Abstract
We propose a novel shape optimization algorithm for region-based active contour models. Region-based active contours are preferred for many segmentation problems, because they incorporate more global information by aggregating cues or statistics over the distinct regions defined by the contour configuration. This makes them effective in a diverse array of segmentation scenarios, also more robust to contour initializations, Unfortunately they are also more expensive computationally, because a significant part of the optimization involves repeated integrations of the image features over the regions through the many iterations of the contour updates. Accordingly, we aim to decrease the overall computational cost of region-based active contours by reducing the cost of an individual iteration, and the total number of iterations. To this end, we first develop a Lagrangian curve representation that is spatially adaptive and economical in terms of the number of nodes used. Then we perform the shape sensitivity analysis of the general form of the region-based segmentation energy. In particular, we compute the second variation or the shape Hessian of the energy, and we use this to compute fast descent directions for the contours to significantly reduce the computational cost. Our implementation builds on a finite element discretization of the whole framework, including the contours. This results in efficient velocity computations in linear time with respect to the number of contour nodes.
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
Aubert, G., Barlaud, M., Faugeras, O., Jehan-Besson, S.: Image segmentation using active contours: calculus of variations or shape gradients? SIAM J. Appl. Math. 63(6) (2003)
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001)
Charpiat, G., Maurel, P., Pons, J.-P., Keriven, R., Faugeras, O.: Generalized gradients: Priors on minimization flows. International Journal of Computer Vision 73, 325–344 (2007)
Cremers, D., Rousson, M., Deriche, R.: A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape. IJCV 72, 195–215 (2007)
Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Advances in Design and Control (2001)
Doğan, G., Morin, P., Nochetto, R.H.: A variational shape optimization approach for image segmentation with a Mumford-Shah functional. SIAM J. Sci. Comp. 30(6) (2008)
Hintermüller, M., Ring, W.: A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64(2), 442–467 (2003/04)
Hintermüller, M., Ring, W.: An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imaging Vision 20(1–2) (2004)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer-Verlag, New York (1999)
Paragios, N., Deriche, R.: Geodesic active regions: A new framework to deal with frame partition problems in computer vision. J. Vis. Commun. Image Represent. 13(1–2) (2002)
Sokołowski, J., Zolésio, J.-P.: Introduction to Shape Optimization, volume 16 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1992)
Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Sobolev active contours. International Journal of Computer Vision 73, 345–366 (2007). doi:10.1007/s11263-006-0635-2
Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM Journal on Optimization 14(4), 1043–1056 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Doğan, G. (2015). Fast Minimization of Region-Based Active Contours Using the Shape Hessian of the Energy. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-18461-6_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18460-9
Online ISBN: 978-3-319-18461-6
eBook Packages: Computer ScienceComputer Science (R0)