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Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing

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Abstract

In the present paper, we prove a substantially improved version of the Pontryagin maximum principle for convex multidimensional control problems of Dieudonné-Rashevsky type. Although the range of the operator describing the first-order PDE system involved in this problem has infinite codimension, we obtain first-order necessary conditions in a completely analogous form as in the one-dimensional case. Furthermore, the adjoint variables are subjected to a Weyl decomposition.

We reformulate two basic problems of mathematical image processing (determination of optical flow and shape from shading problem) within the framework of optimal control, which gives the possibility to incorporate hard constraints in the problems. In the convex case, we state the necessary optimality conditions for these problems.

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References

  1. Cesari, L.: Optimization with partial differential equations in Dieudonné-Rashevsky form and conjugate problems. Arch. Ration. Mech. Anal. 33, 339–357 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  2. Funk, P.: Variationsrechnung und ihre Anwendung in Physik und Technik. Grundlehren, vol. 94. Springer, Berlin (1962)

    MATH  Google Scholar 

  3. Lur’e, K.A. (Лурье, К.А.): Оптимальное управление в задачах математической физики. Nauka, Moskva (1975)

    Google Scholar 

  4. Wagner, M.: Erweiterungen des mehrdimensionalen Pontrjaginschen Maximumprinzips auf meßbare und beschränkte sowie distributionelle Steuerungen. PhD Thesis, Universität Leipzig (1996)

  5. Sauer, E.: Schub und Torsion bei elastischen prismatischen Balken. Verlag Wilhelm Ernst & Sohn, Berlin (1980) (Mitteilungen aus dem Institut für Massivbau der TH Darmstadt 29)

    Google Scholar 

  6. Ting, T.W.: Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech. 19, 531–551 (1969)

    MATH  MathSciNet  Google Scholar 

  7. Ting, T.W.: Elastic-plastic torsion problem III. Arch. Ration. Mech. Anal. 34, 228–244 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chmelka, F., Melan, E.: Einführung in die Festigkeitslehre, 5th edn. Springer, Wien (1976)

    Google Scholar 

  9. Andrejewa, J.A., Klötzler, R.: Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. Teil I. Z. Angew. Math. Mech. 64, 35–44 (1984)

    Article  MathSciNet  Google Scholar 

  10. Andrejewa, J.A., Klötzler, R.: Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. Teil II. Z. Angew. Math. Mech. 64, 147–153 (1984)

    Article  MathSciNet  Google Scholar 

  11. Dacorogna, B., Marcellini, P.: General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178, 1–37 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dacorogna, B., Marcellini, P.: Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal. 152, 404–446 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dacorogna, B., Marcellini, P.: Implicit Partial Differential Equations. Birkhäuser, Boston (1999)

    MATH  Google Scholar 

  14. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren, vol. 170. Springer, Berlin (1971)

    MATH  Google Scholar 

  15. Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen—Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden (2005)

    MATH  Google Scholar 

  16. Klötzler, R.: Optimal transportation flows. J. Anal. Appl. 14, 391–401 (1995)

    MATH  Google Scholar 

  17. Klötzler, R.: Strong duality for transportation flow problems. J. Anal. Appl. 17, 225–228 (1998)

    MATH  Google Scholar 

  18. Pickenhain, S.: Beiträge zur Theorie mehrdimensionaler verallgemeinerter Steuerungsprobleme. Habilitation Thesis, Universität Leipzig (1991)

  19. Pickenhain, S.: Sufficiency conditions for weak local minima in multidimensional optimal control problems with mixed control-state restrictions. Z. Anal. Anw. 11, 559–568 (1992)

    MATH  MathSciNet  Google Scholar 

  20. Pickenhain, S.: Duality theory for state-constrained control problems governed by a first order PDE system. Control Cybern. 31, 833–845 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Pickenhain, S., Tammer, K.: Sufficient conditions for local optimality in multidimensional control problems with state restrictions. Z. Anal. Anw. 10, 397–405 (1991)

    MATH  MathSciNet  Google Scholar 

  22. Pickenhain, S., Wagner, M.: Optimal control problems with a first-order PDE system—necessary and sufficient optimality conditions. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 159–172. Springer, Berlin (2001)

    Google Scholar 

  23. Wagner, M.: Transportation flow problems with Radon measure variables. Discuss. Math. Differ. Incl. Control Optim. 20, 93–111 (2000)

    MATH  MathSciNet  Google Scholar 

  24. Klötzler, R.: On a general conception of duality in optimal control. In: Fábera, J. (ed.) Equadiff IV. Proceedings of the Czechoslovak Conference on Differential Equations and their Applications held in Prague, August 22–26, 1977. Lecture Notes in Mathematics, vol. 703, pp. 189–196. Springer, Berlin (1979)

    Chapter  Google Scholar 

  25. Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Zuhair Nashed, M., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 251–268. AMS, Providence (2002)

    Google Scholar 

  26. Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in PDE-based computation of image motion. Int. J. Comput. Vis. 45, 245–264 (2001)

    Article  MATH  Google Scholar 

  27. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  28. Chipot, M., March, R., Vitulano, D.: Numerical analysis of oscillations in a nonconvex problem related to image selective smoothing. J. Comput. Appl. Math. 136, 123–133 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Brune, C.: Berechnung des Optischen Flusses und Kantenerkennung mit Optimierungsmethoden. Diploma Thesis, Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik (2007)

  30. Brune, C., Maurer, H., Wagner, M.: Edge detection within optical flow via multidimensional control. Brandenburgische Technische Universität Cottbus, Preprint-Reihe Mathematik, Preprint Nr. M-02/2008 (2008, submitted)

  31. Wagner, M.: Mehrdimensionale Steuerungsprobleme mit quasikonvexen Integranden. Habilitation Thesis, Brandenburgische Technische Universität Cottbus (2006)

  32. Franek, L.: Anwendung optimaler Steuerungsprobleme mit L ∞-Steuerbeschränkung auf ein Modellproblem der Bildverarbeitung. Diploma Thesis, Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik (2007)

  33. Franek, M.: Bildentrauschung und Kantenerkennung mit L p-Regularisierung und Gradientenbeschränkung bei Graustufenbildern. Diploma Thesis, Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik (2007)

  34. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  35. Ioffe, A.D., Tichomirow, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)

    MATH  Google Scholar 

  36. Ginsburg, B., Ioffe, A.D.: The maximum principle in optimal control of systems governed by semilinear equations. In: Mordukhovich, B.S., Sussmann, H.J. (eds.) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. IMA Volumes in Mathematics and its Applications, vol. 78, pp. 81–110. Springer, New York (1996)

    Google Scholar 

  37. Wagner, M.: Pontryagin’s maximum principle for Dieudonné-Rashevsky type problems involving Lipschitz functions. Optimization 46, 165–184 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Rund, H.: Sufficiency conditions for multiple integral control problems. J. Optim. Theory Appl. 13, 125–138 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  39. Klötzler, R.: On Pontryagin’s maximum principle for multiple integrals. Beitr. Anal. 8, 67–75 (1976)

    Google Scholar 

  40. Pickenhain, S., Wagner, M.: Piecewise continuous controls in Dieudonné-Rashevsky type problems. J. Optim. Theory Appl. 127, 145–163 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  41. Pickenhain, S.: Maximum principles for multidimensional relaxed control problems. In: Kall, P. (ed.) System Modelling and Optimization. Proceedings of the Fifteenth IFIP Conference held at the University of Zurich, Zurich, September 2–6, 1991. Lecture Notes in Control and Information Sciences, vol. 180, pp. 424–432. Springer, Berlin (1992)

    Google Scholar 

  42. Pickenhain, S., Wagner, M.: Critical points in relaxed deposit problems. In: Ioffe, A., Reich, S., Shafrir, I. (eds.) Calculus of Variations and Optimal Control, Technion 98, vol. II. Research Notes in Mathematics, vol. 411, pp. 217–236. Chapman & Hall/CRC, Boca Raton (2000)

    Google Scholar 

  43. Pickenhain, S., Wagner, M.: Pontryagin principle for state-constrained control problems governed by a first-order PDE system. J. Optim. Theory Appl. 107, 297–330 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  44. Klötzler, R.: Pontryagin’s maximum principle for multiple integrals. In: Kall, P. (ed.) System Modelling and Optimization. Proceedings of the Fifteenth IFIP Conference held at the University of Zurich, Zurich, September 2–6, 1991. Lecture Notes in Control and Information Sciences, vol. 180, pp. 323–333. Springer, Berlin (1992)

    Chapter  Google Scholar 

  45. Klötzler, R., Pickenhain, S.: Pontryagin’s maximum principle for multidimensional control problems. In: Bulirsch, R., Miele, A., Stoer, J., Well, K.H. (eds.) Optimal Control. Calculus of Variations, Optimal Control Theory and Numerical Methods. Papers from the Second Conference held at the University of Freiburg, Freiburg, May 26–June 1, 1991. International Series of Numerical Mathematics, vol. 111, pp. 21–30. Birkhäuser, Basel (1993)

    Google Scholar 

  46. Klötzler, R., Pickenhain, S.: Stability and maximum principle for multiple integral control problems. Schwerpunktprogramm der Deutschen Forschungsgemeinschaft—Anwendungsbezogene Optimierung und Steuerung, Report Nr. 505, Universität Leipzig, Fachbereich Mathematik/Informatik (1994)

  47. Pickenhain, S.: A pointwise maximum principle in optimal control with multiple integrals. Optimization 38, 343–355 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  48. Pickenhain, S., Wagner, M.: Minimizing sequences in class-qualified deposit problems. Set-Valued Anal. 14, 105–120 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  49. Cellina, A., Perrotta, S.: On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient. SIAM J. Control Optim. 36, 1987–1998 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  50. Ishii, H., Loreti, P.: Relaxation in an L ∞-optimization problem. Proc. R. Soc. Edinb. Ser. A 133, 599–615 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  51. Ishii, H., Loreti, P.: Relaxation of Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 169, 265–304 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  52. Klee, V.L.: Separation properties of convex cones. Proc. Am. Math. Soc. 6, 313–318 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  53. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic/Elsevier, Amsterdam (2007)

    Google Scholar 

  54. Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York (1988)

    MATH  Google Scholar 

  55. Fichtenholz, G.M.: Differential- und Integralrechnung, Band III, 12th edn. Harri Deutsch, Thun (1992)

    Google Scholar 

  56. Simader, C., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in L q-spaces for bounded and exterior domains. In: Galdi, G.P. (ed.) Mathematical Problems Relating to the Navier-Stokes Equation, pp. 1–35. World Scientific, Singapore (1992)

    Google Scholar 

  57. Simader, C., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics, vol. 360. Longman, Harlow (1996)

    MATH  Google Scholar 

  58. Mitrea, D.: Sharp L p-Hodge decompositions for Lipschitz domains in ℝ2. Adv. Differ. Equ. 7, 343–364 (2002)

    MATH  MathSciNet  Google Scholar 

  59. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Grundlehren, vol. 130. Springer, Berlin (1966)

    MATH  Google Scholar 

  60. Florack, L., Niessen, W., Nielsen, M.: The intrinsic structure of optic flow incorporating measurement duality. Int. J. Comput. Vis. 27, 263–286 (1998)

    Article  Google Scholar 

  61. Brox, T., Bruhn, A., Weickert, J.: Variational motion segmentation with level sets. In: Bischof, H., Leonardis, A., Pinz, A. (eds.) Computer Vision—ECCV 2006. Lecture Notes in Computer Science, vol. 3951, pp. 471–483. Springer, Berlin (2006)

    Chapter  Google Scholar 

  62. Grossauer, H.: Inpainting of movies using optical flow. In: Scherzer, O. (ed.) Mathematical Models for Registration and Applications to Medical Imaging. Mathematics in Industry, vol. 10, pp. 151–162. Springer, Berlin (2006)

    Chapter  Google Scholar 

  63. Hinterberger, W., Scherzer, O.: Models for image interpolation based on the optical flow. Computing 66, 231–247 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  64. Hinterberger, W., Scherzer, O., Schnörr, C., Weickert, J.: Analysis of optical flow models in the framework of the calculus of variations. Numer. Funct. Anal. Optim. 23, 69–89 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  65. Slesareva, N., Bruhn, A., Weickert, J.: Optic flow goes stereo: a variational method for estimating discontinuity-preserving dense disparity maps. In: Kropatsch, W., Sablatnig, R., Hanbury, A. (eds.) Pattern Recognition. 27th DAGM Symposium, Vienna, Austria, August 31–September 2, 2005, Proceedings. Lecture Notes in Computer Science, vol. 3663, pp. 33–40. Springer, Berlin (2005)

    Google Scholar 

  66. Aubert, G., Deriche, R., Kornprobst, P.: Computing optical flow via variational techniques. SIAM J. Appl. Math. 60, 156–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  67. Kornprobst, P., Deriche, R., Aubert, G.: Image sequence analysis via partial differential equations. J. Math. Imaging Vis. 11, 5–26 (1999)

    Article  MathSciNet  Google Scholar 

  68. Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17, 185–203 (1981)

    Article  Google Scholar 

  69. Enkelmann, W.: Investigations of multigrid algorithms for the estimation of optical flow fields in image sequences. Comput. Vis. Graph. Image Process. 43, 150–177 (1988)

    Article  Google Scholar 

  70. Borzì, A., Ito, K., Kunisch, K.: An optimal control approach to optical flow computation. Int. J. Numer. Methods Fluids 40, 231–240 (2002)

    Article  MATH  Google Scholar 

  71. Borzì, A., Ito, K., Kunisch, K.: Optimal control formulation for determining optical flow. SIAM J. Sci. Comput. 24, 818–847 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  72. Alvarez, L., Weickert, J., Sánchez, J.: Reliable estimation of dense optical flow fields with large displacements. Int. J. Comput. Vis. 39, 41–56 (2000)

    Article  MATH  Google Scholar 

  73. Barnes, I., Zhang, K.: Instability of the eikonal equation and Shape from Shading. M2AN Math. Model. Numer. Anal. 34, 127–138 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  74. Piechullek, C.: Oberflächenrekonstruktion mit Hilfe einer Mehrbild-Shape-from-Shading-Methode. Verlag der Bayerischen Akademie der Wissenschaften in Kommission bei der C.H. Beck’schen Verlagsbuchhandlung München, München (2000). (Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften, Dissertationen, Reihe C, Heft Nr. 518)

    Google Scholar 

  75. Lions, P.L., Rouy, E., Tourin, A.: Shape-from-Shading, viscosity solutions and edges. Numer. Math. 64, 323–353 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  76. Horn, B.K.P., Brooks, M.J.: The variational approach to Shape from Shading. Comput. Vis. Graph. Image Process. 33, 174–208 (1986)

    Article  Google Scholar 

  77. Ikeuchi, K., Horn, B.K.P.: Numerical Shape from Shading and occluding boundaries. Artif. Intell. 17, 141–184 (1981)

    Article  Google Scholar 

  78. Zheng, Q., Chellappa, R.: Estimation of illuminant direction, albedo, and shape from shading. IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991)

    Article  Google Scholar 

  79. Durou, J.-D., Falcone, M., Sagona, M.: A survey of numerical methods for Shape from Shading. IRIT, Université Paul Sabatier, Toulouse, Rapport de recherche IRIT No. 2004-2-R (2004)

  80. Zhang, R., Tsai, P.-S., Cryer, J.E., Shah, M.: Shape from shading: a survey. IEEE Trans. Pattern Anal. Mach. Intel. 21, 690–706 (1999)

    Article  Google Scholar 

  81. Chambolle, A.: A uniqueness result in the theory of stereo vision: coupling Shape from Shading and binocular information allows unambiguous depth reconstruction. Ann. Inst. H. Poincaré—Anal. Non Linéaire 11, 1–16 (1994)

    MATH  MathSciNet  Google Scholar 

  82. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren, vol. 317. Springer, Berlin (1998)

    MATH  Google Scholar 

  83. Rösch, A., Tröltzsch, F.: Sufficient second-order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints. SIAM J. Control Optim. 42, 138–154 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  84. Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15, 616–634 (2005)

    Article  MATH  Google Scholar 

  85. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

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Wagner, M. Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing. J Optim Theory Appl 140, 543–576 (2009). https://doi.org/10.1007/s10957-008-9460-9

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