@CONFERENCE\{IMM2010-05868, author = "M. S. Hansen and R. Larsen and N. V. Christensen", title = "{CURL-GRADIENT} {IMAGE} {WARPING} - Introducing Deformation Potentials for Medical Image Registration using Helmholtz Decomposition", year = "2010", pages = "179-185", booktitle = "{VISAPP} 2009 - Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, Lisboa, Portugal,", volume = "1", series = "", editor = "", publisher = "", organization = "", address = "", url = "http://www2.compute.dtu.dk/pubdb/pubs/5868-full.html", abstract = "mage registration is becoming an increasingly important tool in medical image analysis, and the need to understand deformations within and between subjects often requires analysis of obtained deformation fields. The current paper presents a novel representation of the deformation field based on the Helmholtz decomposition of vector fields. The two decomposed potential fields form a curl free field and a divergence free field. The representation has already proven its worth in fluid modelling and electrostatics, and we show it also has desirable features in image registration and morphometry in particular. The potentials are shown to a offer decoupling of the two potential fields in both elastic and fluid image registration. For morphometry applications, we show that when decomposing the deformation field in symmetric and antisymmetric parts, the vector potential alone describes the vorticity, and the scalar gradient potential gives a first-order approximation to the determinant of the Jacobian. We provide some insight into the behavior of curl and divergence representation of the warp field by constructed examples and by a demonstration on real medical image data. Our theoretical findings are readily observable in our empirical experiment, which further illustrates the benefit of the parametrization." }