A control problem in medical image processing

• Applicant: Professor Dr. Peter Maaß
• Subject Area: Mathematics
• Term: from 2005 to 2009
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 18078967

• Final Report Year: 2008

Final Report Abstract

The project was motivated by a control problem in image processing. This lead to the problem of analyzing • a parameter identification problem for degenerate parabolic equations • defining appropriate, parameter-dependent function spaces extending the general regularization theory for non-linear operator equations to degenerate parameter identification problems in order to apply regularization methods with sparsity constraints to parabolic control problem in image processing (mammography screening, X-ray images). The results can be summarized as follows. • An existence and uniqueness result for the above mentioned parameter identification problems in weighted Sobolv-spaces has been obtained, • first order optimality conditions were obtained for he optimization problem (quadratic functional), • an optimized presentation scenario of mammography images was generated and tested with real life data. This was the main application of the optimal control problem in medical imaging.

Publications

• An optimal control problem in image processing. PAMM, 6(1): 859 - 860, 2006
  K. Bredies, D. A. Lorenz and P. Maass
  
• An optimal control problem in medical image processing. In: F. Ceragioli, A. Dontchev, H. Furuta, K. Marti, L. Pandolfi, editors, Systems, Control, Modeling and Optimization, Proceedings of the 22nd IFIP TC 7 Conference, pages 249 - 260. IFIP, Springer, July 2006
  K. Bredies, D. A. Lorenz and P. Maass
  
• On the minimization of non-convex, non-differentiable functionals with an application to SPECT. In: Oberwolfach Report: Mathematical Methods in Tomography, 34: 18-22, 2006
  T. Bonesky, K. Bredies, D. A. Lorenz and P. Maass
  
• A generalized conditional gradient method for non-linear operator equations with sparsity constraints. Inverse Problems, 23: 2041 - 2058, 2007
  K. Bredies, T. Bonesky, D. A. Lorenz and P. Maass
  
• Iterative soft-thresholding converges linearly. Preprint Series of the DFG SPP 1114, University of Bremen, 2007
  K. Bredies and D. A. Lorenz
  
• Optimal control of degenerate parabolic equations in image processing. PhD thesis, University of Bremen, 2007
  K. Bredies
  
• Solving variational problems in image processing via projections - a common view on TV-denoising and wavelet shrinkage. Zeitschrift für angewandte Mathematik und Mechanik,81(1): 247-256, 2007
  D. A. Lorenz
  
• Inverse problems and parameter identification in image processing. In: R. Dahlhaus, J. Kurths, P. Maass and J. Timmer, editors, Mathematical Methods in Time Series Analysis and Digital Image Processing, pages 111 - 151. Springer, 2008
  J. F. Acker, B. Berkels, K. Bredies, M. S. Diallo, M. Droske, C. S. Garbe, M. Holschneider, J. Hron, C. Kondermann, M. Kulesh, P. Maass, N. Olischläger, H. O. Peitgen, T. Preusser, M. Rumpf, K. Schaller, F. Scherbaum and S. Turek