Abstract
We introduce a method for volumetric cardiac motion analysis using variational optical flow computation involving the prior with the fractional order differentiations. The order of the differentiation of the prior controls the continuity class of the solution. Fractional differentiations is a typical tool for edge detection of images. As a sequel of image analysis by fractional differentiation, we apply the theory of fractional differentiation to a temporal image sequence analysis. Using the fractional order differentiations, we can estimate the orders of local continuities of optical flow vectors. Therefore, we can obtain the optical flow vector with the optimal continuity at each point.
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References
Sorzano, C.Ó.S., Thévenaz, P., Unser, M.: Elastic registration of biological images using vector-spline regularization. IEEE Tr. Biomedical Engineering 52, 652–663 (2005)
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)
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)
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory And Applications of Differentiation and Integration to Arbitrary Order, Dover Books on Mathematics. Dover, New York (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)
Suter, D.: Motion estimation and vector spline. In: Proceedings of CVPR 1994, pp. 939–942 (1994)
Suter, D., Chen, F.: Left ventricular motion reconstruction based on elastic vector splines. IEEE Tr. Medical Imaging, 295–305 (2000)
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)
Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artificial Intelligence 17, 185–204 (1981)
Timoshenko, S.P.: History of Strength of Materials. Dover, New York (1983)
Grenander, U., Miller, M.: Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics 4, 617–694 (1998)
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)
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)
Modersitzki, J.: Numerical Methods for Image Registration. Oxford Univ. Pr., Oxford (2004)
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)
Debbi, L.: Explicit solutions of some fractional partial differential equations via stable subordinators. J. of Applied Mathematics and Stochastic Analysis, Article ID 93502 2006, 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)
Papenberg, N., Bruhn, A., Brox, T., Didas, S., Weickert, J.: Highly accurate optic flow computation with theoretically justified warping. IJCV 67, 141–158 (2006)
Ortiguera, M.D.: Riesz potential operations and inverses via fractional centred derivatives. International Journal of Mathematics and Mathematical Sciences, Article ID 48391, 1–12 (2008)
Krueger, W.M., Phillips, K.: The geometry of differential operator with application to image processing. PAMI 11, 1252–1264 (1989)
Unser, M., Blu, T.: Fractional spline and wavelets. SIAM Review 43, 43–67 (2000)
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Kashu, K., Kameda, Y., Narita, M., Imiya, A., Sakai, T. (2010). Continuity Order of Local Displacement in Volumetric Image Sequence. In: Fischer, B., Dawant, B.M., Lorenz, C. (eds) Biomedical Image Registration. WBIR 2010. Lecture Notes in Computer Science, vol 6204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14366-3_5
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DOI: https://doi.org/10.1007/978-3-642-14366-3_5
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