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Optimal extended optical flow subject to a statistical constraint. (English) Zbl 1186.92027

Summary: This work is motivated by the necessity to improve heart image tracking. This technique is related to the ability of generating an apparent continuous motion, which is observable through the variation of intensity from a starting image to an ending one. Given two images \(\rho_0\) and \(\rho_1\), we calculate an evolution process \(\rho(t,\cdot)\) which transports \(\rho_0\) to \(\rho_1\) by using the optimal extended optical flow. Such a strategy is found to be well suited for heart image tracking, provided the motion is controlled by a statistical model. We use viability theory to give sufficient conditions to handle the optimal extended optical flow subject to a point-wise statistical constraint by using Parzen’s approximation. The strategy is implemented in a 1D case and the numerical results which are presented show the efficiency of the proposed strategy.

MSC:

92C55 Biomedical imaging and signal processing
62H35 Image analysis in multivariate analysis
92C50 Medical applications (general)
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[1] Lynch, M.; Ghita, O.; Whelan, P. F., Segmentation of the left ventricle of the heart in \(3 D + t\) MRI data using an optimised non-rigid temporal model, IEEE TMI, 2, 195-203 (2008)
[3] Wolberg, G., Recent advances in image morphing, (Proc. Int. Conf. Comput. Graph. (1996), IEEE), 64-71
[4] Beier, T.; Neely, S., Feature-based image metamorphosis, (Proc. SIGGRAPH, vol. 26 (1992)), 35-42
[5] Aubert, G.; Deriche, R.; Kornprobst, P., Computing optical flow problem via variational techniques, SIAM J. Appl. Math., 80, 156-182 (1999) · Zbl 0942.35057
[6] Keeling, S.; Ring, W., Medical image registration and interpolation by optical flow with maximal rigidity, J. Math. Imaging Vis., 47-65 (2005) · Zbl 1478.94053
[8] Calow, D.; Kruger, N.; Worgotter, F.; Lappe, M., Statistics of optic flow for self-motion through natural scenes, Dynamic Perception, 133-138 (2004)
[9] Fermuller, C.; Shulman, D.; Aloimonos, Y., The statistics of optical flow, CVIU, 82, 1, 1-32 (2001) · Zbl 1011.68557
[10] Roth, S.; Black, M. J., On the spatial statistics of optical flow, Int. J. Comput. Vis., 74, 1, 33-50 (2007)
[11] Benamou, J.; Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 375-393 (2000) · Zbl 0968.76069
[12] Villani, C., Topics in optimal transportation, (Graduate Studies in Math, vol. 58 (2003), Amer. Math. Soc. Providence) · Zbl 1106.90001
[13] Benamou, J.; Brenier, Y.; Guittet, K., The Monge-Kantorovich mass transfer and its computational fluid Mechanics formulation, Int. J. Numer. Math. Fluids, 1-2, 21-30 (2002) · Zbl 1058.76586
[14] Delhay, B.; Lötjönen, J.; Clarysse, P.; Katila, T.; Magnin, I. E., A dynamic 3-D cardiac surface model from MR images, Comput. Cardiol. (2005)
[16] Tsybakov, A., (Introduction à l’estimation non paramétrique. Introduction à l’estimation non paramétrique, Mathématiques et applications, vol. 41 (2003), Springer) · Zbl 1029.62034
[17] Evans, L. C., (Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (2002), AMS)
[18] Crouzeix, M.; Mignot, A. L., Analyse Numerique Des Equations Differentielles (1989), Masson: Masson Paris · Zbl 0709.65054
[19] Aubin, T., Viability Theory (1991), Birkhauser: Birkhauser Bale · Zbl 0755.93003
[20] Carja, O.; Necula, M.; Vrabie, I. I., (Viability, Invariance and Applications. Viability, Invariance and Applications, Mathematics Studies, vol. 207 (2007), North-Holland) · Zbl 1239.34068
[21] Rokafellar, R. T.; Wets, J. B., Variational Analysis (2005), Springer Verlag: Springer Verlag Berlin
[23] Najman, L.; Cousty, J.; Couprie, M.; Talbot, H.; Clément-Guinaudeau, S.; Goissen, T.; Garot, J., An open, clinically-validated database of \(3 D + t\) cine-MR images of the left ventricle with associates manual and automated segmentation, Insight Journal (2007)
[24] Clarysse, P.; Basset, C.; Khouas, L.; Croisille, P.; Friboulet, D.; Odet, C.; Magnin, I. E., 2D spatial and temporal displacement field fitting from cardiac MR tagging, Med. Image Anal., 3, 253-268 (2000)
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