Abstract
In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.
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Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On the Laplace-Beltrami operator and brain surface flattening. IEEE Trans. Med. Imaging 18(8), 700–711 (1999)
Durrleman, S., Pennec, X., Trouve, A., Thompson, P., Ayache, N.: Measuring brain variability via sulcal lines registration: A diffeomorphic approach. In: Medical Image Computing and Computer-Assisted Intervention (MICCAI 2007). Lecture Notes in Comput. Sci., vol. 4791, pp. 675–682. Springer, Berlin (2007)
Durrleman, S., Pennec, X., Trouve, A., Thompson, P., Ayache, N.: Inferring brain variability from diffeomorphic deformations of currents: An integrative approach. Med. Image Anal. 12, 626–637 (2008)
Fischl, B., Sereno, M.I., Tootell, R.B., Dale, A.M.: High-resolution intersubject averaging and a coordinate system for the cortical surface. Hum. Brain Mapp. 8, 272–284 (1999)
Gardiner, F.: Quasiconformal Teichmüller Theory. American Mathematics Society, Providence (2000)
Glaunès, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. J. Math. Imaging Vis. 20, 179–200 (2004)
Gu, X., Wang, Y., Chan, T., Thompson, P., Yau, S.-T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23(8), 949–958 (2004)
Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph. 6(8), 181–189 (2000)
Hurdal, M., Stephenson, K.: Cortical cartography using the discrete conformal approach of circle packings. NeuroImage 23, S119–S128 (2004)
Hurdal, M.K., Stephenson, K.: Discrete conformal methods for cortical brain flattening. Neuroimage 45, 86–98 (2009)
Morra, Z.T.J.H., Apostolova, L., Green, A., Toga, A., Thompson, P.: Comparison of adaboost and support vector machines for detecting Alzheimer’s disease through automated hippocampal segmentation. IEEE Trans. Med. Imaging 29(1), 30–43 (2010)
Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface Ricci flow. IEEE Trans. Vis. Comput. Graph. 14(5), 1030–1043 (2008)
Joshi, S., Miller, M.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9(8), 1357–1370 (2000)
Ju, L., Stern, J., Rehm, K., Schaper, K., Hurdal, M., Rottenberg, D.: Cortical surface flattening using least square conformal mapping with minimal metric distortion. In: IEEE International Symposium on Biomedical Imaging, pp. 77–80 (2004)
Lehto, O., Virtanen, K.: Quasiconformal Conformal Mappings in the Plane. Springer, New York (1973)
Leow, A., Yu, C., Lee, S., Huang, S., Protas, H., Nicolson, R., Hayashi, K., Toga, A., Thompson, P.: Brain structural mapping using a novel hybrid implicit/explicit framework based on the level-set method. NeuroImage 24(3), 910–927 (2005)
Lepore, N., Leow, P.T.A.D.: Landmark matching on the sphere using distance functions. In: IEEE International Symposium on Biomedical Imaging (ISBI2006), April 6–9 (2006)
Lord, N.A., Ho, J., Vemuri, B., Eisenschenk, S.: Simultaneous registration and parcellation of bilateral hippocampal surface pairs for local asymmetry quantification. IEEE Trans. Med. Imaging 26(4), 471–478 (2007)
Lui, L., Wang, Y., Chan, T., Thompson, P.: Brain anatomical feature detection by solving partial differential equations on general manifolds. Discrete Contin. Dyn. Syst., Ser. B 7(3), 605–618 (2007)
Lui, L., Wang, Y., Chan, T., Thompson, P.: Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl. Numer. Math. 5, 847–858 (2007)
Lui, L.M., Thiruvenkadam, S., Wang, Y., Chan, T., Thompson, P.: Optimized conformal parameterization of cortical surfaces using shape based matching of landmark curves. SIAM J. Imaging Sci. 3(1), 52–78 (2010)
Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21(3), 362–371 (2002)
Lyttelton, O., Bouchera, M., Robbinsa, S., Evans, A.: An unbiased iterative group registration template for cortical surface analysis. NeuroImage 34, 1535–1544 (2007)
Morra, J., et al.: Automated mapping of hippocampal atrophy in 1-year repeat MRI data in 490 subjects with Alzheimer’s disease, mild cognitive impairment, and elderly controls. Neuroimage 45(1), S3–15 (2009)
Morra, J., Tu, Z., Apostolova, L., Green, A., Avedissian, C., Madsen, S., Parikshak, N., Toga, A., Jack, C., Schuff, N., Weiner, M., Thompson, P.: Automated 3d mapping of hippocampal atrophy and its clinical correlates in 400 subjects with Alzheimer’s disease, mild cognitive impairment and elderly controls. NeuroImage, Special Issue on Mathematics in Brain Imaging, published online, PMID: 19041724 (2008)
Morra, J., Tu, Z., Apostolova, L., Green, A., Avedissian, C., Madsen, S., Parikshak, N., Toga, A., Jack, C., Schuff, N., Weiner, M., Thompson, P.: Automated mapping of hippocampal atrophy in 1-year repeat MRI data in 490 subjects with Alzheimer’s disease, mild cognitive impairment and elderly controls. NeuroImage, Special Issue on Mathematics in Brain Imaging, published online, PMID: 19041724 (2008)
Morra, J., Tu, Z., Apostolova, L., Green, A., Avedissian, C., Madsen, S., Parikshak, N., Toga, A., Jack, C., Schuff, N., Weiner, M., Thompson, P.: Validation of a fully automated 3d hippocampal segmentation method using subjects with Alzheimer’s disease, mild cognitive impairment and elderly controls. NeuroImage 43(1), 59–68 (2008)
Morra, J., Tu, Z., Apostolova, L., Green, A., Avedissian, C., Madsen, S., Parikshak, N., Toga, A., Jack, C., Schuff, N., Weiner, M., Thompson, P.: Automated mapping of hippocampal atrophy in 1-year repeat MRI data in 490 subjects with Alzheimer’s disease, mild cognitive impairment and elderly controls. Hum. Brain Mapp. 30(9), 2766–2788 (2009)
Ogren, J., Bragin, A., Wilson, C., Hoftman, G., Lin, J., Dutton, R., Fields, T., Toga, A., Thompson, P., Engel, J., Staba, R.: 3D hippocampal atrophy maps distinguish two common temporal lobe onset seizure patterns. Epilepsia 50(6), 1361–1370 (2009)
Ogren, J., Wilson, C., Bragin, A., Lin, J., Salamon, N., Dutton, R., Luders, E., Fields, T., Fried, I., Toga, A., Thompson, P., Engel, J., Staba, R.: 3d surface maps link local atrophy and fast ripples in human epileptic hippocampus. Ann. Neurol. 66(6), 783–791 (2009)
Shi, Y., Thompson, P.M., Dinov, I., Osher, S., Toga, A.W.: Direct cortical mapping via solving partial differential equations on implicit surfaces. Med. Image Anal. 11(3), 207–223 (2007)
Thompson, P., Hayashi, K., Sowell, E., Gogtay, N., Giedd, J., Rapoport, J., de Zubicaray, G., Janke, A., Rose, S., Semple, J., Doddrell, D., Wang, Y., van Erp, T.C.T.G.M., Toga, A.: Mapping cortical change in Alzheimer’s disease, brain development, and schizophrenia. NeuroImage, Special Issue on Mathematics in Brain Imaging, Suppl 1:S2–18, 23 (2004)
Thompson, P.M., Hayashi, K.M., de Zubicaray, G.I., Janke, A.L., Rose, S.E., Semple, J., Hong, M.S., Herman, D.H., Gravano, D., Doddrell, D.M., Toga, A.W.: Mapping hippocampal and ventricular change in Alzheimer’s disease. NeuroImage 22(4), 1754–1766 (2004)
Tosun, D., Rettmann, M., Prince, J.: Mapping techniques for aligning sulci across multiple brains. Med. Image Anal. 8, 295–309 (2004)
Wang, Y., Chiang, M.-C., Thompson, P.M.: Automated surface matching using mutual information applied to Riemann surface structures. In: Proceeding in Medical Image Computing and Computer-Assisted Intervention—MICCAI 2005, vol. 2, pp. 666–674 (2005)
Wang, Y., Lui, L., Gu, X., Hayashi, K., Chan, T., Toga, A., Thompson, P., Yau, S.: Brain surface conformal parameterization using Riemann surface structure. IEEE Trans. Med. Imaging 26(6), 853–865 (2007)
Wang, Y., Lui, L.M., Chan, T.F., Thompson, P.M.: Optimization of brain conformal mapping with landmarks. In: Proceeding in Medical Image Computing and Computer-Assisted Intervention—MICCAI 2005, pp. 675–683 (2005)
Zeineh, M.M., Engel, S.A., Thompson, P.M., Bookheimer, S.Y.: Dynamics of the hippocampus during encoding and retrieval of face-name pairs. NeuroImage 299(5606), 577–580 (2003)
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An erratum to this article can be found at http://dx.doi.org/10.1007/s10915-011-9541-z.
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Lui, L.M., Wong, T.W., Zeng, W. et al. Optimization of Surface Registrations Using Beltrami Holomorphic Flow. J Sci Comput 50, 557–585 (2012). https://doi.org/10.1007/s10915-011-9506-2
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DOI: https://doi.org/10.1007/s10915-011-9506-2