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International Journal of Computer Vision

, Volume 84, Issue 1, pp 63–79 | Cite as

A Moving Grid Framework for Geometric Deformable Models

  • Xiao Han
  • Chenyang Xu
  • Jerry L. PrinceEmail author
Article

Abstract

Geometric deformable models based on the level set method have become very popular in the last decade. To overcome an inherent limitation in accuracy while maintaining computational efficiency, adaptive grid techniques using local grid refinement have been developed for use with these models. This strategy, however, requires a very complex data structure, yields large numbers of contour points, and is inconsistent with the implementation of topology-preserving geometric deformable models (TGDMs). In this paper, we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models. In addition to the development of a consistent moving grid geometric deformable model framework, our main contributions include the introduction of a new grid nondegeneracy constraint, the design of a new grid adaptation criterion, and the development of novel numerical methods and an efficient implementation scheme. The overall method is simpler to implement than using grid refinement, requiring no large, complex, hierarchical data structures. It also offers an extra benefit of automatically reducing the number of contour vertices in the final results. After presenting the algorithm, we demonstrate its performance using both simulated and real images.

Keywords

Adaptive grid method Geometric deformable model Deformation moving grid Topology preservation Level set method 

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References

  1. Adalsteinsson, D., & Sethian, J. A. (1995). A fast level set method for propagating interfaces. Journal of Computational Physics, 118, 269–277. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bochev, P., Liao, G., & dela Pena, G. (1996). Analysis and computation of adaptive moving grids by deformation. Numerical Methods for Partial Differential Equations, 12, 489–506. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Cao, W., Huang, W., & Russell, R. D. (2002). A moving mesh method based on the geometric conservation law. SIAM Journal on Scientific Computing, 24, 118–142. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Caselles, V., Catte, F., Coll, T., & Dibos, F. (1993). A geometric model for active contours in image processing. Numerische Mathematik, 66, 1–31. zbMATHCrossRefMathSciNetGoogle Scholar
  5. Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. International Journal of Computer Vision, 22, 61–79. zbMATHCrossRefGoogle Scholar
  6. Cohen, L. D. (1991). On active contour models and balloons. CVGIP: Image Understanding, 53, 211–218. zbMATHCrossRefGoogle Scholar
  7. Droske, M., Meyer, B., Schaller, C., & Rumpf, M. (2001). An adaptive level set method for medical image segmentation. In M. F. Insana & R. M. Leahy (Eds.), LNCS: Vol. 2082. Proc. IPMI 2001 (pp. 416–422). Berlin: Springer. Google Scholar
  8. Han, X., Xu, C., & Prince, J. L. (2003a). A 2D moving grid geometric deformable model. In Proc. of CVPR (CVPR’03), Madison, Wisconsin (pp. I:153–160). June 2003. Google Scholar
  9. Han, X., Xu, C., & Prince, J. L. (2003b). Topology preserving geometric deformable models for brain reconstruction. In S. Osher & N. Paragios (Eds.), Geometric level set methods in imaging, vision and graphics. New York: Springer. Google Scholar
  10. Han, X., Xu, C., & Prince, J. L. (2003c). A topology preserving level set method for geometric deformable models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25, 755–768. CrossRefGoogle Scholar
  11. Han, X., Pham, D., Tosun, D., Rettmann, M., Xu, C., & Prince, J. L. (2004). CRUISE: Cortical reconstruction using implicit surface evolution. NeuroImage, 23, 997–1012. CrossRefGoogle Scholar
  12. Han, X., Xu, C., & Prince, J. L. (2006). A moving grid framework for geometric deformable models (ECE technical report). Johns Hopkins University, Baltimore, MD. http://iacl.ece.jhu.edu/~xhan/MMGDM.pdf.
  13. Ivanenko, S. A. (1999). Harmonic mappings. In J. F. Thompson & B. K. Soni (Eds.), Handbook of grid generation (pp. 81–43). Boca Raton: CRC Press. Google Scholar
  14. Kao, C. Y., Osher, S., & Tsai, Y. R. (2002). Fast sweeping methods for a class of static Hamilton-Jacobi equations (Technical Report UCLA-CAM-02-66). Institute for Pure and Applied Mathematics (IPAM), UCLA. Google Scholar
  15. Karaçali, B., & Davatzikos, C. (2006). Simulation of tissue atrophy using a topology preserving transformation model. IEEE Transactions on Medical Imaging, 25, 649–652. CrossRefGoogle Scholar
  16. Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes: Active contour models. International Journal of Computer Vision, 1, 312–333. Google Scholar
  17. Khoo, S., Wang, B., Lim, K., & Wang, M. (2007). An extended level set method for shape and topology optimization. Journal of Computational Physics, 221, 395–421. zbMATHCrossRefMathSciNetGoogle Scholar
  18. Knupp, P., & Steinberg, S. (1994). Fundamentals of grid generation. Boca Raton: CRC Press. zbMATHGoogle Scholar
  19. Kong, T. Y., & Rosenfeld, A. (1989). Digital topology: introduction and survey. CVGIP: Image Understanding, 48, 357–393. Google Scholar
  20. Le Guyader, C., & Vese, L. A. (2008). Self-repelling snakes for topology-preserving segmentation models. IEEE Transactions on Image Processing, 17, 767–779. CrossRefGoogle Scholar
  21. Liao, G., & Xue, J. (2006). Moving meshes by the deformation method. Journal of Computational and Applied Mathematics, 83–92. Google Scholar
  22. Liao, G., Pan, T., & Shu, J. (1994). Numerical grid generator based on Moser’s deformation method. Numerical Methods for Partial Differential Equations, 10, 21–31. zbMATHCrossRefMathSciNetGoogle Scholar
  23. Liao, G., de la Pena, G., & Liao, G. (1999). A deformation method for moving mesh generation. In Proc. 8th int. meshing roundtable, South Lake Tahoe, CA (pp. 155–162). Google Scholar
  24. Liao, G., Liu, F., de la Pena, G., Peng, D., & Osher, S. (2000). Level-set-based deformation methods for adaptive grids. Journal of Computational Physics, 159, 103–122. zbMATHCrossRefMathSciNetGoogle Scholar
  25. Lorensen, W. E., & Cline, H. E. (1987). Marching cubes: A high-resolution 3D surface construction algorithm. ACM Computer Graphics, 21(4), 163–170. CrossRefGoogle Scholar
  26. Malladi, R., Sethian, J. A., & Vemuri, B. C. (1995). Shape modeling with front propagation: A level set approach. IEEE Transactions on PAMI, 17, 158–175. Google Scholar
  27. Milne, B. (1995). Adaptive level set methods interfaces. Ph.D. dissertation, Dept. of Math., UC Berkeley. Google Scholar
  28. Oishi, T., Takamatsu, J., Zheng, B., Ishikawa, R., & Ikeuchi, K. (2008). 6-DOF pose estimation from single ultrasound image using 3D IP models. In Proc. IEEE comput. vision pattern recognit. workshop (pp. 1–8). Google Scholar
  29. Osher, S., & Shu, C. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. Journal of Computational Physics, 83, 32–78. zbMATHCrossRefMathSciNetGoogle Scholar
  30. Peng, D., Merriman, B., Osher, S., Zhao, H., & Kang, M. (1999). A PDE-based fast local level set method. Journal of Computational Physics, 155, 410–438. zbMATHCrossRefMathSciNetGoogle Scholar
  31. Press, W. A., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1995). Numerical recipes in C (2nd ed.). New York: Cambridge University Press. Google Scholar
  32. Ségonne, F. (2008). Active contours under topology control—genus preserving level sets. International Journal of Computer Vision, 79(2), 107–117. Google Scholar
  33. Sethian, J. A. (1999). Level set methods and fast marching methods (2nd ed.). Cambridge: Cambridge University Press. zbMATHGoogle Scholar
  34. Sethian, J. A., & Vladimirsky, A. (2001). Ordered upwind methods for static Hamilton-Jacobi equations. Proceedings of National Academy of Sciences, 98(20), 11069–11074. zbMATHCrossRefMathSciNetGoogle Scholar
  35. Siddiqi, K., Lauziere, Y. B., Tannenbaum, A., & Zucker, S. W. (1998). Area and length minimizing flow for shape segmentation. IEEE Transactions on Image Processing, 7, 433–443. CrossRefGoogle Scholar
  36. Strang, G. (1986). Introduction to applied mathematics. Cambridge: Wellesley Cambridge Press. zbMATHGoogle Scholar
  37. Sundaramoorthi, G., & Yezzi, A. (2007). Global regularizing flows with topology preservation for active contours and polygons. IEEE Transactions on Image Processing, 16, 803–812. CrossRefMathSciNetGoogle Scholar
  38. Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., & Welcome, M. L. (1999). An adaptive level set approach for incompressible two-phase flow. Journal of Computational Physics, 148, 81–124. zbMATHCrossRefMathSciNetGoogle Scholar
  39. Terzopoulos, D., & Vasilescu, M. (1991). Sampling and reconstruction with adaptive meshes. In Proc. CVPR’91, Lahaina, HI (pp. 70–75). Google Scholar
  40. Tsai, Y. R., Cheng, L., Osher, S., & Zhao, H. (2001). Fast sweeping algorithms for a class of Hamilton-Jacobi equations (Technical Report UCLA-CAM-01-27). Institute for Pure and Applied Mathematics (IPAM), UCLA. Google Scholar
  41. Ushakova, O. V. (2001). Conditions of nondegeneracy of three-dimensional cells. A formula of a volume of cells. SIAM Journal of Scientific Computation, 1274–1290. Google Scholar
  42. Vasilescu, M., & Terzopoulos, D. (1992). Adaptive meshes and shells. In Proc. CVPR’92, Champaign, IL (pp. 829–832). Google Scholar
  43. Xie, X., & Mirmehdi, M. (2007). Implicit active model using radial basis function interpolated level sets. In Proc. 17th British machine vision conf. (pp. 1040–1049). Google Scholar
  44. Xu, C., & Prince, J. L. (1998). Snakes, shapes, and gradient vector flow. IEEE Transactions on Image Processing, 7(3), 359–369. zbMATHCrossRefMathSciNetGoogle Scholar
  45. Xu, C., Yezzi, A., & Prince, J. L. (2001). On the relationship between parametric and geometric active contours. In The 34th Asilomar conference on signals, systems, and computers, Pacific Grove, USA (pp. 483–489). Google Scholar
  46. Xu, M., Thompson, P. M., & Toga, A. W. (2004). An adaptive level set segmentation on a triangulated mesh. IEEE Transactions on Medical Imaging, 23, 191–201. CrossRefGoogle Scholar
  47. Yezzi, A., Kichenassamy, S., Olver, P., & Tannenbaum, A. (1997). A geometric snake models for segmentation of medical imagery. IEEE Transactions on Medical Imaging, 16, 199–209. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.CMS Software, Elekta Inc.St. LouisUSA
  2. 2.Siemens Corporate ResearchPrincetonUSA
  3. 3.Center for Imaging Science, Department of Electrical and Computer EngineeringJohns Hopkins UniversityBaltimoreUSA

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