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Generalized Gradients: Priors on Minimization Flows

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Abstract

This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our application-specific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.

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References

  • Akhiezer, N.I. and Glazman, I.M. 1981. Theory of Linear Operators in Hilbert Space. Pitman.

  • Bertalmío, M., Cheng, L.T., Osher, S., and Sapiro, G. 2001. Variational problems and partial differential equations on implicit surfaces. Journal of Computational Physics, 174(2):759–780.

    Article  MathSciNet  Google Scholar 

  • Bertalmio, M., Sapiro, G., Cheng, L.T., and Osher, S. 2001. Variational problems and PDE’s on implicit surfaces. In IEEE Workshop on Variational and Level Set Methods, (Ed.), Vancouver, Canada, pp. 186–193,

  • Bonnans, J.F., Gilbert, J.C., Lemarechal, C., and Sagastizabal, C.A. 2002. Numerical Optimization: Theoretical and Practical Aspects. Springer-Verlag.

  • Boykov, Y. and Kolmogorov, V. 2003. Computing geodesics and minimal surfaces via graph cuts. In International Conference on Computer Vision, vol. 1, pp. 26–33.

  • Caselles, V., Kimmel, R., and Sapiro, G. 1997. Geodesic active contours. The International Journal of Computer Vision, 22(1):61–79.

    Article  Google Scholar 

  • Charpiat, G., Faugeras, O., and Keriven, R. 2005. Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations of Computational Mathematics, 5(1):1–58.

    Article  MathSciNet  Google Scholar 

  • Charpiat, G., Keriven, R., Pons, J.P., and Faugeras, O. 2005. Designing spatially coherent minimizing flows for variational problems based on active contours. In 10th International Conference on Computer Vision, Beijing, China.

  • Dervieux, A. and Thomasset, F. 1979. A finite element method for the simulation of Rayleigh-Taylor instability. Lecture Notes in Mathematics, 771:145–159.

  • DoCarmo, M.P. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.

  • Duan, Y., Yang, L., Qin, H., and Samaras, D. 2004. Shape reconstruction from 3D and 2D data using PDE-based deformable surfaces. In European Conference on Computer Vision, vol. 3, pp. 238–251.

  • Faugeras, O. and Keriven, R. 1998. Variational principles, surface evolution, PDE’s, level set methods and the stereo problem. IEEE Transactions on Image Processing, 7(3):336–344.

    Article  MathSciNet  Google Scholar 

  • Goldlücke, B. and Magnor, M. 2004. Space-time isosurface evolution for temporally coherent 3D reconstruction. In International Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 350–355.

  • Goldlücke, B. and Magnor, M. 2004. Weighted minimal hypersurfaces and their applications in computer vision. In European Conference on Computer Vision, vol. 2, pp. 366–378.

  • Jin, H., Soatto, S., and Yezzi, A.J. 2003. Multi-view stereo beyond Lambert. In International Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 171–178.

  • Kass, M., Witkin, A., and Terzopoulos, D. 1987. Snakes: Active contour models. The International Journal of Computer Vision, 1(4):321–331.

    Article  Google Scholar 

  • Kolmogorov, V. and Zabih, R. 2002. Multi-camera scene reconstruction via graph cuts. In European Conference on Computer Vision, vol. 3, pp. 82–96.

  • Kolmogorov, V. and Zabih, R. 2004. What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(2):147–159.

    Article  Google Scholar 

  • Maurel, P., Keriven, R., and Faugeras, O. 2006. Reconciling landmarks and level sets. In International Conference on Pattern Recognition.

  • Michor, P.W. and Mumford, D. 2005. Riemannian geometries of space of plane curves. Preprint.

  • Osher, S. and Fedkiw, R. 2002. The Level Set Method and Dynamic Implicit Surfaces. Springer-Verlag.

  • Osher, S. and Paragios, N. (Eds.) 2003. Geometric Level Set Methods in Imaging, Vision and Graphics. Springer Verlag.

  • Osher, S. and Sethian, J.A. 1988. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79(1):12–49.

    Article  MathSciNet  Google Scholar 

  • Overgaard, N.C. and Solem, J.E. 2005. An analysis of variational alignment of curves in images. In International Conference on Scale Space and PDE Methods in Computer Vision, pp. 480–491.

  • Paragios, N. and Deriche, R. 2005. Geodesic active regions and level set methods for motion estimation and tracking. Computer Vision and Image Understanding, 97(3):259–282.

    Article  Google Scholar 

  • Peng, D., Merriman, B., Osher, S., Zhao, H.-K., and Kang, M. 1999. A PDE-based fast local level set method. Journal of Computational Physics, 155(2):410–438.

    Article  MathSciNet  Google Scholar 

  • Pons, J.-P., Hermosillo, G., Keriven, R., and Faugeras, O. 2003. How to deal with point correspondences and tangential velocities in the level set framework. In International Conference on Computer Vision, vol. 2, pp. 894–899.

  • Rudin, W. 1966. Real and Complex Analysis. McGraw-Hill.

  • 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 University Press.

  • Solem, J.E. and Overgaard, N.C. 2005. A geometric formulation of gradient descent for variational problems with moving surfaces. In International Conference on Scale Space and PDE Methods in Computer Vision, pp. 419–430.

  • Sundaramoorthi, G., Yezzi, A.J., and Mennucci, A. 2005. Sobolev active contours. In IEEE Workshop on Variational and Level Set Methods, Beijing, China, pp. 109–120.

  • Trouvé, A. 1998. Diffeomorphisms groups and pattern matching in image analysis. The International Journal of Computer Vision, 28(3):213–221,

    Article  Google Scholar 

  • Yezzi, A.J. and Mennucci, A.C.G. 2005. Metrics in the space of curves. Preprint.

  • Yezzi, A.J. and Soatto, S. 2003. Deformotion: Deforming motion, shape average and the joint registration and approximation of structures in images. The International Journal of Computer Vision, 53(2):153–167.

    Article  Google Scholar 

  • Younes, L. 1998. Computable elastic distances between shapes. SIAM Journal of Applied Mathematics, 58(2):565–586.

    Article  MathSciNet  Google Scholar 

  • Zhao, H.-K., Osher, S., Merriman, B., and Kang, M. 2000. Implicit and non-parametric shape reconstruction from unorganized points using a variational level set method. Computer Vision and Image Understanding, 80(3):295–314.

    Article  Google Scholar 

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Correspondence to G. Charpiat.

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Charpiat, G., Maurel, P., Pons, JP. et al. Generalized Gradients: Priors on Minimization Flows. Int J Comput Vision 73, 325–344 (2007). https://doi.org/10.1007/s11263-006-9966-2

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  • DOI: https://doi.org/10.1007/s11263-006-9966-2

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