Abstract
Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation of gradients for functionals to construct gradient descent evolutions for these constrained problems. The result is a generalization of the standard gradient projection method to an infinite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.
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 subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Solem, J.E., Overgaard, N.: A geometric formulation of gradient descent for variational problems with moving surfaces. In: The 5th International Conference on Scale Space and PDE methods in Computer Vision, Scale Space 2005, pp. 419–430. Springer, Hofgeismar (2005)
Damgaard Pedersen, U., Fogh Olsen, O., Holm Olsen, N.: A multiphase variational level set approach for modelling human embryos. In: IEEE Proc. Workshop on Variational, Geometric and Level Set Methods in Computer Vision, pp. 25–32 (2003)
Zhao, H., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Computational Physics 127, 179–195 (1996)
Solem, J.E., Kahl, F., Heyden, A.: Visibility constrained surface evolution. In: International Conference on Computer Vision and Pattern Recognition, San Diego, CA (2005)
Rosen, J.: The gradient projection method for nonlinear programming: Part II, nonlinear constraints. J. Society for Industrial and Applied Mathematics 9, 514–532 (1961)
Thorpe, J.A.: Elementary Topics in Differential Geometry. Springer, Heidelberg (1985)
Dervieux, A., Thomasset, F.: A finite element method for the simulation of Rayleigh–Taylor instability. In: Rautman, R. (ed.) Approximation Methods for Navier–Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 145–158. Springer, Heidelberg (1979)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)
Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. Journal of Computer Vision (1997)
Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces based object segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 394–398 (1997)
Paragios, N., Deriche, R.: Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision. International Journal of Visual Communication and Image Representation (2000)
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)
Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and non-parametric shape reconstruction from unorganized points using a variational level set method. Computer Vision and Image Understanding, 295–319 (2000)
Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Heidelberg (2002)
Lorensen, W., Cline, H.: Marching cubes: a high resolution 3d surface reconstruction algorithm. Computer Graphics (Siggraph 1987 ) 21, 163–169 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Solem, J.E., Overgaard, N.C. (2005). A Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds) Variational, Geometric, and Level Set Methods in Computer Vision. VLSM 2005. Lecture Notes in Computer Science, vol 3752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11567646_28
Download citation
DOI: https://doi.org/10.1007/11567646_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29348-4
Online ISBN: 978-3-540-32109-5
eBook Packages: Computer ScienceComputer Science (R0)