Abstract
Large-scale deformations of a tubular object, or generalized cylinder, are often defined by a target shape for its center curve, typically using a parametric target curve. This task is non-trivial for free-form deformations or direct manipulation methods because it is hard to manually control the centerline by adjusting control points. Most skeleton-based methods are no better, again due to the small number of manually adjusted control points. In this paper, we propose a method to deform a generalized cylinder based on its skeleton composed of a centerline and orthogonal cross sections. Although we are not the first to use such a skeleton, we propose a novel skeletonization method that tries to minimize the number of intersections between neighboring cross sections by means of a relative curvature condition to detect intersections. The mesh deformation is first defined geometrically by deforming the centerline and mapping the cross sections. Rotation minimizing frames are used during mapping to control twisting. Secondly, given displacements on the cross sections, the deformation is decomposed into finely subdivided regions. We limit distortion at these vertices by minimizing an elastic thin shell bending energy, in linear time. Our method can handle complicated generalized cylinders such as the human colon.
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Kim, M.-S.; Park, E.-J.; Lee, H.-Y. Modelling and animation of generalized cylinders with variable radius offset space curves. The Journal of Visualization and Computer Animation Vol. 5, 189–207, 1994.
Ballard, D. H.; Brown, C. M. Computer Vision. Prentice-Hall, Inc., 1982.
Shani, U.; Ballard, D. H. Splines as embeddings for generalized cylinders. Computer Vision, Graphics, and Image Processing Vol. 27, No. 2, 129–156, 1984.
Chang, T.-I.; Lee, J.-H.; Kim, M.-S.; Hong, S. J. Direct manipulation of generalized cylinders based on B-spline motion. The Visual Computer Vol. 14, Nos. 5–6, 228–239, 1998.
Jüttler, B.; Wagner, M. G. Computer-aided design with spatial rational B-spline motions. Journal of Mechanical Design Vol. 118, No. 2, 193–201, 1996.
O’Donnell, T.; Boult, T. E.; Fang, X.-S.; Gupta, A. The extruded generalized cylinder: A deformable model for object recovery. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 174–181, 1994.
Sederberg, T. W.; Parry, S. R. Free-form deformation of solid geometric models. ACM SIGGRAPH Computer Graphics Vol. 20, No. 4, 151–160, 1986.
Hsu, W. M.; Hughes, J. F.; Kaufman, H. Direct manipulation of free-form deformations. In: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, 177–184, 1992.
Sorkine, O.; Cohen-Or, D.; Lipman, Y.; Alexa, M.; Rössl, C.; Seidel, H.-P. Laplacian surface editing. In: Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, 175–184, 2004.
Yu, Y.; Zhou, K.; Xu, D.; Shi, X.; Bao, H.; Guo, B.; Shum, H.-Y. Mesh editing with Poisson-based gradient field manipulation. In: Proceedings of the ACM SIGGRAPH 2004 Papers, 644–651, 2004.
Sorkine, O. Laplacian mesh processing. In: Proceedings of the Eurographics 2005: State of the Art Reports, 53–70, 2005.
Zhou, K.; Huang, J.; Snyder, J.; Liu, X.; Bao, H.; Guo, B.; Shum, H.-Y. Large mesh deformation using the volumetric graph Laplacian. In: Proceedings of the ACM SIGGRAPH 2005 Papers, 496–503, 2005.
Sumner, R. W.; Popović, J. Deformation transfer for triangle meshes. In: Proceedings of the ACM SIGGRAPH 2004 Papers, 399–405, 2004.
Sorkine, O.; Alexa, M. As-rigid-as-possible surface modeling. In: Proceedings of the 5th Eurographics Symposium on Geometry Processing, 109–116, 2007.
Xu, W.-W.; Zhou, K. Gradient domain mesh deformation—A survey. Journal of Computer Science and Technology Vol. 24, No. 1, 6–18, 2009.
Botsch, M.; Sorkine, O. On linear variational surface deformation methods. IEEE Transactions on Visualization and Computer Graphics Vol. 14, No. 1, 213–230, 2008.
Zhang, S.; Huang, J.; Metaxas, D. N. Robust mesh editing using Laplacian coordinates. Graphical Models Vol. 73, No. 1, 10–19, 2011.
Kavan, L.; Collins, S.; Žára, J.; O’Sullivan, C. Skinning with dual quaternions. In: Proceedings of the Symposium on Interactive 3D Graphics and Games, 39–46, 2007.
Kavan, L.; Sorkine, O. Elasticity-inspired deformers for character articulation. ACM Transactions on Graphics Vol. 31, No. 6, Article No. 196, 2012.
Rohmer, D.; Hahmann, S.; Cani, M.-P. Local volume preservation for skinned characters. Computer Graphics Forum Vol. 27, No. 7, 1919–1927, 2008.
Kho, Y.; Garland, M. Sketching mesh deformations. In: Proceedings of the Symposium on Interactive 3D Graphics and Games, 147–154, 2005.
Yoshizawa, S.; Belyaev, A. G.; Seidel, H.-P. Skeletonbased variational mesh deformations. Computer Graphics Forum Vol. 26, No. 3, 255–264, 2007.
Aujay, G.; Hétroy, F.; Lazarus, F.; Depraz, C. Harmonic skeleton for realistic character animation. In: Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 151–160, 2007.
Antiga, L.; Ene-Iordache, B.; Remuzzi, A. Centerline computation and geometric analysis of branching tubular surfaces with application to blood vessel modeling. In: Proceedings of the 11th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision, 13–16, 2003.
Zeng, W.; Marino, J.; Gurijala, K. C.; Gu, X.; Kaufman, A. Supine and prone colon registration using quasiconformal mapping. IEEE Transactions on Visualization and Computer Graphics Vol. 16, No. 6, 1348–1357, 2010.
Zhou, Y.; Yin, K.; Huang, H.; Zhang, H.; Gong, M.; Cohen-Or, D. Generalized cylinder decomposition. ACM Transactions on Graphics Vol. 34, No. 6, Article No. 171, 2015.
Damon, J. Swept regions and surfaces: Modeling and volumetric properties. Theoretical Computer Science Vol. 392, Nos. 1–3, 66–91, 2008.
Wang, W.; Jüttler, B.; Zheng, D.; Liu, Y. Computation of rotation minimizing frames. ACM Transactions on Graphics Vol. 27, No. 1, Article No. 2, 2008.
Bergou, M.; Wardetzky, M.; Harmon, D.; Zorin, D.; Grinspun, E. A quadratic bending model for inextensible surfaces. In: Proceedings of the Eurographics Symposium on Geometry Processing, 227–230, 2006.
Zhao, Q.; Price, T.; Pizer, S.; Niethammer, M.; Alterovitz, R.; Rosenman, J. Surface registration in the presence of missing patches and topology change. In: Proceedings of the Medical Image Understanding and Analysis Conference, 2015.
Ma, R.; Zhao, Q.; Wang, R.; Damon, J.; Rosenman, J.; Pizer, S. Skeleton-based generalized cylinder deformation under the relative curvature condition. In: Proceedings of the Pacific Graphics, 2018.
Yoshizawa, S.; Belyaev, A. G.; Seidel, H.-P. Free-form skeleton-driven mesh deformations. In: Proceedings of the 8th ACM Symposium on Solid Modeling and Applications, 247–253, 2003.
Shi, X.; Zhou, K.; Tong, Y.; Desbrun, M.; Bao, H.; Guo, B. Mesh puppetry: Cascading optimization of mesh deformation with inverse kinematics. In: Proceedings of the ACM SIGGRAPH 2007 Papers, Article No. 81, 2007.
Lewis, J. P.; Cordner, M.; Fong, N. Pose space deformation: A unified approach to shape interpolation and skeleton-driven deformation. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, 165–172, 2000.
Weber, O.; Sorkine, O.; Lipman, Y.; Gotsman, C. Context-aware skeletal shape deformation. Computer Graphics Forum Vol. 26, No. 3, 265–273, 2007.
Wei, Z.; Rossignac, J. Fleshing: Spine-driven bending with local volume preservation. Computer Graphics Forum Vol. 32, No. 2pt3, 295–304, 2013.
Angelidis, A.; Singh, K. Kinodynamic skinning using volume-preserving deformations. In: Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 129–140, 2007.
Vaillant, R.; Barthe, L.; Guennebaud, G.; Cani, M.-P.; Rohmer, D.; Wyvill, B.; Gourmel, O.; Paulin, M. Implicit skinning: Real-time skin deformation with contact modeling. ACM Transactions on Graphics Vol. 32, No. 4, Article No. 125, 2013.
Verroust, A.; Lazarus, F. Extracting skeletal curves from 3D scattered data. The Visual Computer Vol. 16, No. 1, 15–25, 2000.
Mortara, M.; Patanè, G. Shape-covering for skeleton extraction. International Journal of Shape Modeling Vol. 8, No. 2, 139–158, 2002.
Chuang, J.-H.; Tsai, C.-H.; Ko, M.-C. Skeletonisation of three-dimensional object using generalized potential field. IEEE Transactions on Pattern Analysis and Machine Intelligence Vol. 22, No. 11, 1241–1251, 2000.
Antiga, L.; Ene-Iordache, B.; Remuzzi, A. Computational geometry for patient-specific reconstruction and meshing of blood vessels from MR and CT angiography. IEEE Transactions on Medical Imaging Vol. 22, No. 5, 674–684, 2003.
Antiga, L.; Piccinelli, M.; Botti, L.; Ene-Irodache, B.; Remuzzi, A.; Steinman, D. An image-based modeling framework for patient-specific computational hemodynamics. Medical & Biological Engineering & Computing Vol. 46, 1097–1112, 2008.
Wang, G.; McFarland, E. G.; Brown, B. P.; Zhang, Z.; Vannier, M. Curved cross-section based system and method for gastrointestinal tract unraveling. U.S. Patent 6,212,420. 2001.
Damon, J. Lorentzian geodesic flows and interpolation between hypersurfaces in Euclidean spaces. Preliminary preprint in MIDAG paper collection. 2018.
Hong, W.; Gu, X.; Qiu, F.; Jin, M.; Kaufman, A. Conformal virtual colon flattening. In: Proceedings of the ACM Symposium on Solid and Physical Modeling, 85–93, 2006.
Halier, S.; Angenent, S.; Tannenbaum, A.; Kikinis, R. Nondistorting flattening maps and the 3-D visualization of colon CT images. IEEE Transactions on Medical Imaging Vol. 19, No. 7, 665–670, 2000.
Peng, J.; Kristjansson, D.; Zorin, D. Interactive modeling of topologically complex geometric detail. In: Proceedings of the ACM SIGGRAPH 2004 Papers, 635–643, 2004.
Gingold, Y.; Secord, A.; Han, J. Y.; Grinspun, E.; Zorin, D. A discrete model for inelastic deformation of thin shells. Technical Report. Courant Institute of Mathematical Sciences, 2004.
Grinspun, E. A discrete model of thin shells. In: Proceedings of the ACM SIGGRAPH 2005 Courses, Article No. 4, 2005.
Meyer, M.; Desbrun, M.; Schröder, P.; Barr, A. H. Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III. Mathematics and Visualization. Hege, H. C.; Polthier, K. Eds. Springer Berlin Heidelberg, 35–57, 2003.
Wardetzky, M.; Mathur, S.; Kälberer, F.; Grinspun, E. Discrete Laplace operators: No free lunch. In: Proceedings of the 5th Eurographics Symposium on Geometry Processing, 33–37, 2007.
Nadeem, S.; Marino, J.; Gu, X.; Kaufman, A. Corresponding supine and prone colon visualization using eigenfunction analysis and fold modeling. IEEE Transactions on Visualization and Computer Graphics Vol. 23, No. 1, 751–760, 2017.
Nadeem, S.; Gu, X. D.; Kaufman, A. E. LMap: Shapepreserving local mappings for biomedical visualization. IEEE Transactions on Visualization and Computer Graphics Vol. 24, No. 12, 3111–3122, 2018.
Wang, H.; Li, L.; Han, H.; Shi, R.; Song, B.; Peng, H.; Liu, Y.; Gu, X.; Wang, Y.; Liang, Z. A 2.5D colon wall flattening model for CT-based virtual colonoscopy. In: Machine Learning in Medical Imaging. Lecture Notes in Computer Science, Vol, 8184. Wu, G.; Zhang, D.; Shen, D.; Yan, P.; Suzuki, K.; Wang, F. Eds. Springer Cham, 203–210, 2013.
Marino, J.; Zeng, W.; Gu, X.; Kaufman, A. Context preserving maps of tubular structures. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 12, 1997–2004, 2011.
Marino, J.; Kaufman, A. Planar visualization of treelike structures. IEEE Transactions on Visualization and Computer Graphics Vol. 22, No. 1, 906–915, 2016.
Xu, X.; Reinhardt, J. M.; Hu, Q.; Bakall, B.; Tlucek, P. S.; Bertelsen, G.; Abràmoff, M. D. Retinal vessel width measurement at branchings using an improved electric field theory-based graph approach. PLoS ONE Vol. 7, No. 11, e49668, 2012.
Mortara, M.; Patané, G.; Spagnuolo, M.; Falcidieno, B.; Rossignac, J. Blowing bubbles for multi-scale analysis and decomposition of triangle meshes. Algorithmica Vol. 38, No. 1, 227–248, 2004.
Mortara, M.; Patané, G.; Spagnuolo, M.; Falcidieno, B.; Rossignac, J. Plumber: A method for a multi-scale decomposition of 3D shapes into tubular primitives and bodies. In: Proceedings of the 9th ACM Symposium on Solid Modeling and Applications, 339–344, 2004.
Sangalli, L. M.; Secchi, P.; Vantini, S.; Veneziani, A. Efficient estimation of three-dimensional curves and their derivatives by free-knot regression splines, applied to the analysis of inner carotid artery centrelines. Journal of the Royal Statistical Society: Series C (Applied Statistics) Vol. 58, No. 3, 285–306, 2009.
Wampler, K. Fast and reliable example-based mesh IK for stylized deformations. ACM Transactions on Graphics Vol. 35, No. 6, Article No. 235, 2016.
Gao, L.; Lai, Y.-K.; Liang, D.; Chen, S.-Y.; Xia, S. Efficient and flexible deformation representation for data-driven surface modeling. ACM Transactions on Graphics Vol. 35, No. 5, Article No. 158, 2016.
Acknowledgements
We gratefully thank Dr. Saad Nadeem and Dr. Arie Kaufman from Stony Brook University and Dr. Sarah McGill from UNC Medical School for sharing their results, data, and suggestions. This work was supported by National Institutes of Health grant R01 CA158925.
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Ruibin Ma is a Ph.D. candidate in the Department of Computer Science, University of North Carolina at Chapel Hill. His research focuses on medical image analysis and computer vision. He received his bachelor degree from the Department of Electronic Engineering, Tsinghua University, in 2016.
Qingyu Zhao received his Ph.D. degree from the Computer Science Department at the University of North Carolina at Chapel Hill in 2017. He is currently a postdoctoral research fellow in the Department of Psychiatry and Behavioral Sciences, Stanford University. His research focuses on geometric modeling, non-linear statistics, and machine learning from medical images, with emphasis on neuroimaging applications.
Rui Wang received his bachelor and master degrees in computer science from the University of Missouri at Columbia in 2010 and 2012, respectively. He is currently a Ph.D. candidate in the Department of Computer Science, University of North Carolina at Chapel Hill. His primary research is on the combination of 3D vision and deep learning, and on their applications in endoscopic image analysis.
James Damon received his Ph.D. degreee in mathematics from Harvard University in 1972. He is an Emeritus Professor in Mathematics at the University of North Carolina at Chapel Hill, and a Fellow of the Amer. Math. Soc. He has also been a Simons Fellow, a Fulbright Lecturer (Santiago, Chile), an SERC Senior Visiting Fellow (Univ. Liverpool), and a Fulbright Scholar (Math. Inst. of Univ. Warwick). His areas of interest include singularity theory and its applications to bifurcation theory, computer imaging, and shape and position modeling for medical imaging.
Julian Rosenman is a Professor of Radiation Oncology at the University of North Carolina at Chapel Hill, and a cancer doctor (M.D.) who has treated cancer patients there since 1981. Before going to medical school, he received his Ph.D. degree from the University of Texas in Theoretical Physics. He has had an adjunct appointment in Computer Science since 1984. His major interest is in medical image analysis and display to improve cancer diagnosis and treatment. His major academic accomplishment was as part of a small team that developed and employed a fully 3D radiation treatment planning system that has been widely copied by commercial systems. 3D treatment planning is now standard-of-care around the world.
Stephen Pizer is a Kenan Professor at the University of North Carolina at Chapel Hill in Computer Science, Radiation Oncology, Biomedical Engineering, and Radiology and also has active collaborations in Psychiatry there. At UNC since 1967, he was also co-founder of Morphormics, Inc. (now part of Accuray), which specializes in software for segmentation and registration for radiation oncology. His research, centered since 1962 on medical image computing and display, presently focuses on image and object shape statistics, segmentation, registration, and 3D visualization of medical images. Other research directions have included interactive 3D graphics, human vision, image quality analysis, contrast enhancement, and image restoration. He is co-author of a book on medial representations and author of 2 books on numerical computing, as well as over 360 published chapters and journal and proceedings articles. He is a Fellow of the MICCAI Society.
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Ma, R., Zhao, Q., Wang, R. et al. Deforming generalized cylinders without self-intersection by means of a parametric center curve. Comp. Visual Media 4, 305–321 (2018). https://doi.org/10.1007/s41095-018-0127-7
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DOI: https://doi.org/10.1007/s41095-018-0127-7