Abstract
Active contours and active surfaces are means of model-driven segmentation. Their use enforces closed and smooth boundaries for each segmentation irrespective of the image content. They are particularly useful if such properties cannot be derived everywhere from the data.
In this chapter, we will discuss explicit and implicit active contours, their definition, parameterization, and properties. Different fitting methods for active contours will be presented in detail since their understanding is necessary to understand parameterization and stability issues.
Concepts, notions and definitions introduced in this chapter
- Explicit active contours and surfaces
- Snakes
- Intrinsic and extrinsic attributes of active contours
- Optimization of explicit active contours
- Constraints for the evolution of explicit active contours
- Implicit active contours in the level set framework
- Stationary and dynamic level sets
- Level set evolution by front propagation
- Geodesic active contours, variational level sets
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The procedure is similar to optical flow computation by simultaneous minimization of the error for the Horn–Schunck constraint and the difference between neighboring displacement vectors.
- 2.
The advantage over using a pixel or voxel grid is that unique surface representations for a surface passing the cell exist. This is important for an active contour because it is a surface in 2D or 3D. Using a pixel or voxel grid requires an additional mechanism to resolve ambiguities, which are known, for instance, from the Marching Cube algorithm on a voxel grid.
- 3.
The chain rule for a function g of several functions f 1,f 2,…,f n is
$$\frac{d[ g( f_{1}( t),f_{2}( t ),\ldots,f_{n}( t ) ) ]}{dt} =\frac{dg}{df_{1}}\frac{df_{1}}{dt} + \frac{dg}{df_{2}}\frac{df_{2}}{dt}+ \cdots+\frac{dg}{df_{n}}\frac{df_{n}}{dt}.$$ - 4.
It is a solution under some specializing assumptions that are necessary to define a unique optimum because the formulation contains a free parameter. Details are described in Caselles et al. (1997). It does not restrict the use of this formalism to find an optimal active contour.
References
Adalsteinsson D, Sethian JA (1995) A fast level set method for propagating interfaces. J Comput Phys 118(2):269–277
Caselles V, Kimmel R, Sapiro G (1997) Geodesic active contours. Int J Comput Vis, 21(1):61–79
Chan TF, Vese LA (1999) Active contours without edges. IEEE Trans Image Process 10(2):266–277
Chan TF, Vese LA (2001) A level set algorithm for minimizing the Mumford–Shah functional in image processing. In: IEEE workshop on variational and level set methods in computer vision, pp 161–168
Cohen LD (1991) On active contour models and balloons. CVGIP, Image Underst 53(2):211–218
Cohen LD, Cohen I (1993) Finite-element methods for active contour models and balloons for 2-d and 3-d images. IEEE Trans Pattern Recogn Mach Intell 15(11):1131–1147
Cohen LD, Kimmel R (1997) Global minimum for active contour models: a minimal path approach. Int J Comput Vis 24(1):57–78
Davatzikos CA, Prince JL (1995) An active contour model for mapping the cortex. IEEE Trans Med Imaging 14(2):65–80
Han X, Xu C, Prince JL (2003) A topology preserving level set method for geometric deformable models. IEEE Trans Pattern Anal Mach Intell 25(6):755–768
Honea DM, Snyder WE, Hanson KM (1999) Three-dimensional active surface approach to lymph node segmentation. Proc SPIE 3661(2):1003–1011
Ladak HM, Milner JS, Steinman DA (2000) Rapid three-dimensional segmentation of the carotid bifurcation from serial MR images. J Biomech Eng 122(1):96–99
Leventon ME, Grimson WEL, Faugeras O (2000) Statistical shape influence in geodesic active contours. In: IEEE comp soc conf computer vision and pattern recognition (CVPR’00), vol 1, pp 1316–1323
Kass M, Witkins A, Terzopoulos D (1988) Snakes: active contour models. Int J Comput Vis 1(4):321–331
Kichenassamy S, Kumar A, Olver P, Tannenbaum A, Yezzi A (1995) Gradient flows and geometric active contour models. In: 5th intl conf computer vision (ICCV’95), pp 810–817
McInerney T, Terzopoulos D (1995a) A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis. J Comput Med Imaging Graph 19(1):69–83
McInerney T, Terzopoulos D (1995b) Topologically adaptable snakes. In: Intl conf computer vision ICCV, pp 840–845
McInerney T, Terzopoulos D (2000) T-snakes: topologically adaptable snakes. Med Image Anal 4(1):73–91
Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685
Neuenschwander WM, Fua P, Iverson L, Székely G, Kübler O (1997) Ziplock snakes. Int J Comput Vis 25(3):191–201
Osher S, Sethian J (1988) Fronts propagating with curvature-dependent speed—algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79:12–49
Osher S, Paragios N (2003) Geometric level set methods in imaging, vision, and graphics. Springer, Berlin, 2003
Paragios N (2002) Geodesic active regions and level set methods for supervised texture segmentation. Int J Comput Vis 46(3):223–247
Paragios N (2003) A level set approach for shape-driven segmentation and tracking of the left ventricle. IEEE Trans Med Imaging 22(6):773–776
Paragios N, Mellina-Gottardo O, Ramesh V (2004) Gradient vector flow fast geodesic active contours. IEEE Trans Pattern Anal Mach Intell 26(3):402–407
Park J, Keller JM (2001) Snakes on the watershed. IEEE Trans Pattern Anal Mach Intell 23(10):1201–1205
Rousson M, Paragios N (2002) Shape priors for level set representations. In: 7th European conference on computer vision ECCV 2002. LNCS, vol 2351, pp 416–418
Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Appl Math 93(4):1591–1595
Sethian JA (1998) Level set and fast marching methods. Cambridge University Press, Cambridge
Tsai A, Yezzi A, Willsky AS (2001) Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans Image Process 10(8):1169–1186
Westin CF, Lorigo LM, Faugeras O, Grimson WEL, Dawson S, Norbash A, Kikinis R (2000) Segmentation by adaptive geodesic active contours. In: Medical image computing and computer-assisted intervention—MICCAI 2000. LNCS, vol 1935, pp 266–275
Xu X, Prince JL (1998) Snakes, shapes, and gradient vector flow. IEEE Trans Image Process 7(3):359–369
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Toennies, K.D. (2012). Active Contours and Active Surfaces. In: Guide to Medical Image Analysis. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-2751-2_9
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2751-2_9
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2750-5
Online ISBN: 978-1-4471-2751-2
eBook Packages: Computer ScienceComputer Science (R0)