Abstract
We introduce variational optical flow computation involving priors with fractional order differentiations. Fractional order differentiations are typical tools in signal processing and image analysis. The zero-crossing of a fractional order Laplacian yields better performance for edge detection than the zero-crossing of the usual Laplacian. The order of the differentiation of the prior controls the continuity class of the solution. Therefore, using the square norm of the fractional order differentiation of optical flow field as the prior, we develop a method to estimate the local continuity order of the optical flow field at each point. The method detects the optimal continuity order of optical flow and corresponding optical flow vector at each point. Numerical results show that the Horn-Schunck type prior involving the n + ε order differentiation for 0 < ε< 1 and an integer n is suitable for accurate optical flow computation.
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References
Papenberg, N., Bruhn, A., Brox, T., Didas, S., Weickert, J.: Highly accurate optic flow computation with theoretically justified warping. IJCV 67, 141–158 (2006)
Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Visual Communication and Image Representation 18, 240–252 (2007)
Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. of Computational Physics 220, 813–823 (2007)
Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 55, 293–318 (1992)
Davis, J.A., Smith, D.A., McNamara, D.E., Cottrell, D.M., Campos, J.: Fractional derivatives-analysis and experimental implementation. Applied Optics 32, 5943–5948 (2001)
Tseng, C.-C., Pei, S.-C., Hsia, S.-C.: Computation of fractional derivatives using Fourier transform and digital FIR differentiator. Signal Processing 80, 151–159 (2000)
Zhang, J., Wei, Z.-H.: Fractional variational model and algorithm for image denoising. In: Proceedings of 4th International Conference on Natural Computation, vol. 5, pp. 524–528 (2008)
Mathieu, B., Melchior, P., Oustaloup, A., Ceyral, Cn.: Fractional differentiation for edge detection. Signal Processing 83, 2421–2432 (2003)
Sabatier, J., Agrawel, O.P., Tenreiro Machado, I.A.: Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering. Springer, Netherlands (2007)
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory And Applications of Differentiation And Integration to Arbitrary Order (Dover Books on Mathematics). Dover (2004)
Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, London (1999)
Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artificial Intelligence 17, 185–204 (1981)
Beauchemin, S.S., Barron, J.L.: The computation of optical flow. ACM Computer Surveys 26, 433–467 (1995)
Nagel, H.-H., Enkelmann, W.: An investigation of smoothness constraint for the estimation of displacement vector fields from image sequences. IEEE Trans. on PAMI 8, 565–593 (1986)
Nagel, H.-H.: On the estimation of optical flow:Relations between different approaches and some new results. Artificial Intelligence 33, 299–324 (1987)
Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons and Fractals 31, 1248–1255 (2007)
Murio, D.A.: Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP. Inverse Problems in Science and Engineering 17, 229–243 (2009)
Gorenfloa, R., Abdel-Rehimb, E.A.: Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. of Computational and Applied Mathematics 205, 871–881 (2007)
Debbi. L., Explicit solutions of some fractional partial differential equations via stable subordinators. J. of Applied Mathematics and Stochastic Analysis, Article ID 93502, 1–18 (2006)
Debbi, L.: On some properties of a high order fractional differential operator which is not in general selfadjoint. Applied Mathematical Sciences 1, 1325–1339 (2007)
Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen. 38, L679–L684 (2005)
Duits, R., Felsberg, M., Florack, L.M.J., Platel, B.: α scale spaces on a bounded domain. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 502–518. Springer, Heidelberg (2003)
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Kashu, K., Kameda, Y., Imiya, A., Sakai, T., Mochizuki, Y. (2009). Computing the Local Continuity Order of Optical Flow Using Fractional Variational Method. In: Cremers, D., Boykov, Y., Blake, A., Schmidt, F.R. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2009. Lecture Notes in Computer Science, vol 5681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03641-5_12
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DOI: https://doi.org/10.1007/978-3-642-03641-5_12
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