Abstract
Many different approaches have been proposed for segmenting vessels, or more generally tubular-like structures from 2D/3D images. In this work we propose to reconstruct the boundaries of 2D/3D tubular structures by continuously deforming an initial distance function following the Partial Differential Equation (PDE)-based diffusion model derived from a minimal volume-like variational formulation. The gradient flow for this functional leads to a non-linear curvature motion model. An anisotropic variant is provided which includes a diffusion tensor aimed to follow the tube geometry. Space discretization of the PDE model is obtained by finite volume schemes and semi-implicit approach is used in time/scale. The use of an efficient strategy to apply the linear system iterative solver allows us to reduce significantly the numerical effort by limiting the computation near the structure boundaries. We illustrate how the proposed method works to segment 2D/3D images of synthetic and medical real data representing branching tubular structures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Antiga, L., Ene-Iordache, B., Remuzzi, A.: Computational geometry for patient-specific reconstruction and meshing of blood vessels from MR and CT angiography. IEEE Transaction on Medical Imaging 22(5), 674–684 (2003)
Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proc. ICCV 1995, Cambridge, MA, pp. 694–699 (1995)
Cohen, L.D., Deschamps, T.: Segmentation of 3D tubular objects with adaptive front propagation and minimal tree extraction for 3D medical imaging. Computer Methods in Biomechanics and Biomedical Engineering 10(4), 289–305 (2007)
Corsaro, S., Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit co-volume method in 3D image segmentation. SIAM Journal on Scientific Computing 28(6), 2248–2265 (2006)
Dekanic, K., Loncari, S.: 3D vascular tree segmentation using level set deformable model. In: Proceeding of the 5th International Symposium on image and signal Processing and analysis (2007)
Deschamps, T., Schwartz, P., Trebotich, D., Colella, P., Saloner, D., Malladi, R.: Vessel segmentation and blood flow simulation using level set and embedded boundary method. International Congress Series, vol. 1268, pp. 75–80 (2004)
Drblikova, O., Mikula, K.: Semi-implicit Diamond-cell Finite volume Scheme for 3D Nonlinear Tensor Diffusion in Coherence Enhancing Image Filtering. In: Eymard, R., Herard, J.M. (eds.) Finite Volumes for Complex Applications V: Problems and Perspectives, ISTE and WILEY, London, pp. 343–350 (2008)
Eymard, R., Gallouet, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook for Numerical Analysis, vol. 7. Elsevier, Amsterdam (2000)
Franchini, E., Morigi, S., Sgallari, F.: Composed Segmentation of Tubular Structures by an Anisotropic PDE Model. In: Tai, X.-C., et al. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 75–86. Springer, Heidelberg (2009)
Gooya, A., Liao, H., et al.: A variational method for geometric regularization of vascular segmentation in medical images. IEEE Transaction on Image Processing 17(8), 1295–1312 (2008)
Handlovi cova, A., Mikula, K., Sgallari, F.: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math. 93, 675–695 (2003)
Hassan, H., Farag, A.A.: Cerebrovascular segmentation for MRA data using levels set. International Congress Series, vol. 1256, pp. 246–252 (2003)
Kirbas, C., Quek, F.: A review of vessel extraction techniques and algorithms. ACM Computing Surveys 36(2), 81–121 (2004)
Le Veque, R.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
Meijering, E., Niessen, W., Weickert, J., Viergever, M.: Diffusion-enhanced visualization and quantification of vascular anomalies in three-dimensional rotational angiography: Results of an in-vitro evaluation. Medical Image Analysis 6(3), 215–233 (2002)
Mikula, K., Sarti, A., Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation. In: Suri, J., et al. (eds.) Handbook of Medical Image Analysis: Segmentation and Registration Models, pp. 583–626. Springer, New York (2005)
Osher, S., Fedwik, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 53. Springer, New York (2002)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)
Reichel, L., Ye, Q.: Breakdown-free GMRES for singular systems. SIAM J. Matrix Anal. Appl. 26, 1001–1021 (2005)
Saad, Y.: Iterative methods for sparse linear systems. PWS Publ. Comp. (1996)
Scherl, H., et al.: Semi automatic level set segmentation and stenosis quatification of internal carotid artery in 3D CTA data sets. Medical Image Analysis 11, 21–34 (2007)
Sarti, A., Malladi, R., Sethian, J.A.: Subjective Surfaces: A Geometric Model for Boundary Completion. International Journal of Computer Vision 46(3), 201–221 (2002)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry. Fluid Mechanics, Computer Vision, and Material Science. Cambridge University Press, Cambridge (1999)
Walkington, N.J.: Algorithms for computing motion by mean curvature. SIAM J. Numer. Anal. 33(6), 2215–2238 (1996)
Weickert, J.: Coherence-enhancing diffusion of colour images. Image and Vision Computing 17, 201–212 (1999)
Weickert, J., Scharr, H.: A scheme for coherence enhancing diffusion filtering with optimized rotation invariance. Journal of Visual Communication and Image Representation 13(1/2), 103–118 (2002)
Westin, C.-F., Lorigo, L.M., Faugeras, O.D., Grimson, W.E.L., Dawson, S., Norbash, A., Kikinis, R.: Segmentation by Adaptive Geodesic Active Contours. In: Proceedings of MICCAI 2000, Third International Conference on Medical Image Computing and Computer-Assisted Intervention, October 11-14, pp. 266–275 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Franchini, E., Morigi, S., Sgallari, F. (2010). Segmentation of 3D Tubular Structures by a PDE-Based Anisotropic Diffusion Model. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-11620-9_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11619-3
Online ISBN: 978-3-642-11620-9
eBook Packages: Computer ScienceComputer Science (R0)