Abstract
In this note, we present a method for flattening anatomical surfaces such as branched vessels and intestinal tracts in an area-preserving way. This method is based on the theory of optimal mass transport and conformal mapping of surfaces. The flattened representations differ minimally from conformality in a certain precise sense. Potential applications include the detection and visualization of pathologies such as stenoses and polyps.
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Angenent, S., Haker, S., Tannenbaum, A.: Minimizing flows for the Monge-Kantorovich problem. To appear in SIAM J. Math. Analysis (2003)
Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On the Laplace-Beltrami operator and brain surface flattening. IEEE Trans. on Medical Imaging 18, 700–711 (1999)
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Com. Pure Appl. Math. 64, 375–417 (1991)
Carman, G.J., Drury, H.A., Van Essen, D.C.: Computational methods for reconstructing and unfolding the cerebral cortex. Cerb. Cortex 5(6), 506–517 (1995)
Gangbo, W., McCann, R.: The geometry of optimal transportation. Acta. Math. 177, 113–161 (1996)
Gangbo, W.: An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mechanics Anal. 128, 381–399 (1994)
Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R.: Nondistorting flattening maps and the 3D visualization of colon CT images. IEEE Trans. on Medical Imaging 19, 665–670 (2000)
Haker, S., Tannenbaum, A.: Mass-preserving maps for registration and visual tracking. IEEE proceeding on Decision and Control, 4812–4817 (2000)
Kantorovich, L.V.: On a problem of Monge. Uspekhi Mat. Nauk 3, 225–226 (1948)
Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965)
Nehari, Z.: Conformal mapping. Dover Publications, New York (1975)
Paik, D., Beaulieu, C., Jeffrey, R., Karadi, C., Napel, S.: Visualization modes for CT colonography using cylindrical and planar map projections. J. Comput. Assist. Tomogr. 24(2), 179–188 (2000)
Schwartz, E., Shaw, A., Wolfson, E.: A numerical solution to the generalized mapmaker’s problem: flattening noncovex polyhedral surfaces. IEEE Trans. Pattern Anal. Machine Intell. 11, 1005–1008 (1989)
Wandell, B., Engel, S., Hel-Or, H.: Creating images of the flattened cortical sheet. Invest. Opth. and Vis. Sci. 36(S612) (1996)
Wang, G., Dave, S., Brown, B., Zhang, Z., mcFarland, E., Haler, J., Vannier, M.: Colon unraveling based on electrical field – Recent progress and further work. In: SPIE Proc. 3660 (1999)
Zhu, L., Haker, S., Tannenbaum, A., Bouix, S., Siddiqi, K.: Angle-preserving mappings for the visualization of multi-branched vessels. In: Proc. Image Processing 2002, vol. 2, pp. 945–948 (2002)
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Zhu, L., Haker, S., Tannenbaum, A. (2003). Area-Preserving Mappings for the Visualization of Medical Structures. In: Ellis, R.E., Peters, T.M. (eds) Medical Image Computing and Computer-Assisted Intervention - MICCAI 2003. MICCAI 2003. Lecture Notes in Computer Science, vol 2879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39903-2_35
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DOI: https://doi.org/10.1007/978-3-540-39903-2_35
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