February  2013, 7(1): 1-25. doi: 10.3934/ipi.2013.7.1

Shape spaces via medial axis transforms for segmentation of complex geometry in 3D voxel data

1. 

Computational Science Center, University of Vienna, Nordbergstraße 15, A-1090 Wien, and Institute of Mathematics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria

2. 

Institute for Software Technology, Graz University of Technology, Austria

3. 

Institute of Mathematics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, and Department of Digitization and Digital Preservation, University of Innsbruck, Innrain 52, A-6020 Innsbruck, Austria

4. 

Geom e.U. Softwareentwicklung, Brockmanngasse 15, A-8010 Graz, Austria

5. 

Radon Institute of Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria

Received  July 2011 Revised  May 2012 Published  February 2013

In this paper we construct a shape space of medial ball representations from given shape training data using methods of Computational Geometry and Statistics. The ultimate goal is to employ the shape space as prior information in supervised segmentation algorithms for complex geometries in 3D voxel data. For this purpose, a novel representation of the shape space (i.e., medial ball representation) is worked out and its implications on the whole segmentation pipeline are studied. Such algorithms have wide applications for industrial processes and medical imaging, when data are recorded under varying illumination conditions, are corrupted with high noise or are occluded.
Citation: Jochen Abhau, Oswin Aichholzer, Sebastian Colutto, Bernhard Kornberger, Otmar Scherzer. Shape spaces via medial axis transforms for segmentation of complex geometry in 3D voxel data. Inverse Problems & Imaging, 2013, 7 (1) : 1-25. doi: 10.3934/ipi.2013.7.1
References:
[1]

, Medical image computing and computer assisted intervention., Available from: , (). Google Scholar

[2]

J. Abhau, W. Hinterberger and O. Scherzer, Segmenting surfaces of arbitrary topology: A two-step approach,, in, (6437). Google Scholar

[3]

R. K. Ahuja, T. L. Magnanti and J. B. Orlin, "Network Flows: Theory, Algorithms, and Applications,", Prentice Hall, (1993). Google Scholar

[4]

O. Aichholzer, F. Aurenhammer, T. Hackl, M. Kornberger, B. M. Peternell and H. Pottmann, Approximating boundary-triangulated objects with balls,, in, (2007), 130. Google Scholar

[5]

O. Aichholzer, F. Aurenhammer and B. Kornberger, Stable piecewise-linear approximations of 3d medial axes,, Manuscript., (). Google Scholar

[6]

O. Aichholzer, F. Aurenhammer, B. Kornberger, S. Plantinga, G. Rote, A. Sturm and G. Vegter, Recovering structure from $r$-sampled objects,, in, 28 (2009), 1349. Google Scholar

[7]

N. Amenta and M. Bern, Surface reconstruction by Voronoi filtering,, Discrete & Computational Geometry, 22 (1999), 481. doi: 10.1007/PL00009475. Google Scholar

[8]

N. Amenta and R. Kolluri, Accurate and efficient unions of balls,, In, (2000), 119. doi: 10.1145/336154.336193. Google Scholar

[9]

H. Blum and R. N. Nagel, Shape description using weighted symmetric axis features,, Pattern Recognition, 10 (1978), 167. Google Scholar

[10]

J.-D. Boissonnat and M. Teillaud, eds., "Effective Computational Geometry for Curves and Surfaces,", Mathematics and Visualization, (2007). Google Scholar

[11]

F. L. Bookstein, "Morphometric tools for landmark data: Geometry and biology,", Cambridge University Press, (1997). Google Scholar

[12]

X. Bresson, P. Vandergheynst and J. P. Thiran, A variational model for object segmentation using boundary information and shape prior driven by the mumford-shah functional,, Int. J. Comput. Vision, 28 (2006), 145. Google Scholar

[13]

T. Chan and L. Vese, Active contours without edges,, IEEE Trans. Image Process., 10 (2001), 266. Google Scholar

[14]

G. Charpiat, O. Faugeras and R. Keriven, Approximations of shape metrics and application to shape warping and empirical shape statistics,, Foundations of Computational Mathematics, 5 (2004), 1. doi: 10.1007/s10208-003-0094-x. Google Scholar

[15]

Q. Chen, Z. M. Zhou, M. Tang, P. A. Heng and D. S. Xia, Shape statistics variational approach for the outer contour segmentation of left ventricle mr images,, IEEE Trans. Inf. Technol. Biomed., 10 (2006), 588. Google Scholar

[16]

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs and E. A. Geiser, Using prior shapes in geometric active contours in a variational framework,, Int. J. Comput. Vision, 50 (2002), 315. Google Scholar

[17]

S. Colutto, F. Fr¨¹hauf, M Fuchs and O. Scherzer, The CMA-ES on Riemannian manifolds to reconstruct shapes in 3D voxel images,, IEEE Transactions on Evolutionary Computation, 14 (2010), 227. Google Scholar

[18]

D. Cremers, T. Kohlberger and C. Schnoerr, Shape statistics in kernel space for variational image segmentation,, Pattern Recognition, 36 (2003), 1929. Google Scholar

[19]

D. Cremers, F. Tischh0Š1user, J. Weickert and Ch. Schn0‹2rr, Diffusion snakes: Introducing statistical shape knowledge into the Mumford-Shah functional,, Int. J. Comput. Vision, 50 (2002), 295. Google Scholar

[20]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis,", Wiley Series in Probability and Statistics: Probability and Statistics, (1998). Google Scholar

[21]

H. Edelsbrunner, Deformable smooth surface design,, Discrete Comp. Geom., 21 (1999), 87. doi: 10.1007/PL00009412. Google Scholar

[22]

H. Edelsbrunner and E. P. M¨¹cke, Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms,, in, (1988), 118. doi: 10.1145/73393.73406. Google Scholar

[23]

W. Fang and K. L. Chan, Incorporating shape prior into geodesic active contours for detecting partially occluded object,, Pattern Recognition, 40 (2007), 2163. Google Scholar

[24]

P. T. Fletcher, S. Joshi, C. Ju and S. M. Pizer, Principal geodesic analysis for the study of nonliner statistics of shape,, IEEE Trans. Med. Imag., 23 (2004), 995. Google Scholar

[25]

D. S. Fritsch, S. M. Pizer, L. Yu, V. Johnson and E. L. Chaney, Localization and segmentation of medical image objects using deformable shape loci,, Lecture Notes in Computer Science, 1230 (1997), 127. Google Scholar

[26]

M. Fuchs and S. Gerber, Variational shape detection in microscope images based on joint shape and image feature statistics,, in, (2008), 1. Google Scholar

[27]

M. Gastaud, M. Barlaut and G. Aubert, Combining shape prior and statistical features for active contour segmentation,, IEEE Trans. Circuits and Systems, 14 (2004), 726. Google Scholar

[28]

N. Hansen, S. D. M¨¹ller and P. Koumoutsakos, Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES),, Evolutionary Computation, 11 (2003), 1. Google Scholar

[29]

K. Jafari-Khouzani, K. Elisevich, S. Patel and H. Soltanian-Zadeh, Dataset of magnetic resonance images of nonepileptic subjects and temporal lobe epilepsy patients for validation of hippocampal segmentation techniques,, Neuroinformatics, (2011). Google Scholar

[30]

S. Joshi, S. Pizer, P. T. Fletcher, A. Thall and G. Tracton, Multi-scale 3-d deformable model segmentation based on medical description,, in, (2001), 64. Google Scholar

[31]

T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, An efficient k-means clustering algorithm: Analysis and implementation,, IEEE Trans. Pattern Anal. Mach. Intell., 24 (2002), 881. Google Scholar

[32]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vision, 1 (1987), 321. Google Scholar

[33]

D. G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces,, Bull. Lond. Math. Soc., 16 (1984), 81. doi: 10.1112/blms/16.2.81. Google Scholar

[34]

J. T. Kent, K. V. Mardia and C. C. Taylor, Matching problems for unlabelled configurations,, Bioinf. Images Wavelets, (2004), 33. Google Scholar

[35]

J. T. Kent, K. V. Mardia and C. C. Taylor, Matching unlabelled configurations and protein bioinformatics,, to appear., (). Google Scholar

[36]

S. Kern and N. Hansen, Evaluating the cma evolution strategy on multimodal test functions,, in, 3242 (2004), 282. Google Scholar

[37]

N. Kruithof and G. Vegter, Meshing skin surfaces with certified topology,, Comput. Geom., 36 (2007), 166. doi: 10.1016/j.comgeo.2006.01.003. Google Scholar

[38]

M. E. Leventon, W. E. L. Grimson and O. Faugeras, Statistical shape influence in geodesic active contours, in "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR-00),", Los Alamitos, (2000), 316. Google Scholar

[39]

K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis,", Academic Press, (1977). Google Scholar

[40]

G. McLachlan and T. Krishnan, "The EM Algorithm and Extensions,", Second edition, (2008). doi: 10.1002/9780470191613. Google Scholar

[41]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. Google Scholar

[42]

S. M. Pizer, P. T. Fletcher, S. Joshi, A. Thall, J. Z. Chen, Y. Fridman, D. S. Fritsch, A. G. Gash, J. M. Glotzer, M. R. Jiroutek, C. Lu, K. E. Muller, G. Tracton, P. Yushkevich and E. L. Chaney, Deformable m-reps for 3d medical image segmentation,, Int. J. Comput. Vision, 55 (2003), 85. Google Scholar

[43]

S. M. Pizer, A. L. Thall and D. T. Chen, M-reps: A new object representation for graphics,, Technical Report TR99-030, (1999), 99. Google Scholar

[44]

M. A. Puso and T. A. Laursen, A 3-d contact smoothing algorithm method using gregory patches,, Numer. Meth. in Engineering, (2002). Google Scholar

[45]

M. Rousson and N. Paragios, Shape priors for level set representations,, in, (2002), 78. Google Scholar

[46]

M. Rousson and N. Paragios, Prior knowledge, level set representations & visual grouping,, Int. J. Comput. Vision, 76 (2007), 231. Google Scholar

[47]

K. Siddiqi and S. M., Pizer, eds., "Medial representations. Mathematics, Algorithms and Applications,", Computational Imaging and Vision, 37 (2008). doi: 10.1007/978-1-4020-8658-8. Google Scholar

[48]

C. G. Small, "The Statistical Theory of Shape,", Springer Series in Statistics, (1996). doi: 10.1007/978-1-4612-4032-7. Google Scholar

[49]

S. Suri, Bipartite matching and the hungarian method,, 2006. Available from: , (). Google Scholar

[50]

A. Tsai, A. Yezzi, C. Tempany, D. Tucker, A. Fan, W. E. L. Grimson and A. Willsky, A shape-based approach to the segmentation of medical imagery using level sets,, IEEE Trans. Med. Imag., 22 (2003), 137. Google Scholar

[51]

A. Witkin, M. Kass and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vision, 1 (1987), 321. Google Scholar

[52]

P. Yushkevich, P. T. Fletcher, S. C. Joshi, A. Thall and S. M. Pizer, Continuous medial representations for geometric object modeling in 2D and 3D,, Image and Vision Computing, 21 (2003), 17. Google Scholar

[53]

P. A. Yushkevich, J. Piven, C. Hazlett, H.and G. Smith, R. Ho, S., J. C. Gee and G. Gerig, User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability,, Neuroimage, 31 (2006), 1116. Google Scholar

show all references

References:
[1]

, Medical image computing and computer assisted intervention., Available from: , (). Google Scholar

[2]

J. Abhau, W. Hinterberger and O. Scherzer, Segmenting surfaces of arbitrary topology: A two-step approach,, in, (6437). Google Scholar

[3]

R. K. Ahuja, T. L. Magnanti and J. B. Orlin, "Network Flows: Theory, Algorithms, and Applications,", Prentice Hall, (1993). Google Scholar

[4]

O. Aichholzer, F. Aurenhammer, T. Hackl, M. Kornberger, B. M. Peternell and H. Pottmann, Approximating boundary-triangulated objects with balls,, in, (2007), 130. Google Scholar

[5]

O. Aichholzer, F. Aurenhammer and B. Kornberger, Stable piecewise-linear approximations of 3d medial axes,, Manuscript., (). Google Scholar

[6]

O. Aichholzer, F. Aurenhammer, B. Kornberger, S. Plantinga, G. Rote, A. Sturm and G. Vegter, Recovering structure from $r$-sampled objects,, in, 28 (2009), 1349. Google Scholar

[7]

N. Amenta and M. Bern, Surface reconstruction by Voronoi filtering,, Discrete & Computational Geometry, 22 (1999), 481. doi: 10.1007/PL00009475. Google Scholar

[8]

N. Amenta and R. Kolluri, Accurate and efficient unions of balls,, In, (2000), 119. doi: 10.1145/336154.336193. Google Scholar

[9]

H. Blum and R. N. Nagel, Shape description using weighted symmetric axis features,, Pattern Recognition, 10 (1978), 167. Google Scholar

[10]

J.-D. Boissonnat and M. Teillaud, eds., "Effective Computational Geometry for Curves and Surfaces,", Mathematics and Visualization, (2007). Google Scholar

[11]

F. L. Bookstein, "Morphometric tools for landmark data: Geometry and biology,", Cambridge University Press, (1997). Google Scholar

[12]

X. Bresson, P. Vandergheynst and J. P. Thiran, A variational model for object segmentation using boundary information and shape prior driven by the mumford-shah functional,, Int. J. Comput. Vision, 28 (2006), 145. Google Scholar

[13]

T. Chan and L. Vese, Active contours without edges,, IEEE Trans. Image Process., 10 (2001), 266. Google Scholar

[14]

G. Charpiat, O. Faugeras and R. Keriven, Approximations of shape metrics and application to shape warping and empirical shape statistics,, Foundations of Computational Mathematics, 5 (2004), 1. doi: 10.1007/s10208-003-0094-x. Google Scholar

[15]

Q. Chen, Z. M. Zhou, M. Tang, P. A. Heng and D. S. Xia, Shape statistics variational approach for the outer contour segmentation of left ventricle mr images,, IEEE Trans. Inf. Technol. Biomed., 10 (2006), 588. Google Scholar

[16]

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs and E. A. Geiser, Using prior shapes in geometric active contours in a variational framework,, Int. J. Comput. Vision, 50 (2002), 315. Google Scholar

[17]

S. Colutto, F. Fr¨¹hauf, M Fuchs and O. Scherzer, The CMA-ES on Riemannian manifolds to reconstruct shapes in 3D voxel images,, IEEE Transactions on Evolutionary Computation, 14 (2010), 227. Google Scholar

[18]

D. Cremers, T. Kohlberger and C. Schnoerr, Shape statistics in kernel space for variational image segmentation,, Pattern Recognition, 36 (2003), 1929. Google Scholar

[19]

D. Cremers, F. Tischh0Š1user, J. Weickert and Ch. Schn0‹2rr, Diffusion snakes: Introducing statistical shape knowledge into the Mumford-Shah functional,, Int. J. Comput. Vision, 50 (2002), 295. Google Scholar

[20]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis,", Wiley Series in Probability and Statistics: Probability and Statistics, (1998). Google Scholar

[21]

H. Edelsbrunner, Deformable smooth surface design,, Discrete Comp. Geom., 21 (1999), 87. doi: 10.1007/PL00009412. Google Scholar

[22]

H. Edelsbrunner and E. P. M¨¹cke, Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms,, in, (1988), 118. doi: 10.1145/73393.73406. Google Scholar

[23]

W. Fang and K. L. Chan, Incorporating shape prior into geodesic active contours for detecting partially occluded object,, Pattern Recognition, 40 (2007), 2163. Google Scholar

[24]

P. T. Fletcher, S. Joshi, C. Ju and S. M. Pizer, Principal geodesic analysis for the study of nonliner statistics of shape,, IEEE Trans. Med. Imag., 23 (2004), 995. Google Scholar

[25]

D. S. Fritsch, S. M. Pizer, L. Yu, V. Johnson and E. L. Chaney, Localization and segmentation of medical image objects using deformable shape loci,, Lecture Notes in Computer Science, 1230 (1997), 127. Google Scholar

[26]

M. Fuchs and S. Gerber, Variational shape detection in microscope images based on joint shape and image feature statistics,, in, (2008), 1. Google Scholar

[27]

M. Gastaud, M. Barlaut and G. Aubert, Combining shape prior and statistical features for active contour segmentation,, IEEE Trans. Circuits and Systems, 14 (2004), 726. Google Scholar

[28]

N. Hansen, S. D. M¨¹ller and P. Koumoutsakos, Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES),, Evolutionary Computation, 11 (2003), 1. Google Scholar

[29]

K. Jafari-Khouzani, K. Elisevich, S. Patel and H. Soltanian-Zadeh, Dataset of magnetic resonance images of nonepileptic subjects and temporal lobe epilepsy patients for validation of hippocampal segmentation techniques,, Neuroinformatics, (2011). Google Scholar

[30]

S. Joshi, S. Pizer, P. T. Fletcher, A. Thall and G. Tracton, Multi-scale 3-d deformable model segmentation based on medical description,, in, (2001), 64. Google Scholar

[31]

T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, An efficient k-means clustering algorithm: Analysis and implementation,, IEEE Trans. Pattern Anal. Mach. Intell., 24 (2002), 881. Google Scholar

[32]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vision, 1 (1987), 321. Google Scholar

[33]

D. G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces,, Bull. Lond. Math. Soc., 16 (1984), 81. doi: 10.1112/blms/16.2.81. Google Scholar

[34]

J. T. Kent, K. V. Mardia and C. C. Taylor, Matching problems for unlabelled configurations,, Bioinf. Images Wavelets, (2004), 33. Google Scholar

[35]

J. T. Kent, K. V. Mardia and C. C. Taylor, Matching unlabelled configurations and protein bioinformatics,, to appear., (). Google Scholar

[36]

S. Kern and N. Hansen, Evaluating the cma evolution strategy on multimodal test functions,, in, 3242 (2004), 282. Google Scholar

[37]

N. Kruithof and G. Vegter, Meshing skin surfaces with certified topology,, Comput. Geom., 36 (2007), 166. doi: 10.1016/j.comgeo.2006.01.003. Google Scholar

[38]

M. E. Leventon, W. E. L. Grimson and O. Faugeras, Statistical shape influence in geodesic active contours, in "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR-00),", Los Alamitos, (2000), 316. Google Scholar

[39]

K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis,", Academic Press, (1977). Google Scholar

[40]

G. McLachlan and T. Krishnan, "The EM Algorithm and Extensions,", Second edition, (2008). doi: 10.1002/9780470191613. Google Scholar

[41]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. Google Scholar

[42]

S. M. Pizer, P. T. Fletcher, S. Joshi, A. Thall, J. Z. Chen, Y. Fridman, D. S. Fritsch, A. G. Gash, J. M. Glotzer, M. R. Jiroutek, C. Lu, K. E. Muller, G. Tracton, P. Yushkevich and E. L. Chaney, Deformable m-reps for 3d medical image segmentation,, Int. J. Comput. Vision, 55 (2003), 85. Google Scholar

[43]

S. M. Pizer, A. L. Thall and D. T. Chen, M-reps: A new object representation for graphics,, Technical Report TR99-030, (1999), 99. Google Scholar

[44]

M. A. Puso and T. A. Laursen, A 3-d contact smoothing algorithm method using gregory patches,, Numer. Meth. in Engineering, (2002). Google Scholar

[45]

M. Rousson and N. Paragios, Shape priors for level set representations,, in, (2002), 78. Google Scholar

[46]

M. Rousson and N. Paragios, Prior knowledge, level set representations & visual grouping,, Int. J. Comput. Vision, 76 (2007), 231. Google Scholar

[47]

K. Siddiqi and S. M., Pizer, eds., "Medial representations. Mathematics, Algorithms and Applications,", Computational Imaging and Vision, 37 (2008). doi: 10.1007/978-1-4020-8658-8. Google Scholar

[48]

C. G. Small, "The Statistical Theory of Shape,", Springer Series in Statistics, (1996). doi: 10.1007/978-1-4612-4032-7. Google Scholar

[49]

S. Suri, Bipartite matching and the hungarian method,, 2006. Available from: , (). Google Scholar

[50]

A. Tsai, A. Yezzi, C. Tempany, D. Tucker, A. Fan, W. E. L. Grimson and A. Willsky, A shape-based approach to the segmentation of medical imagery using level sets,, IEEE Trans. Med. Imag., 22 (2003), 137. Google Scholar

[51]

A. Witkin, M. Kass and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vision, 1 (1987), 321. Google Scholar

[52]

P. Yushkevich, P. T. Fletcher, S. C. Joshi, A. Thall and S. M. Pizer, Continuous medial representations for geometric object modeling in 2D and 3D,, Image and Vision Computing, 21 (2003), 17. Google Scholar

[53]

P. A. Yushkevich, J. Piven, C. Hazlett, H.and G. Smith, R. Ho, S., J. C. Gee and G. Gerig, User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability,, Neuroimage, 31 (2006), 1116. Google Scholar

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