L∞ fitting for inverse problems with uniform noise

For inverse problems where the data are corrupted by uniform noise such as arising from quantization errors, the L∞ norm is a more robust data-fitting term than the standard L2 norm. Well-posedness and regularization properties for linear inverse problems with L∞ data fitting are shown, and the automatic choice of the regularization parameter is discussed. After introducing an equivalent reformulation of the problem and a Moreau–Yosida approximation, a superlinearly convergent semi-smooth Newton method becomes applicable for the numerical solution of L∞ fitting problems. Numerical examples illustrate the performance of the proposed approach as well as the qualitative behavior of L∞ fitting.

[1]  T. Grund,et al.  Optimal control of a linear elliptic equation with a supremum norm functional , 2001 .

[2]  Bangti Jin,et al.  A Semismooth Newton Method for L1 Data Fitting with Automatic Choice of Regularization Parameters and Noise Calibration , 2010, SIAM J. Imaging Sci..

[3]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[4]  J. Zowe,et al.  Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems , 1979 .

[5]  Hamid Gharavi,et al.  Special issue on sensor networks and applications , 2003 .

[6]  Per Christian Hansen,et al.  Regularization Tools version 4.0 for Matlab 7.3 , 2007, Numerical Algorithms.

[7]  Yongduek Seo,et al.  A fast method to minimize L∞ error norm for geometric vision problems , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[8]  O. Scherzer,et al.  A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators , 2007 .

[9]  Jack Williams,et al.  Nonlinear Chebyshev fitting from the solution of ordinary differential equations , 1993, Numerical Algorithms.

[10]  Bangti Jin,et al.  A Semismooth Newton Method for Nonlinear Parameter Identification Problems with Impulsive Noise , 2012, SIAM J. Imaging Sci..

[11]  B. Kummer.  Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis , 1992 .

[12]  U. Prüfert,et al.  The minimization of a maximum‐norm functional subject to an elliptic PDE and state constraints , 2009 .

[13]  S. Manesis,et al.  A Survey of Applications of Wireless Sensors and Wireless Sensor Networks , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[14]  Richard I. Hartley,et al.  L-8Minimization in Geometric Reconstruction Problems , 2004, CVPR.

[15]  O. Seleznjev,et al.  Stochastic structure of asymptotic quantization errors , 2006 .

[16]  Christiane Pöschl,et al.  An overview on convergence rates for Tikhonov regularization methods for non-linear operators , 2009 .

[17]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[18]  Xiaojun Chen,et al.  Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations , 2000, SIAM J. Numer. Anal..

[19]  Anton Schiela.  A Simplified Approach to Semismooth Newton Methods in Function Space , 2008, SIAM J. Optim..

[20]  Bernard Widrow,et al.  Quantization Noise: Quantization of System Parameters , 2008 .

[21]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[22]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[23]  Jack Williams,et al.  Least squares and Chebyshev fitting for parameter estimation in ODEs , 1993, Adv. Comput. Math..

[24]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[25]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[26]  Zhi-Quan Luo,et al.  Distributed Estimation Using Reduced-Dimensionality Sensor Observations , 2005, IEEE Transactions on Signal Processing.

[27]  David E. Culler,et al.  Analysis of wireless sensor networks for habitat monitoring , 2004 .

[28]  S. Osher,et al.  Convergence rates of convex variational regularization , 2004 .

[29]  Peter Grabner,et al.  Laplace operators on fractals and related functional equations , 2012, 1206.1211.

[30]  Ivan P. Gavrilyuk,et al.  Lagrange multiplier approach to variational problems and applications , 2010, Math. Comput..

[31]  K. Kunisch,et al.  Minimal invasion: An optimal L∞ state constraint problem , 2011 .

[32]  Kristy Sim,et al.  Removing outliers using the L∞ Norm , 2006, CVPR 2006.

[33]  U. Prüfert,et al.  The minimization of an L^{\infty}-functional subject to an elliptic PDE and state constraints , 2008 .

[34]  Ya'acov Ritov,et al.  Discussion: The Dantzig selector: statistical estimation when $p$ is much larger than $n$ , 2008, 0803.3130.

[35]  E. Resmerita.  Regularization of ill-posed problems in Banach spaces: convergence rates , 2005 .

[36]  Jens Flemming,et al.  Convergence rates in constrained Tikhonov regularization: equivalence of projected source conditions and variational inequalities , 2011 .

[37]  S. Kindermann.  CONVERGENCE ANALYSIS OF MINIMIZATION-BASED NOISE LEVEL-FRE E PARAMETER CHOICE RULES FOR LINEAR ILL-POSED PROBLEMS , 2011 .

[38]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[39]  Bernard Widrow,et al.  Quantization Noise: Quantization in Signal Processing, Feedback Control, and Computations , 2008 .

[40]  O. Scherzer,et al.  Error estimates for non-quadratic regularization and the relation to enhancement , 2006 .

[41]  Bangti Jin,et al.  A duality-based splitting method for ` 1-TV image restoration with automatic regularization parameter choice ∗ , 2009 .

[42]  David M. Doolin,et al.  Wireless sensors for wildfire monitoring , 2005, SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring.

[43]  P.K. Varshney,et al.  Target Location Estimation in Sensor Networks With Quantized Data , 2006, IEEE Transactions on Signal Processing.