References & Further Reading

References

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996
The standard graduate-level reference for regularization theory. Covers the full spectrum from Hadamard well-posedness through Tikhonov, spectral, and iterative regularization with rigorous convergence analysis. Chapters 2–6 are the primary source for this chapter.
P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010
Excellent computational perspective on inverse problems. Particularly strong on the SVD, the L-curve, and practical implementation of regularization methods. The Regularization Tools MATLAB toolbox accompanies this book.
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005
Bridges the classical (deterministic) regularization theory with the Bayesian approach. The Bayesian interpretation of Tikhonov regularization and the treatment of the posterior covariance follow this reference.
A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston & Sons, 1977
The foundational monograph by Tikhonov himself. Introduces the variational formulation of regularization and the concept of regularizing families. Of historical importance and still surprisingly readable.
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 2nd edition ed., 2011
Mathematically rigorous treatment aimed at graduate students. Covers compact operators, singular value decomposition, regularization theory, and inverse scattering.
V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, 1984
The original monograph introducing the discrepancy principle for parameter selection. The theoretical treatment of the principle's optimality is developed here.
F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, 2001
Rigorous treatment of the Radon transform, Fourier slice theorem, and tomographic reconstruction. The limited-angle tomography example in Section 2.1 follows this reference.
M. Benning and M. Burger, Modern Regularisation Methods for Inverse Problems, Acta Numerica, 2018
A comprehensive survey covering classical regularization, variational methods including TV and sparsity-promoting penalties, and learning-based approaches. An excellent bridge to the advanced topics in Chapters 3–4.
R. Tibshirani, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistical Society B, 1996
The original LASSO paper. Introduces the $\ell_1$ penalised regression estimator and discusses its sparsity-inducing properties.
S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic Decomposition by Basis Pursuit, SIAM Journal on Scientific Computing, 1998
Introduces Basis Pursuit (the noiseless LASSO) and Basis Pursuit Denoising in the signal processing context. Parallel independent development of the LASSO idea.
E. J. Candes and T. Tao, Near-Optimal Signal Recovery From Random Projections, IEEE Transactions on Information Theory, 2006
Proves the exact recovery guarantee for $\ell_1$ minimization under the Restricted Isometry Property (RIP). The foundation of modern compressed sensing theory.
D. L. Donoho, Compressed Sensing, IEEE Transactions on Information Theory, 2006
Companion paper to Candes–Tao, establishing the compressed sensing framework and the role of sparsity in dimensionality reduction.
L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms, Physica D, 1992
The original ROF model paper introducing total variation regularization for image denoising. One of the most influential papers in computational imaging.
A. B. Bakushinsky and M. Yu. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Springer, 2004
Comprehensive treatment of iterative regularization methods for nonlinear inverse problems, including convergence theory for IRGNM and Levenberg–Marquardt.
M. Hanke, A Regularizing Levenberg–Marquardt Scheme, Inverse Problems, 1997
The original convergence analysis of the Levenberg–Marquardt method as a regularization scheme for nonlinear ill-posed problems.
M. Pastorino, Microwave Imaging, Wiley, 2010
Comprehensive reference for microwave tomography including the Born iterative method, distorted-Born iterative method, and practical implementation aspects.
G. H. Golub, M. Heath, and G. Wahba, Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 1979
The original GCV paper in the ridge regression context. The GCV criterion for Tikhonov regularization parameter selection is developed here.
G. Wahba, Spline Models for Observational Data, SIAM, 1990
Develops the theory of GCV in the spline smoothing context. Chapter 4 covers the regularization parameter selection problem in depth.
C. M. Stein, Estimation of the Mean of a Multivariate Normal Distribution, Annals of Statistics, 1981
The original SURE paper. Proves the unbiasedness of the risk estimator for linear estimators of a Gaussian mean.
Y. C. Eldar and M. Mishali, Robust Recovery of Signals From a Structured Union of Subspaces, IEEE Transactions on Information Theory, 2010
Develops the group sparsity framework and recovery guarantees for joint sparse recovery from multiple measurements.
I. Stojanovic, W. C. Karl, and M. Unser, Compressed Sensing of Monostatic and Multistatic SAR, IEEE Journal of Selected Topics in Signal Processing, 2010
Applies compressed sensing and LASSO to SAR image formation, demonstrating super-resolution for sparse scenes.
P. C. Hansen and D. P. O'Leary, The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems, SIAM Journal on Scientific Computing, 1992
The definitive paper on the L-curve criterion for regularization parameter selection, including the curvature formula and its computational implementation.