betke, M., Makris, N.C.: Recognition, resolution, and complexity of objects subject to affine transformations. Int. J. Comput. Vis. 44(1), 5–40 (2001)
Berthilsson, R., Heyden, A., Sparr, G.: Recursive structure and motion from image sequences using shape and depth spaces. In: Proceedings IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 444–449 (1997)
Carne, T.K.: The geometry of shape spaces. Proc. Lond. Math. Soc. 16, 407–432 (1989)
Chui, H., Rangarajan, A.: A new point matching algorithm for non-rigid registration. Comput. Vis. Image Underst. 89(2–3), 114–141 (2003)
Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)
Faugeras, O.: Three Dimensional Computer Vision. MIT Press, Cambridge (1993)
Fletcher, P.T., Venkatasubramanian, S., Joshi, S.: Robust statistics on Riemannian manifolds via the geometric median. In: Proceedings CVPR (2008)
Hagedoorn, M., Veltkamp, R.C.: Reliable and efficient pattern matching using an affine invariant metric. Int. J. Comput. Vis. 31(2/3), 203–225 (1999)
Kendall, D.G.: Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)
Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley, Chichester (1999)
Klein, G.J., Reutter, R.W., Huesman, R.H.: Four-dimensional affine registration models for respiratory-gated PET. IEEE Trans. Nucl. Sci. 48, 756–760 (2001)
Lamdan, Y., Schwartz, J.T., Wolfson, H.J.: Affine invariance model based object recognition. IEEE Trans. Robot. Autom. 6, 578–589 (1990)
Mardia, K.V., Patrangenaru, V., Derado, G., Patrangenaru, V.P.: Reconstruction of planar scenes from multiple views using affine and projective shape manifolds. In: 2003 IEEE Workshop on Statistical Signal Processing, pp. 298–301, October 2003
Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton (1974)
Ness, L.: A stratification of the null cone via the moment map. Am. J. Math. 106, 1281–1329 (1984)
Patrangenaru, V., Mardia, K.V.: Affine shape analysis and image analysis. In: Aykroyd, R.G., Mardia, K.V., Langdon, M.J. (eds.) Stochastic Geometry, Biological Structure and Images, pp. 57–62 (2003)
Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)
Rueckert, D., Sonoda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Nonrigid registration using free-form deformations: application to breast MR images. IEEE Trans. Med. Imaging 18, 712–721 (1999)
Sepiashvili, D., Moura, J.M.F., Ha, V.H.S.: Affine permutation symmetry: invariance and shape space. In: 2003 IEEE Workshop on Statistical Signal Processing, pp. 307–310 (2003)
Small, C.G.: The Statistical Theory of Shape. Springer, New York (1996)
Sparr, G.: Simultaneous reconstruction of scene structure and camera locations from uncalibrated image sequences. In: Proceedings of the 13th International Conference on Pattern Recognition, vol. 1, pp. 328–333, August 1996
Triggs, B.: Autocalibration from planar scenes. In: Burkhardt, H., Neumann, B. (eds.) Proceedings of the Fifth European Conference on Computer Vision. Lecture Notes in Computer Science. Springer, Berlin (1998)
Tieng, Q.M., Boles, W.W.: Wavelet-based affine invariant representation: a tool for recognizing planar objects in 3D space. IEEE Trans. Pattern Anal. Mach. Intell. 19(8), 846–857 (1997)
Yang, Z., Cohen, F.S.: Image registration and object recognition using affine invariants and convex hulls. IEEE Trans. Image Process. 8(7), 934–946 (1999)
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Groisser, D., Tagare, H.D. On the Topology and Geometry of Spaces of Affine Shapes. J Math Imaging Vis 34, 222–233 (2009). https://doi.org/10.1007/s10851-009-0143-4
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DOI: https://doi.org/10.1007/s10851-009-0143-4