Abstract
Poor convergence to concave shapes is a main limitation of snakes as a standard segmentation and shape modelling technique. The gradient of the external energy of the snake represents a force that pushes the snake into concave regions, as its internal energy increases when new inflexion points are created. In spite of the improvement of the external energy by the gradient vector flow technique, highly non convex shapes can not be obtained, yet. In the present paper, we develop a new external energy based on the geometry of the curve to be modelled. By tracking back the deformation of a curve that evolves by minimum curvature flow, we construct a distance map that encapsulates the natural way of adapting to non convex shapes. The gradient of this map, which we call curvature vector flow (CVF), is capable of attracting a snake towards any contour, whatever its geometry. Our experiments show that, any initial snake condition converges to the curve to be modelled in optimal time.
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
Caselles, V., Catte, F., Coll, T., Dibos, F.: A geometric model for active contours. Numerische Mathematik 66, 1–31 (1993)
Caselles, V., Kimmel, R.: G. Sapiro Geodesic Active Contours. Int. J. Comp. Vision
Cohen, L.D., Kimmel, R.: Global minimum for active contour models: A minimal path approach. Int.Journal Comp. Vision 24(1), 57–78 (1997)
Gil, D., Radeva, P.: Regularized curvature flow. CVC Tech. Report no 63 (2002)
Gil, D., Radeva, P.: Curvature based Distance Maps. CVC Tech. Report no 70 (2003)
Gil, D., Radeva, P.: Anisotropic Contour Completion, ICIP 2003 (submmited)
Evans, L.C.: Partial Differential equations. In: Berkeley Math. Lect. Notes, vol. 3B
Forsey, D.R., Bartels, R.H.: Surface Fitting with Hierarchical Splines. Computer Graphics (April 1995)
Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differential Geometry 26, 285–314 (1986)
Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differential Geometry 23, 69–96 (1986)
Gage, M.: Curve shortening makes convex curves circular. Invent. Math 76, 357–364 (1984)
Guichard, F., Morel, J.M.: Mathematical Models in Image Processing. Advanced Courses on Mathematical Aspects on Image Processing
Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Scheweitzer, J., Stuetzle, W.: Piecewise smooth surface reconstruction. In: Proc. ACM SIGGRAPH, pp. 295–302 (July 1994)
Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active Contour Models. Int.Journal of Computer Vision 1, 321–331 (1987)
Knoll, Ch., Alcañiz, M., Grau, V., Montserrat, C., Juan, M.C.: Outlining of the prostate using snakes with shapes restrictions based on the wavelet transform. Pattern Recognition 32, 1767–1781 (1999)
Rudin, W.: Complex and Real Analysis. McGraw-Hill, Inc.New York
Malladi, R., Sethian, J.A.: Image Processing: Flows under min-max curvature and mean curvature. Graph. Models and Image Process 58(2) (March 1996)
Sapiro, G., Kimia, B.B., Kimmel, R., Shaked, D., Bruckstein, A.: Implementing continuous-scale morphology. Pattern Recognition 26(9) (1992)
Siddiqi, K., Tannenbaum, A., Zucker, S.W.: A Hanmiltonian Aaproach to the Eikonal Equation. In: Hancock, E.R., Pelillo, M. (eds.) EMMCVPR 1999. LNCS, vol. 1654, pp. 1–13. Springer, Heidelberg (1999)
Sun, C., Pallotino, S.: Circular shortest path on regular grids. In: Asian Conference on Computer Vision, Melbourne, Australia, pp. 852–857 (January 2002)
Tari, Z.S.G., Shah, J., Pien, H.: Extraction of shape skeletons from grayscale images. Comp. Vision and Image Understanding 66, 133–146 (1997)
Xu, C., Prince, J.L.: Snakes, shapes and gradient vector flow. IEEE Trans. on Image Proc. 7(3), 359–369 (1998)
Paragios, N., Mellina-Gottardo, O., Ramesh, V.: Gradient Vector Flow Fast Geodesic Active Contours. In: ICCV-WS 1999 (2001)
Xu, C., Prince, J.L.: Generalized gradient vector flow external forces for active contours. Signal Processing, An International Journal 71(2), 132–139 (1998)
Zhang, D., Herbert, M.: Harmonic shape images: a representation for 3-d free-form surfaces based on energy minimization. In: Hancock, E.R., Pelillo, M. (eds.) EMMCVPR 1999. LNCS, vol. 1654, pp. 30–43. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gil, D., Radeva, P. (2003). Curvature Vector Flow to Assure Convergent Deformable Models for Shape Modelling. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2003. Lecture Notes in Computer Science, vol 2683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45063-4_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-45063-4_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40498-9
Online ISBN: 978-3-540-45063-4
eBook Packages: Springer Book Archive