Optical flow and advection on 2-Riemannian manifolds: A common framework

IEEE Transactions on Pattern Analysis and Machine Intelligence

Volumn 30, Issue 6, 2008, Pages 1081-1092

  Lefèvre, Julien ^a   Baillet, Sylvain ^a  

^a Centre National de la Recherche Scientifique   (France)

Author keywords

Advection; Brain imaging; Differential geometry; EEG; Finite element method; MEG; Optical flow; Partial differential equations; Propagation; Riemannian manifold

Indexed keywords

COMPUTATIONAL GEOMETRY; FINITE ELEMENT METHOD; OPTICAL FLOWS; PARTIAL DIFFERENTIAL EQUATIONS; PATTERN RECOGNITION;

BRAIN IMAGING; DIFFERENTIAL GEOMETRY; MOTION EXTRACTION; PATTERN ANALYSIS; RIEMANNIAN MANIFOLD;

MEDICAL IMAGING;

ALGORITHM; ARTICLE; ARTIFICIAL INTELLIGENCE; AUTOMATED PATTERN RECOGNITION; COMPUTER ASSISTED DIAGNOSIS; IMAGE ENHANCEMENT; IMAGE SUBTRACTION; METHODOLOGY; MOTION; REPRODUCIBILITY; SENSITIVITY AND SPECIFICITY; SIGNAL PROCESSING; VIDEORECORDING;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; MOTION; PATTERN RECOGNITION, AUTOMATED; REPRODUCIBILITY OF RESULTS; SENSITIVITY AND SPECIFICITY; SIGNAL PROCESSING, COMPUTER-ASSISTED; SUBTRACTION TECHNIQUE; VIDEO RECORDING;

EID: 43249119866     PISSN: 01628828     EISSN: None     Source Type: Journal    
DOI: 10.1109/TPAMI.2008.51     Document Type: Article

References (47)

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. F. 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. Knerif, 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 or 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 PDF-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.