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Journal of Scientific Computing

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03-03-2017

Segmentation of Three-Dimensional Images with Parametric Active Surfaces and Topology Changes

Published in: Journal of Scientific Computing | Issue 3/2017

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Abstract

In this paper, we introduce a novel parametric finite element method for segmentation of three-dimensional images. We consider a piecewise constant version of the Mumford–Shah and the Chan–Vese functionals and perform a region-based segmentation of 3D image data. An evolution law is derived from energy minimization problems which push the surfaces to the boundaries of 3D objects in the image. We propose a parametric scheme which describes the evolution of parametric surfaces. An efficient finite element scheme is proposed for a numerical approximation of the evolution equations. Since standard parametric methods cannot handle topology changes automatically, an efficient method is presented to detect, identify and perform changes in the topology of the surfaces. One main focus of this paper are the algorithmic details to handle topology changes like splitting and merging of surfaces and change of the genus of a surface. Different artificial images are studied to demonstrate the ability to detect the different types of topology changes. Finally, the parametric method is applied to segmentation of medical 3D images.

previous article A Three-Stage Approach for Segmenting Degraded Color Images: Smoothing, Lifting and Thresholding (SLaT)

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The author acknowledges the National Cancer Institute and the Foundation for the National Institutes of Health, and their critical role in the creation of the free publicly available LIDC/IDRI Database.

Ardon, R., Cohen, L.D., Yezzi, A.: A new implicit method for surface segmentation by minimal paths: applications in 3D medical images. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 3757, pp. 520–535. Springer, Berlin (2005)CrossRef

Armato III, S.G., McLennan, G., Bidaut, L., McNitt-Gray, M.F., Meyer, C.R., Reeves, A.P., Zhao, B., Aberle, D.R., Henschke, C.I., Hoffman, E.A., Kazerooni, E.A., MacMahon, H., van Beek, E.J.R., Yankelevitz, D., Biancardi, A.M., Bland, P.H., Brown, M.S., Engelmann, R.M., Laderach, G.E., Max, D., Pais, R.C., Qing, D.P.Y., Roberts, R.Y., Smith, A.R., Starkey, A., Batra, P., Caligiuri, P., Farooqi, A., Gladish, G.W., Jude, C.M., Munden, R.F., Petkovska, I., Quint, L.E., Schwartz, L.H., Sundaram, B., Dodd, L.E., Fenimore, C., Gur, D., Petrick, N., Freymann, J., Kirby, J., Hughes, B., Vande Casteele, A., Gupte, S., Sallam, M., Heath, M.D., Kuhn, M.H., Dharaiya, E., Burns, R., Fryd, D.S., Salganicoff, M., Anand, V., Shreter, U., Vastagh, S., Croft, B.Y., Clarke, L.P.: The Lung Image Database Consortium (LIDC) and Image Database Resource Initiative (IDRI): A completed reference database of lung nodules on CT scans. Med. Phys. 38(2), 915–931 (2011)CrossRef

Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2006)MATH

Aujol, J.F., Kang, S.H.: Color image decomposition and restoration. J. Vis. Commun. Image Represent. 17(4), 916–928 (2006)CrossRef

Balažovjech, M., Mikula, K., Petrášová, M., Urbán, J.: Lagrangean method with topological changes for numerical modelling of forest fire propagation. In: Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, pp. 42–52. Vysoké Tatry, Podbansk’v, Slovakia (2012)

Bänsch, E., Morin, P., Nochetto, R.H.: A finite element method for surface diffusion: the parametric case. J. Comput. Phys. 203(1), 321–343 (2005)MathSciNetMATHCrossRef

Barrett, J.W., Garcke, H., Nürnberg, R.: A variational formulation of anisotropic geometric evolution equations in higher dimensions. Numer. Math. 109(1), 1–44 (2008)MathSciNetMATHCrossRef

Barrett, J.W., Garcke, H., Nürnberg, R.: On the parametric finite element approximation of evolving hypersurfaces in \({\mathbb{R}}^3\). J. Comput. Phys. 227(9), 4281–4307 (2008)MathSciNetMATHCrossRef

Barrett, J.W., Garcke, H., Nürnberg, R.: Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31(1), 225–253 (2008)MathSciNetMATHCrossRef

10.

Benninghoff, H., Garcke, H.: Efficient image segmentation and restoration using parametric curve evolution with junctions and topology changes. SIAM J. Imaging Sci. 7(3), 1451–1483 (2014)MathSciNetMATHCrossRef

11.

Benninghoff, H., Garcke, H.: Segmentation and restoration of images on surfaces by parametric active contours with topology changes. J. Math. Imaging Vis. 55(1), 105–124 (2015)

12.

Brakke, K.A.: The surface evolver. Exp. Math. 1(2), 141–165 (1992)MathSciNetMATHCrossRef

13.

Brochu, T., Bridson, R.: Robust topological operations for dynamic explicit surfaces. SIAM J. Sci. Comput. 31(4), 2472–2493 (2009)MathSciNetMATHCrossRef

14.

Brown, E.S., Chan, T.F., Bresson, X.: Completely convex formulation of the Chan–Vese image segmentation model. Int. J. Comput. Vis. 98(1), 103121 (2012)MathSciNetMATHCrossRef

15.

Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)MATHCrossRef

16.

Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces: a geometric three dimensional segmentation approach. Numer. Math. 77, 423–451 (1997)MathSciNetMATHCrossRef

17.

Chambolle, A., Cremers, D., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetMATHCrossRef

18.

Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)MathSciNetMATHCrossRef

19.

Chan, T.F., Kang, S.H., Shen, J.: Total variation denoising and enhancement of color images based on the CB and HSV color models. J. Vis. Commun. Image Represent. 12(4), 422–435 (2001)CrossRef

20.

Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)MATHCrossRef

21.

Cohen, L.D.: On active contour models and balloons. Comput. Vis. Graph. Image Process. Image Underst. 53(2), 211–218 (1991)MATH

22.

Cohen, L.D., Cohen, I.: Finite element methods for active contour models and balloons for 2D and 3D images. IEEE Trans. Pattern Anal. Mach. Intell. 15(11), 1131–1147 (1993)CrossRef

23.

Cremers, D., Schnörr, C., Weickert, J.: Diffusion-snakes: combining statistical shape knowledge and image information in a variational framework. In: Proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 137–144. Vancouver (2001)

24.

Cremers, D., Rousson, M., Deriche, R.: Review of statistical approaches to level set segmentation: integrating color, texture, motion, and shape. Int. J. Comput. Vis. 72(2), 195–215 (2007)

25.

Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)MathSciNetMATHCrossRef

26.

Doǧan, G., Morin, P., Nochetto, R.H.: A variational shape optimization approach for image segmentation with a Mumford–Shah functional. SIAM J. Sci. Comput. 30(6), 3028–3049 (2008)MathSciNetMATHCrossRef

27.

Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R. (eds.) Partial Differential Equations and Calculus of Variations, Vol. 1357 of Lecture Notes in Mathematics, p. 142155. Springer, Berlin, Heidelberg (1988)

28.

Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(1), 603–611 (1991)MathSciNetMATHCrossRef

29.

Grady, L., Alvino, C.V.: The piecewise smooth Mumford–Shah functional on an arbitrary graph. IEEE Trans. Image Process. 18(11), 2547–2561 (2009)MathSciNetCrossRef

30.

Hintermüller, M., Ring, W.: An inexact Newton-CG-type active contour approach for the minimization of the Mumford–Shah functional. J. Math. Imaging Vis. 20(1–2), 19–42 (2004)MathSciNetMATHCrossRef

31.

Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)MATHCrossRef

32.

Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83–97 (1955)MathSciNetMATHCrossRef

33.

Li, H., Yezzi, A., Cohen, L.D.: Fast 3D brain segmentation using dual-front active contours with optional user-interaction. In: Liu, Y., Jiang, T., Zhang, C. (eds.) Computer Vision for Biomedical Image Applications. Lecture Notes in Computer Science, vol. 3765, pp. 335–345. Springer, Berlin (2005)CrossRef

34.

Lu, C., Cao, Y., Mumford, D.: Surface evolution under curvature flows. J. Vis. Commun. Image Represent. 13, 6581 (2002)CrossRef

35.

Malladi, R., Sethian, J.A., Vemuri, B.C.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)CrossRef

36.

McInerney, T., Terzopoulos, D.: Topology adaptive deformable surfaces for medical image volume segmentation. IEEE Trans. Med. Imaging 18(10), 840–850 (1999)CrossRef

37.

McInerney, T., Terzopoulos, D.: T-snakes: topology adaptive snakes. Med. Image Anal. 4, 7391 (2000)CrossRef

38.

Mikula, K., Peyriéras, N., Remešíková, M.: Stašová: segmentation of 3D cell membrane images by PDE methods and its applications. Comput. Biol. Med. 41(6), 326–339 (2011)CrossRef

39.

Mikula, K., Peyriéras, N., Špir, R.: Numerical algorithm for tracking cell dynamics in 4D biomedical images. Discrete Contin. Dyn. Syst. Ser. S 8(5), 953–967 (2015)

40.

Mikula, K., Urbán, J.: New fast and stable Lagrangean method for image segmentation. In: Proceedings of the 5th International Congress on Image and Signal Processing (CISP 2012), pp. 834–842. Chongquing, China (2012)

41.

Mille, J.: Narrow band region-based active contours and surfaces for 2D and 3D segmentation. Comput. Vis. Image Underst. 113(9), 946–965 (2009)CrossRef

42.

Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)MathSciNetMATHCrossRef

43.

Nieuwenhuis, C., Toppe, E., Cremers, D.: A survey and comparison of discrete and continuous multi label optimization approachesfor the Potts model. Int. J. Comput. Vis. 104(3), 223–240 (2014)MATHCrossRef

44.

Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetMATHCrossRef

45.

Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford–Shah functional. In: Proceedings of the 12th IEEE International Conference on Computer Vision (ICCV 2009), pp. 1133–1140 (2009)

46.

Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM J. Imaging Sci. 3(4), 11221145 (2010)MathSciNetMATHCrossRef

47.

Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D.: A convex formulation of continuous multi-label problems. In: Proceedings of the European Conference on Computer Vision (ECCV) (2008)

48.

Reeves, A.P., Biancardi, A.M.: The Lung Image Database Consortium (LIDC) Nodule Size Report, Release: 2011-10-27. http://www.via.cornell.edu/lidc/ (2011)

49.

Reeves, A.P., Biancardi, A.M., Apanasovich, T.V., Meyer, C.R., MacMahon, H., van Beek, E.J., Kazerooni, E.A., Yankelevitz, D., McNitt-Gray, M.F., McLennan, G., Armato III, S.G., Henschke, C.I., Aberle, D.R., Croft, B.Y., Clarke, L.P.: The Lung Image Database Consortium (LIDC): a comparison of different size metrics for pulmonary nodule measurements. Acad. Radiol. 14(12), 1475–1485 (2007)CrossRef

50.

Ronfard, R.: Region-based strategies for active contour models. Int. J. Comput. Vis. 13(2), 229–251 (1994)CrossRef

51.

Rousseau, O., Bourgault, Y.: Heart Segmentation with an Iterative Chan–Vese Algorithm. http://hal.archives-ouvertes.fr/hal-00403627/en/ (2009)

52.

Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. No. 167 in Applied Mathematical Sciences. Springer, New York (2009)

53.

Sharma, N., Aggarwal, L.M.: Automated medical image segmentation techniques. J. Med. Phys. 35(1), 3–14 (2010)CrossRef

54.

Shen, T., Huang, X.: 3D medical image segmentation by multiple-surface active volume models. In: Yang, G.Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) Medical Image Computing and Computer-Assisted Intervention MICCAI 2009. Lecture Notes in Computer Science, vol. 5762, pp. 1059–1066. Springer, Berlin (2009)CrossRef

55.

Tang, B., Sapiro, G., Caselles, V.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10(5), 701–707 (2002)MATHCrossRef

56.

Tsai, A., Yezzi, A., Willsky, A.S.: Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation and magnification. IEEE Trans. Image Process. 10(8), 1169–1186 (2001)MATHCrossRef

57.

Udupa, J.K., Herman, G.T.: 3D Imaging in Medicine, 2nd edn. CRC Press Inc., Boca Raton, FL (1999)

58.

Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002)MATHCrossRef

59.

Yezzi Jr., A., Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imaging 16(2), 199–209 (1997)CrossRef

60.

Zhang, Y., Matuszewski, B.J., Shark, L.K., Moore, C.J.: Medical image segmentation using new hybrid level-set method. In: Proceedings of IEEE International Conference on Biomedical Visualisation MEDi08VIS, pp. 71–76. IEEE, London (2008)

Title: Segmentation of Three-Dimensional Images with Parametric Active Surfaces and Topology Changes
Publication date: 03-03-2017
Published in: Journal of Scientific Computing / Issue 3/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0401-3

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