Geodesic Warps by Conformal Mappings
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications in e.g. medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D’Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphisms, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps composed of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations.
- Arnold, V. I., & Khesin, B. A. (1998). Topological methods in hydrodynamics. Volume 125 of applied mathematical sciences. New York: Springer.
- Beg, M. (2003). Variational and computational methods for flows of diffeomorphisms in image matching and growth in computational anatomy. PhD thesis, John Hopkins University.
- Beg, M. F., Miller, M. I., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61, 139–157.
- Bruveris, M., Gay-Balmaz, F., Holm, D. D., & Ratiu, T. S. (2011). The momentum map representation of images. Journal of Nonlinear Science, 21, 115–150.
- Dupuis, P., & Grenander, U. (1998). Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, LVI, 587–600.
- Ebin, D. G., & Marsden, J. E. (1970). Groups of diffeomorphisms and the notion of an incompressible fluid. Annals of Mathematics, 92, 102–163.
- Gay-Balmaz, F., Marsden, J., & Ratiu, T. (2012). Reduced variational formulations in free boundary continuum mechanics. Journal of Nonlinear Science, 22, 463–497.
- Hamilton, R. S. (1982). The inverse function theorem of Nash and Moser. Bulletin of the American Mathematical Society (New Series), 7, 65–222.
- Holm, D. D., & Marsden, J. E. (2005). Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation. In The breadth of symplectic and Poisson geometry. Progress in Mathematics (Vol. 232, pp. 203–235). Boston, MA: Birkhäuser.
- Joshi, S., & Miller, M. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing, 9, 1357–1370.
- Khesin, B., & Wendt, R. (2009). The Geometry of Infinite-dimensional Groups. Volume 51 of a series of modern surveys in mathematics. Berlin: Springer.
- Lang, S. (1999). Fundamentals of differential geometry. Volume 191 of Graduate texts in mathematics. New York: Springer. CrossRef
- Marsden, J. E., & West, M. (2001). Discrete mechanics and variational integrators. Acta Numerica, 10, 357–514. CrossRef
- Marsland, S., McLachlan, R.I., Modin, K., & Perlmutter, M. (2011a). On a geodesic equation for planar conformal template matching. In Proceedings of the 3rd MICCAI workshop on mathematical foundations of computational anatomy (MFCA’11), Toronto.
- Marsland, S., McLachlan, R.I., Modin, K., & Perlmutter, M. (2011b). Application of the hodge decomposition to conformal variational problems. arXiv:1203.4464v1 [math.DG].
- Michor, P. W., & Mumford, D. (2006). Riemannian geometries on spaces of plane curves. Journal of European Mathematical Society (JEMS), 8, 1–48.
- Miller, M. I., & Younes, L. (2001). Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision, 41, 61–84.
- Modin, K., Perlmutter, M., Marsland, S., & McLachlan, R. I. (2011). On Euler–Arnold equations and totally geodesic subgroups. Journal of Geometry and Physics, 61, 1446–1461.
- Sharon, E., & Mumford, D. (2006). 2D-shape analysis using conformal mapping. International Journal of Computer Vision, 70, 55–75.
- Shkoller, S. (1998). Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics. Journal of Functional Analysis, 160, 337–365.
- Thompson, D. (1942). On growth and form. New York: Cambridge University Press.
- Trouvé, A. (1995). An infinite dimensional group approach for physics based models in patterns recognition. Technical report, Ecole Normale Supérieure.
- Trouvé, A. (1998). Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision, 28, 213–221.
- Wallace, A. (2006). D’Arcy Thompson and the theory of transformations. Nature Reviews Genetics, 7, 401–406.
- Younes, L. (2010). Shapes and diffeomorphisms. Applied mathematical sciences. New York: Springer. CrossRef
- Geodesic Warps by Conformal Mappings
International Journal of Computer Vision
Volume 105, Issue 2 , pp 144-154
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
- Additional Links
- Image registration
- Conformal mappings
- Infinite dimensional manifolds
- Geodesic warps
- Industry Sectors
- Author Affiliations
- 1. School of Engineering and Advanced Technology (SEAT), Massey University, Palmerston North, New Zealand
- 2. Institute of Fundamental Sciences (IFS), Massey University, Private Bag 11222, Palmerston North, New Zealand