This Article 
 Bibliographic References 
 Add to: 
Optical Flow and Advection on 2-Riemannian Manifolds: A Common Framework
June 2008 (vol. 30 no. 6)
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.

[1] G. Allaire, Analyse Numérique et Optimisation, first ed. Editions de l'Ecole Polytechnique, 2005.
[2] G. Aubert, R. Deriche, and P. Kornprobst, “Computing Optical Flow via Variational Techniques,” SIAM J. Applied Math., vol. 60, no. 1, pp. 156-182, 1999.
[3] P. Azerad and G. Pousin, “Inégalité de Poincaré Courbe Pour le Traitement Variationnel de l'équation de Transport,” Comptes rendus de l'Académie des Sciences, pp. 721-727, 1996.
[4] S. Baillet, J.C. Mosher, and R.M. Leahy, “Electromagnetic Brain Mapping,” IEEE Signal Processing Magazine, Nov. 2001.
[5] J.L. Barron, D.J. Fleet, and S.S. Beauchemin, “Performance of Optical Flow Techniques,” Int'l J. Computer Vision, vol. 12, pp. 43-77, 1994.
[6] J.L. Barron and A. Liptay, “Measuring 3-D Plant Growth Using Optical Flow,” Bioimaging, vol. 5, pp. 82-86, 1997.
[7] S.S. Beauchemin and J.L. Barron, “The Computation of Optical Flow,” ACM Computing Surveys, vol. 27, no. 3, pp. 433-467, 1995.
[8] D. Bereziat, I. Herlin, and L. Younes, “A Generalized Optical Flow Constraint and Its Physical Interpretation,” Proc. Int'l Conf. Computer Vision and Pattern Recognition, vol. 2, p. 2487, 2000.
[9] M. Bertalmio, L.T. Cheng, S. Osher, and G. Sapiro, “Variational Problems and Partial Differential Equations on Implicit Surfaces,” J. Computational Physics, vol. 174, no. 2, pp. 759-780, 2001.
[10] O. Besson and G. De Montmollin, “Space-Time Integrated Least Squares: A Time-Marching Approach,” Int'l J. Numerical Methods in Fluids, vol. 44, no. 5, pp. 525-543, 2004.
[11] D.L. Book, J.P. Boris, and K. Hain, “Flux-Corrected Transport—II: Generalizations of the Method,” J. Computational Physics, vol. 18, pp. 248-283, 1975.
[12] A. Bossavit, “Whitney Forms : A Class of Finite Elements for Three-Dimensional Computations in Electromagnetism,” IEE Proc., part A, vol. 135, no. 8, pp. 493-500, Nov. 1988.
[13] N.E.H. Bowler, C.E. Pierce, and A. Seed, “Development of a Precipitation Nowcasting Algorithm Based upon Optical Flow Techniques,” J. Hydrology, vol. 288, pp. 74-91, 2004.
[14] A. Cachia, J.F. Mangin, D. Riviere, F. Kherif, N. Boddaert, A. Andrade, D. Papadopoulos-Orfanos, J.B. Poline, I. Bloch, M. Zilbovicius, P. Sonigo, F. Brunelle, and J. Regis, “A Primal Sketch of the Cortex Mean Curvature: A Morphogenesis Based Approach to Study the Variability of the Folding Patterns,” IEEE Trans. Medical Imaging, vol. 22, no. 6, pp. 754-765, June 2003.
[15] U. Clarenz, M. Rumpf, and A. Telea, “Finite Elements on Point Based Surfaces,” Proc. Eurographics Symp. Point-Based Graphics, pp.201-211, 2004.
[16] T. Corpetti, D. Heitz, G. Arroyo, E. Mémin, and A. Santa-Cruz, “Fluid Experimental Flow Estimation Based on an Optical-Flow Scheme,” Experiments in Fluids, vol. 40, no. 1, pp. 80-97, 2006.
[17] T. Corpetti, E. Mémin, and P. Pérez, “Dense Estimation of Fluid Flows,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 3, pp. 365-380, Mar. 2002.
[18] U. Diewald, T. Preusser, and M. Rumpf, “Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 2, pp.139-149, 2000.
[19] M.P. Do Carmo, Riemannian Geometry. Birkhauser, 1993.
[20] O. Druet, E. Hebey, and F. Robert, Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Princeton Univ. Press, chapter background material, pp. 1-12, 2004.
[21] A. Giachetti, M. Campani, and V. Torre, “The Use of Optical Flow for Road Navigation,” IEEE Trans. Robotics and Automation, vol. 14, pp. 34-48, 1998.
[22] J.L. Guermond, “A Finite Element Technique for Solving First Order PDE's in L1,” SIAM J. Numerical Analysis, vol. 42, no. 2, pp.714-737, 2004.
[23] B.K.P. Horn and B.G. Schunck, “Determining Optical Flow,” Artificial Intelligence, vol. 17, pp. 185-204, 1981.
[24] A. Imiya, H. Sugaya, A. Torii, and Y. Mochizuki, “Variational Analysis of Spherical Images,” Proc. Computer Analysis of Images and Patterns, 2005.
[25] T. Inouye, K. Shinosaki, S. Toi, Y. Matsumoto, and N. Hosaka, “Potential Flow of Alpha-Activity in the Human Electroencephalogram,” Neuroscience Letters, vol. 187, pp. 29-32, 1995.
[26] V.K. Jirsa, K.J. Jantzen, A. Fuchs, and J.A.S. Kelso, “Spatiotemporal Forward Solution of the EEG and MEG Using Networkmodeling,” IEEE Trans. Medical Imaging, vol. 21, no. 5, pp. 493-504, 2002.
[27] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge Univ. Press, 1987.
[28] J. Lefèvre, G. Obosinski, and S. Baillet, “Imaging Brain Activation Streams from Optical Flow Computation on 2-Riemannian Manifold,” Proc. 20th Int'l Conf. Information Processing in Medical Imaging, 2007.
[29] C. Lenglet, R. Deriche, and O. Faugeras, Inferring White Matter Geometry from Diffusion Tensor MRI: Application to Connectivity Mapping. Pajdla and Matas, 2004.
[30] H. Liu, T. Hong, M. Herman, T. Camus, and R. Chellapa, “Accuracy versus Efficiency Trade-Off in Optical Flow Algorithms,” Computer Vision and Image Understanding, vol. 72, pp. 271-286, 1998.
[31] L. Lopez-Perez, R. Deriche, and N. Sochen, “The Beltrami Flow over Triangulated Manifolds,” Computer Vision and Math. Methods in Medical and Biomedical Image Analysis: ECCV 2004 Workshops CVAMIA and MMBIA, May 2004, revised selected papers.
[32] L.M. Lui, Y. Wang, and T.F. Chan, “Solving PDEs on Manifold Using Global Conformal Parameterization,” Proc. Third IEEE Workshop Variational, Geometric and Level Set Methods in Computer Vision, pp. 307-319, 2005.
[33] F. Mémoli, G. Sapiro, and S. Osher, “Solving Variational Problems and Partial Differential Equations Mapping into General Target Manifolds,” J. Computational Physics, vol. 195, no. 1, pp. 263-292, 2004.
[34] H.-H. Nagel, “On the Estimation of Optical Flow: Relations between Different Approaches and Some New Results,” Artificial Intelligence, vol. 33, pp. 299-324, 1987.
[35] P.L. Nunez, “Toward a Quantitative Description of Large-Scale Neocortical Dynamic Function and EEG,” Behavioral and Brain Sciences, vol. 23, no. 3, pp. 371-398, 2000.
[36] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian Framework for Tensor Computing,” Int'l J. Computer Vision, vol. 66, no. 1, pp.41-66, 2006.
[37] P. Perrochet and P. Azérad, “Space-Time Integrated Least-Squares: Solving a Pure Advection Equation with a Pure Diffusion Operator,” J. Computational Physics, vol. 117, pp. 183-193, 1995.
[38] J.A. Rossmanith, D.S. Bale, and R.J. LeVeque, “A Wave Propagation Algorithm for Hyperbolic Systems on Curved Manifolds,” J.Computational Physics, vol. 199, pp. 631-662, 2004.
[39] C. Schnorr, “Determining Optical Flow for Irregular Domains by Minimizing Quadratic Functionals of a Certain Class,” Int'l J. Computer Vision, vol. 6, no. 1, pp. 25-38, 1991.
[40] N. Sochen, R. Deriche, and L. Lopez Perez, “The Beltrami Flow over Implicit Manifolds,” Proc. Ninth IEEE Int'l Conf. Computer Vision, p. 832, 2003.
[41] A. Spira and R. Kimmel, “Geometric Curve Flows on Parametric Manifolds,” J. Computational Physics, vol. 223, no. 1, pp. 235-249, 2007.
[42] Z. Sun, G. Bebis, and R. Miller, “On-Road Vehicle Detection: A Review,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 5, pp. 694-711, May 2006.
[43] A. Torii, A. Imiya, H. Sugaya, and Y. Mochizuki, “Optical Flow Computation for Compound Eyes: Variational Analysis of Omni-Directional Views,” Proc. First Int'l Symp. Brain, Vision, and Artificial Intelligence, 2005.
[44] D. Tschumperle and R. Deriche, “Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 4, pp. 506-517, May 2005.
[45] K. Wang, Weiwei, Y. Tong, M. Desbrun, and P. Schroder, “Edge Subdivision Schemes and the Construction of Smooth Vector Fields,” ACM Trans. Graphics, vol. 25, no. 3, pp. 1041-1048, 2006.
[46] J. Weickert and C. Schnorr, “A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion,” Int'l J. Computer Vision, vol. 45, no. 3, pp. 245-264, Dec. 2001.
[47] D. Cosmelli, O. David, J.P. Lachaux, J. Martinerie, L. Garnero, B. Renault, and F. Varela, “Waves of Consciousness: Ongoing Cortical Patterns during Binocular Rivalry,” Neuroimage, vol. 23, no. 1, pp. 128-140, 2004.

Index Terms:
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
Julien Lefèvre, Sylvain Baillet, "Optical Flow and Advection on 2-Riemannian Manifolds: A Common Framework," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 6, pp. 1081-1092, June 2008, doi:10.1109/TPAMI.2008.51
Usage of this product signifies your acceptance of the Terms of Use.