, Volume 81, Issue 2–3, pp 137–160 | Cite as

Signal and image approximation with level-set constraints

  • C. SchnörrEmail author


We present a novel variational approach to signal and image approximation using filter statistics (histograms) as constraints. Given a set of linear filters, we study the problem to determine the closest point to given data while constraining the level-sets of the filter outputs. This criterion and the constraints are formulated as a bilevel optimization problem. We develop an algorithm by representing the lower-level problem through complementarity constraints and by applying an interior-penalty relaxation method. Based on a decomposition of the penalty term into the difference of two convex functions, the resulting algorithm approximates the data by solving a sequence of convex programs. Our approach allows to model and to study the generation of image structure through the interaction of two convex processes for spatial approximation and for preserving filter statistics, respectively.


level-sets image approximation equilibrium constraints complementarity constraints DC-programming 

AMS Subject Classifications

68U10 65K05 65K10 90C33 


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  1. Anitescu M. (2005). On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. SIAM J Optim 15(4): 1203–1236 zbMATHCrossRefMathSciNetGoogle Scholar
  2. An L.T.H. and Tao P.D. (2005). The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann Oper Res 133: 23–46 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Aujol J.-F., Gilboa G., Chan T. and Osher S. (2006). Structure-texture image decomposition – modeling, algorithms, parameter selection. Int J Comp Vision 67(1): 111–136 CrossRefGoogle Scholar
  4. Boyd S. and Vandenberghe L. (2004). Convex optimization. Cambridge University Press, London zbMATHGoogle Scholar
  5. Chambolle A. (2004). An algorithm for total variation minimization and applications. J Math Imaging Vis 20: 89–97 CrossRefMathSciNetGoogle Scholar
  6. Cottle R.W., Pang J.-S. and Stone R.E. (1992). The linear complementarity problem. Academic, Dublin zbMATHGoogle Scholar
  7. Facchinei F. and Pang J.-S. (2003). Finite-dimensional variational inequalities and complementarity problems, vol. I. Springer, New York zbMATHGoogle Scholar
  8. Field D.J. (1999). Wavelets, vision, the statistics of natural scenes. Philos Trans R Soc Lond A 357: 2527–2542 zbMATHCrossRefMathSciNetGoogle Scholar
  9. Graham A. (1981). Kronecker products and matrix calculus with applications. Wiley, New York zbMATHGoogle Scholar
  10. Horst R. and Thoai N.V. (1999). DC programming: overview. J Optim Theory Appl 103(1): 1–43 CrossRefMathSciNetGoogle Scholar
  11. Hu X.M. and Ralph D. (2004). Convergence of a penalty method for mathematical programming with complementarity constraints. J Optim Theory Appl 123(2): 365–390 CrossRefMathSciNetGoogle Scholar
  12. Lee A.B., Pedersen K.S. and Mumford D. (2003). The nonlinear statistics of high-contrast patches in natural images. Int J Comp Vision 54: 83–103 zbMATHCrossRefGoogle Scholar
  13. Leyffer S., Lopez-Calva G. and Nocedal J. (2006). Interior methods for mathematical programs with complementarity constraints. SIAM J Optim 17(1): 52–77 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Luo Z.-Q., Pang J.-S. and Ralph D. (1996). Mathematical programs with equilibrium constraints. Cambridge University Press, London Google Scholar
  15. Raghunathan A.U. and Biegler L.T. (2005). An interior point method for mathematical programs with complementarity constraints (MPCCs). SIAM J Optim 15(3): 720–750 zbMATHCrossRefMathSciNetGoogle Scholar
  16. Rockafellar, R. T., Wets, R. J.-B.: Variational analysis. In: Grundlehren der math. Wissenschaften, vol. 317. Springer, New York (1998)Google Scholar
  17. Rudin L., Osher S. and Fatemi E. (1992). Nonlinear total variation based noise removal algorithms. Phys D 60: 259–268 zbMATHCrossRefGoogle Scholar
  18. Scheel H. and Scholtes S. (2000). Mathematical program with complementarity constraints: stationarity, optimality and sensitivity. Math Oper Res 25: 1–22 zbMATHCrossRefMathSciNetGoogle Scholar
  19. Scholtes S. (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J Optim 11(4): 918–936 zbMATHCrossRefMathSciNetGoogle Scholar
  20. Srivastava A., Liu X. and Grenander U. (2002). Universal analytical forms for modeling image probabilities. IEEE Trans Pattern Anal Mach Intell 24(9): 1200–1214 CrossRefGoogle Scholar
  21. Tao P.D. and An L.T.H. (1998). A D.C. optimization algorithm for solving the trust-region subproblem. SIAM J Optim 8(2): 476–505 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Zhu S.C. and Mumford D. (1997). Prior learning and gibbs reaction-diffusion. IEEE Trans Pattern Anal Mach Intell 19(11): 1236-1250 CrossRefGoogle Scholar
  23. Zhu S.C., Wu Y. and Mumford D. (1998). Filters, random fields and maximum entropy (FRAME). Towards a unified theory for texture modeling. Int J Comp Vision 27(2): 107–126 CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of HeidelbergHeidelbergGermany

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