Abstract
Convex relaxation techniques have become a popular approach to shape optimization as they allow to compute solutions independent of initialization to a variety of problems. In this chapter, we will show that shape priors in terms of moment constraints can be imposed within the convex optimization framework, since they give rise to convex constraints. In particular, the lower-order moments correspond to the overall area, the centroid, and the variance or covariance of the shape and can be easily imposed in interactive segmentation methods. Respective constraints can be imposed as hard constraints or soft constraints. Quantitative segmentation studies on a variety of images demonstrate that the user can impose such constraints with a few mouse clicks, leading to substantial improvements of the resulting segmentation, and reducing the average segmentation error from 12 % to 0.35 %. GPU-based computation times of around 1 second allow for interactive segmentation.
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Blake A, Zisserman A (1987) Visual reconstruction. MIT Press, Cambridge
Boyle JP, Dykstra RL (1986) An method for finding projections onto the intersection of convex sets in Hilbert spaces. In: Lecture Notes in Statistics, vol 37, pp 28–47
Caselles V, Kimmel R, Sapiro G (1995) Geodesic active contours. In: Proc IEEE int conf on computer vision, Boston, USA, pp 694–699
Chan T, Esedoḡlu S, Nikolova M (2006) Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J Appl Math 66(5):1632–1648
Cremers D, Osher SJ, Soatto S (2006) Kernel density estimation and intrinsic alignment for shape priors in level set segmentation. Int J Comput Vis 69(3):335–351
Das P, Veksler O, Zavadsky V, Boykov Y (2008) Semiautomatic segmentation with compact shape prior. Image Vis Comput 27(1–2):206–219
Etyngier P, Segonne F, Keriven R (2007) Shape priors using manifold learning techniques. In: IEEE int conf on computer vision, Rio de Janeiro, October 2007
Foulonneau A, Charbonnier P, Heitz F (2006) Affine-invariant geometric shape priors for region-based active contours. IEEE Trans Pattern Anal Mach Intell 28(8):1352–1357
Greig DM, Porteous BT, Seheult AH (1989) Exact maximum a posteriori estimation for binary images. J R Stat Soc B 51(2):271–279
Grenander U, Chow Y, Keenan DM (1991) Hands: a pattern theoretic study of biological shapes. Springer, New York
Ising E (1925) Beitrag zur theorie des ferromagnetismus. Z Phys 23:253–258
Kass M, Witkin A, Terzopoulos D (1988) Snakes: active contour models. Int J Comput Vis 1(4):321–331
Kichenassamy S, Kumar A, Olver PJ, Tannenbaum A, Yezzi AJ (1995) Gradient flows and geometric active contour models. In: IEEE int conf on computer vision, pp 810–815
Klodt M, Cremers D (2011) A convex framework for image segmentation with moment constraints. In: IEEE int conf on computer vision, Barcelona, Spain
Kolev K, Cremers D (2008) Integration of multiview stereo and silhouettes via convex functionals on convex domains. In: European conference on computer vision (ECCV), Marseille, France, October 2008
Kolev K, Klodt M, Brox T, Cremers D (2009) Continuous global optimization in multiview 3d reconstruction. Int J Comput Vis 84(1):80–96
Lempitsky V, Kohli P, Rother C, Sharp T (2009) Image segmentation with a bounding box prior. In: IEEE int conf on computer vision, Kyoto, Japan
Luenberger DG (1997) Optimization by vector space methods, 1st edn. Wiley, New York
Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685
Osher SJ, Sethian JA (1988) Fronts propagation with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comp Physiol 79:12–49
Papoulis A, Pillai SU (2002) Probability, random variables, and stochastic processes, 4th edn. McGraw-Hill, New York
Rother C, Kolmogorov V, Blake A (2004) GrabCut: interactive foreground extraction using iterated graph cuts. ACM Trans Graph 23(3):309–314
Schoenemann T, Cremers D (2009) A combinatorial solution for model-based image segmentation and real-time tracking. IEEE Trans Pattern Anal Mach Intell 32(7):1153–1164
Unger M, Pock T, Cremers D, Bischof H (2008) Tvseg—interactive total variation based image segmentation. In: British machine vision conference (BMVC), Leeds, UK, September 2008
Veksler O (2008) Star shape prior for graph-cut image segmentation. In: Europ conf on computer vision, pp 454–467
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Klodt, M., Steinbrücker, F., Cremers, D. (2013). Moment Constraints in Convex Optimization for Segmentation and Tracking. In: Farinella, G., Battiato, S., Cipolla, R. (eds) Advanced Topics in Computer Vision. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-5520-1_8
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DOI: https://doi.org/10.1007/978-1-4471-5520-1_8
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