Abstract
This paper analyzes the problem of determining the optimal scaling to prune the medial axis of spurious branches with the use of the Scale Axis Transform (SAT) in \(\mathbb{R}^{2}\). This optimal scaling is found by minimizing the Fréchet distance between the boundary of the true shape and the boundary of the SAT-filtered version of the shape perturbed by noise. To compute the minimum, the noisy shape is filtered using a variety of scalings s > 1 of the SAT algorithm. The optimal scaling is then related to the level of noise used to perturb the true shape. The minimization problem is repeated for various shapes and different noise levels. In applications such as image recognition and registration, the medial axis is very relevant. However, it is highly susceptible to noise along the boundary. The results presented here offer crucial information to automate the de-noising process, by providing a link between the level of noise and the optimal SAT scaling factor.
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
References
Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5, 75–91 (1995)
Amenta, N., Choi, S., Kolluri, R.K.: The power crust. In: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, Ann Arbor pp. 249–266. ACM (2001)
Attali, D., Montanvert, A.: Computing and simplifying 2D and 3D continuous skeletons. Comput. Vis. Image Underst. 67(3), 261–273 (1997)
Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)
Buchin, K., Buchin, M., Wenk, C.: Computing the Fréchet distance between simple polygons in polynomial time. In: Proceedings of the Twenty-Second Annual Symposium on Computational Geometry (SCG ’06), Sedona, pp. 80–87. ACM (2006)
Chazal, F., Lieutier, A.: The λ-medial axis. Graph. Models 67(4), 304–331 (2005)
Feragen, A., Lauze, F., Lo, P., de Bruijne, M., Nielsen, M.: Geometries on spaces of treelike shapes. In: Computer Vision–ACCV 2010, Queenstown, pp. 160–173. Springer (2011)
Feragen, A., Owen, M., Petersen, J., Wille, M.M., Thomsen, L.H., Dirksen, A., de Bruijne, M.: Tree-space statistics and approximations for large-scale analysis of anatomical trees. In: Information Processing in Medical Imaging, Asilomar, pp. 74–85. Springer (2013)
Giesen, J., Miklos, B., Pauly, M., Wormser, C.: The scale axis picture show. In: ACM Video/Multimedia Session of Symposium of Computational Geometry (2009)
Giesen, J., Miklos, B., Pauly, M., Wormser, C.: The scale axis transform. In: Proceedings of the 25th Annual Symposium on Computational Geometry, Aarhus, pp. 106–115. ACM (2009)
Klein, A., Andersson, J., Ardekani, B.A., Ashburner, J., Avants, B., Chiang, M.C., Christensen, G.E., Collins, D.L., Gee, J., Hellier, P., et al.: Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. Neuroimage 46(3), 786–802 (2009)
Leonard, K.: An efficiency criterion for 2D shape model selection. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, vol. 1, pp. 1289–1296. IEEE (2006)
Leonard, K.: Efficient shape modeling: entropy, adaptive coding, and boundary curves-vs-Blum’s medial axis. Int. J. Comput. Vis. 74(2), 183–199 (2007)
Lieutier, A.: Any open bounded subset of \(\mathbb{R}^{n}\) has the same homotopy type as its medial axis. Comput. Aided Design 36(11), 1029–1046 (2004)
Liu, L., Chambers, E.W., Letscher, D., Ju, T.: Extended grassfire transform on medial axes of 2D shapes. Comput. Aided Design 43(11), 1496–1505 (2011)
Mederos, B., Amenta, N., Velho, L., de Figueiredo, L.H.: Surface reconstruction for noisy point clouds. In: Symposium on Geometry Processing, Vienna, pp. 53–62. Citeseer (2005)
Miklós, B.: The scale axis transform. Ph.D. thesis, ETH Zürich (2010)
Miklós, B., Giesen, J., Pauly, M.: Medial axis approximation from inner Voronoi balls: a demo of the Mesecina tool. In: Proceedings of the 23rd Annual Symposium on Computational Geometry, Gyeongju. ACM (2007)
Miklos, B., Giesen, J., Pauly, M.: Discrete scale axis representations for 3D geometry. ACM Trans. Graph. (TOG) 29(4), 101 (2010)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)
Sebastian, T., Klein, P., Kimia, B.: Recognition of shapes by editing shock graphs. In: IEEE International Conference on Computer Vision, Vancouver, vol. 1, pp. 755–755. IEEE Computer Society (2001)
Sebastian, T.B., Klein, P.N., Kimia, B.B.: Recognition of shapes by editing their shock graphs. IEEE Trans. Pattern Anal. Mach. Intell. 26(5), 550–571 (2004)
Shaked, D., Bruckstein, A.M.: Pruning medial axes. Comput. Vis. Image Underst. 69(2), 156–169 (1998)
Siddiqi, K., Shokoufandeh, A., Dickinson, S.J., Zucker, S.W.: Shock graphs and shape matching. Int. J. Comput. Vis. 35(1), 13–32 (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland & The Association for Women in Mathematics
About this paper
Cite this paper
Abiva, J., Larsson, L.J. (2015). Towards Automated Filtering of the Medial Axis Using the Scale Axis Transform. In: Leonard, K., Tari, S. (eds) Research in Shape Modeling. Association for Women in Mathematics Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16348-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-16348-2_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16347-5
Online ISBN: 978-3-319-16348-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)