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Issue No. 06 - June (2008 vol. 30)
ISSN: 0162-8828
pp: 1081-1092
Julien Lefèvre , CNRS, Paris
Sylvain Baillet , CNRS, Paris
Dynamic pattern analysis and motion extraction can be efficiently addressed using optical flow techniques. This article presents a generalization of these questions to non-flat surfaces, where optical flow is tackled through the problem of evolution processes on non-Euclidian domains. The classical equations of optical flow in the Euclidian case are transposed to the theoretical framework of differential geometry. We adopt this formulation for the regularized optical flow problem, prove its mathematical well-posedness and combine it with the advection equation. The optical flow and advection problems are dual: a motion field may be retrieved from some scalar evolution using optical flow; conversely, a scalar field may be deduced from a velocity field using advection. These principles are illustrated with qualitative and quantitative evaluations from numerical simulations bridging both approaches. The proof-of-concept is further demonstrated with preliminary results from time-resolved functional brain imaging data, where organized propagations of cortical activation patterns are evidenced using our approach.
Partial Differential Equations, Finite element methods, Hyperbolic equations, Elliptic equations, Approximation of surfaces and contours, Global optimization, Data and knowledge visualization, Feature extraction or construction, Time-varying imagery, Biology and genetics

S. Baillet and J. Lefèvre, "Optical Flow and Advection on 2-Riemannian Manifolds: A Common Framework," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 30, no. , pp. 1081-1092, 2008.
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