A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity con\-straints

DOI10.1007/s00211-006-0016-3zbMath1101.65056OpenAlexW2023299560MaRDI QIDQ2503143

Gerd Teschke, Ronny Ramlau

Publication date: 14 September 2006

Published in: Numerische Mathematik (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00211-006-0016-3

zbMATH Keywords

nonlinear ill-posed problemTikhonov functionalframeisometric mapLandweber fixed point iterationnonlinear SPECT problemsparse expansionTikhonov projection iterationweighted \(\ell_p\) penalty

Mathematics Subject Classification ID

Nonlinear ill-posed problems (47J06) Numerical solutions to equations with nonlinear operators (65J15) Numerical solutions of ill-posed problems in abstract spaces; regularization (65J20)

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Uses Software

• TIGRA

Cites Work

• Unnamed Item
• Nonlinear total variation based noise removal algorithms
• Harmonic analysis of the space BV.
• Modeling textures with total variation minimization and oscillating patterns in image processing
• Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising
• Ten Lectures on Wavelets
• Simultaneous activity and attenuation emission tomography as a nonlinear ill‐posed problem
• Nonlinear approximation and the space BV[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]
• TIGRA an iterative algorithm for regularizing nonlinear ill-posed problems
• Accurate Attenuation Correction in SPECT Imaging Using Optimization of Bilinear Functions and Assuming an Unknown Spatially-Varying Attenuation Distribution
• A steepest descent algorithm for the global minimization of the Tikhonov functional
• MOROZOV'S DISCREPANCY PRINCIPLE FOR TIKHONOV-REGULARIZATION OF NONLINEAR OPERATORS
• An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
• A new approach towards simultaneous activity and attenuation reconstruction in emission tomography
• Tikhonov replacement functionals for iteratively solving nonlinear operator equations