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Języki publikacji
Abstrakty
Optimal control problems governed by a transport equation are investigated that are motivated by optical flow problems. The control is given by the velocity field, corresponding to the optical flow, while the state corresponds to the brightness of image points. The problem is studied in the setting of spatially BVregular vector fields under very low regularity requirements. Existing stability results for the control-to-state operator are improved and based on this the existence of minimizers for several classes of optimal control problems is proved under mild assumptions on the admissible sets.
Czasopismo
Rocznik
Tom
Strony
261--306
Opis fizyczny
Bibliogr. 31 poz., tab.
Twórcy
autor
- Technische Universität München, Department of Mathematics, Boltzmannstr. 3, 85747 Garching, Germany
autor
- Technische Universität München, Department of Mathematics, Boltzmannstr. 3, 85747 Garching, Germany
Bibliografia
- Aliprantis, C. and Border, K. (2006) Infinite Dimensional Analysis. Springer Verlag.
- Ambrosio, L. (2004) Transport equation and Cauchy problem for BV vector fields. Inventiones mathematicae, 158, 2, 227–260.
- Ambrosio, L., Caffarelli, L., Crandall, M., Evans, L. and Fusco, N. (2008) Calculus of Variations and Nonlinear Partial Differential Equations. Springer Verlag.
- Ambrosio, L., Fusco, N. and Pallara, D. (2000) Functions of Bounded Variations and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford University Press, New York.
- Ambrosio, L., Gigli, N. and Savare, G. (2008) Gradient Flows. Springer Verlag.
- Attouch, H., Buttazzo, G. and Michaille, G. (2014) Variational Analysis in Sobolev and BV Spaces. MPS-SIAM.
- Aubert, G. and Kornprobst, P. (2006) Mathematical Problems in Image Processing. Springer Verlag.
- Borzì, A., Ito, K. and Kunisch, K. (2002) Optimal control formulation for determining optical flow. SIAM Journal on Scientific Computing, 24, 3, 818–847.
- Baker, S., Scharstein, D., Lewis, J., Roth, S., Black M. And Szeliski, R. (2011) A Database and Evaluation Methodology for Optical Flow. International Journal of Computer Vision, 92, 1, 1–31.
- Chen, K. (2011) Optimal control based image sequence interpolation. Mathematical Department, Universität Bremen.
- Chen, K. and Lorenz, D. (2011) Image Sequence Interpolation Using Optimal Control. Journal of Mathematical Imaging and Vision, 41, 3, 222–238.
- Cornet, B. and Martins da Rocha, V.-F. (2004) Fatou’s Lemma for unbounded Gelfand integrable mappings. Working papers series in theoretical and applied economics. Paper no. 200503. University of Kansas.
- Crippa, G. (2007) The flow associated to weakly differentiable vector fields. Ph.D. Thesis, Classe di Scienze Matematiche, Fisiche e Natural, Scuola Normale Superiore di Pisa.
- Crippa, G., Donadello, C. and Spinolo, L. (2014) Initial-boundary value problems for continuity equations with BV coefficients. Journal de mathématiques pures et appliqués, 102, 1, 79–98.
- Crippa, G., Donadello, C. and Spinolo, L. (2014) A note on the initialboundary value problem for continuity equations with rough coefficients. Hyperbolic problems: theory, numerics, applications, 957–966.
- De Lellis, C. (2006/2007) Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio (d’aprés Ambrosio, DiPerna, Lions). Séminaire Bourbaki, 2006/2007, 972.
- Diestel, J., Ruess, W. and Schachermayer, W. (1993) Weak compactness in L1(μ,X). Proceedings of the American Mathematical Society, 118, 2, 447–453.
- DiPerna, R. and Lions, P. (1989) Ordinary differential equations, transport theory and Sobolev spaces. Inventiones mathematicae, 98, 3, 511–547.
- Emmrich, E. (2004) Gew¨ohnliche und Operator-Differentialgleichung. Vieweg+Teubner Verlag.
- Evans, L. and Gariepy, R. (1992) Measure Theory and Fine Properties of Functions. CRC Press.
- Hinterberger, W. and Scherzer, O. (2001) Models for image interpolation based on the optical flow. Computing, 66, 3, 231–247.
- Horn, B. and Schunck, B. (1981) Determining optical flow. Artificial Intelligence, 17, 185–203.
- Jarde, P. (2018) Analysis of optimal control problems for the optical flow equation under mild regularity assumptions. Doctoral thesis, Fakultät für Mathematik, Technische Universität München.
- Lucas, B. and Kanade, T. (1981) An iterative image registration technique with an application to image to stereo vision. Proceedings of the 7th International Joint Conference on Artificial Intelligence, 2, 674–679.
- Moussa, A. (2016) Some variants of the classical Aubin-Lions Lemma. Journal of Evolution Equations, 16, 1, 65–93.
- Murat, F. (2005) Compacit´e par compensation: condition nécessaire et suffisante de continuité faible sous une hypoth`ese de rang constant. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 8, 1, 69–102.
- Okada, S., Ricker, W. and Pérez, E. (2008) Optimal Domain and Integral Extension of Operators. Birkhäuser.
- Pełczyński, A. and Wojciechowski, M. (2003) Spaces of functions with bounded variation and Sobolev spaces without local unconditional structure. Journal für die Reine und Angewandte Mathematik, 558, 109–157.
- Schweizer, B. (2013) Partielle Differentialgleichungen. Springer Verlag.
- Tartar, L. (1979) Compensated compactness and applications to partial differential equations. Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium. Pitman Press, 4, 136–212.
- Werner, D. (2011) Funktionalanalysis. Springer Verlag.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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