Abstract
Background subtraction is an elementary method for detection of foreground objects and their segmentations. Obviously it requires an observation image as well as a background one. In this work we attempt to remove the last requirement by reconstructing the background from the observation image and a guess on the location of the object to be segmented via variational inpainting method. A numerical evaluation of this reconstruction provides a “disocclusion measure” and the correct foreground segmentation region is expected to maximize this measure. This formulation is in fact an optimal control problem, where controls are shapes/regions and states are the corresponding inpaintings. Optimization of the disocclusion measure leads formally to a coupled contour evolution equation, an inpainting equation (the state equation) as well as a linear PDE depending on the inpainting (the adjoint state equation). The contour evolution is implemented in the framework of level sets. Finally, the proposed method is validated on various examples. We focus among others in the segmentation of calcified plaques observed in radiographs from human lumbar aortic regions.
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Aubert, G., & Kornprobst, P. (2006). Applied mathematical sciences: Vol. 147, Mathematical problems in image processing: partial differential equations and the calculus of variations, 2nd edn. New York: Springer.
Aubert, G., Barlaud, M., Jehan-Besson, S., & Faugeras, O. (2003). Image segmentation using active contours: calculus of variations or shape gradients? SIAM Journal of Applied Mathematics, 63(6), 2128–2154.
Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G., & Verdera, J. (2001). Filling-in by joint interpolation of vector fields and gray levels. IEEE Transactions on Image Processing, 10(8), 1200–1211.
Bruhn, A., Weickert, J., Feddern, C., Kohlberger, T., & Schnörr, C. (2005). Variational optical flow computation in real time. IEEE Transactions on Image Processing, 14(5), 608–615.
Caselles, V., Kimmel, R., & Sapiro, G. (1995). Geodesic active contours. In Proceedings of the 5th international conference on computer vision (pp. 694–699). Boston, MA, June 1995. Los Alamitos: IEEE Computer Society Press.
Chan, T., & Shen, J. (2002). Mathematical models for local nontexture inpainting. SIAM Journal of Applied Mathematics, 62(3), 1019–1043.
Chan, T., & Vese, L. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277.
Cohen, L. D., & Cohen, I. (1993). Finite-element methods for active contour models and balloons for 2-D and 3-D images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), 1131–1147.
Cohen, L. D., Bardinet, E., & Ayache, N. (1993). Surface reconstruction using active contour models. In SPIE conference on geometric methods in computer vision, San Diego, CA, USA.
Cremers, D., Rousson, M., & Deriche, R. (2007). A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. The International Journal of Computer Vision, 72(2), April.
Delfour, M. C., & Zolésio, J. P. (1989). Analyse des problèmes de forme par la dérivation des minimax. Annales de l’Institut Henri Poincaré, Section C, S6, 211–227.
Delfour, M. C., & Zolésio, J.-P. (2001). Shapes and geometries. Advances in design and control. Philadelphia: SIAM.
Griewank, A., Juedes, D., & Utke, J. (1996). ADOL–C, a package for the automatic differentiation of algorithms written in C/C++. ACM Transactions on Mathematical Software, 22(2), 131–167.
Gunzburger, M. (2001). Adjoint equation-based methods for control problems in incompressible, viscous flows. Flow, Turbulence and Combustion, 65, 249–272.
Heiler, M., & Schnörr, C. (2005). Natural image statistics for natural image segmentation. The International Journal of Computer Vision, 63(1), 5–19.
Jehan-Besson, S., Barlaud, M., & Aubert, G. (2003). DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation. The International Journal of Computer Vision, 53(1), 45–70.
Kass, M., Witkin, A., & Terzopoulos, D. (1987). Snakes: active contour models. In First international conference on computer vision (pp. 259–268), London, June 1987.
Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., & Yezzi, A. (1995). Gradient flows and geometric active contour models. In Proceedings of the 5th international conference on computer vision (pp. 810–815). Boston, MA, June 1995. Los Alamitos: IEEE Computer Society Press.
Kim, J., Fisher III, J. W., Yezzi, A., Çetin, M., & Willsky, A. S. (2005). A nonparametric statistical method for image segmentation using information theory and curve evolution. IEEE Transactions on Image Processing, 14(10), 1486–1502.
Lauze, F., & de Bruijne, M. (2007). Toward automated detection and segmentation of aortic calcifications from radiographs. In J. Pluim & J.M. Reinhardt (Eds.), Medical imaging—SPIE procSPIE (Vol. 6512), SPIE press.
Lauze, F., & Nielsen, M. (2005). From inpainting to active contours. In N. Paragios et al. (Eds.) LNCS: Vol. 3752. Proceedings of the third IEEE workshop on variational, geometric and level set methods in computer vision, Bejing, China, October 2005 (pp. 97–108). New York: Springer.
Lions, J. L. (1971). Optimal control of systems governed by partial differential equations (trans: Mitter, S. K.). New York: Springer.
Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–684.
Ohta, N. (2001). A statistical approach to background substraction for surveillance systems. In International conference on computer vision (Vol. 2, pp. 481–486). Vancouver, BC, CA, July 2001.
Osher, S., & Paragios, N. (2003). Geometric level set methods in imaging, vision and graphics. New York: Springer.
Paragios, N., & Deriche, R. (2002). Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Journal of Visual Communication and Image Representation, Special Issue on Partial Differential Equations in Image Processing, Computer Vision and Computer Graphics, 13(1/2), 249–268.
Ronfard, R. (1994). Region based strategies for active contour models. The International Journal of Computer Vision, 13(2), 229–251.
Rostaing, N., Dalmas, S., & Galligo, A. (1993). Automatic differentiation in Odyssée. Tellus, 45(5), 558–568.
Roy, T., Debreuve, E., Barlaud, M., & Aubert, G. (2006). Segmentation of a vector field: dominant parameter and shape optimization. Journal of Mathematical Imaging and Vision, 24(2), 259–276.
Rybak, I. V. (2004). Monotone and conservative difference schemes for elliptic equations with mixed derivatives. Mathematical Modelling and Analysis, 9(2), 169–178.
Sethian, J. A. (1999). Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials sciences. Cambridge monograph on applied and computational mathematics. Cambridge: Cambridge University Press.
Ta’asan, S. (1997). Lecture notes on optimization. 1. Introduction to shape design and control (Technical report). Von Karman Institue.
Xu, C., & Prince, J. L. (1997). Gradient vector flow: a new external force for snakes. In International conference on computer vision and pattern recognition (p. 66).
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Lauze, F., Nielsen, M. From Inpainting to Active Contours. Int J Comput Vis 79, 31–43 (2008). https://doi.org/10.1007/s11263-007-0088-2
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DOI: https://doi.org/10.1007/s11263-007-0088-2