Author | Korostelev, A. P. author |
---|---|
Title | Minimax Theory of Image Reconstruction [electronic resource] / by A. P. Korostelev, A. B. Tsybakov |
Imprint | New York, NY : Springer New York, 1993 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2712-0 |
Descript | XII, 258p. online resource |
1. Nonparametric Regression and Change-Point Problems -- 1.1. Introduction -- 1.2. The nonparametric regression problem -- 1.3. Kernel estimators -- 1.4. Locally-polynomial estimators -- 1.5. Piecewise-polynomial estimators -- 1.6. Bias and variance of the estimators -- 1.7. Criteria for comparing the nonparametric estimators -- 1.8. Rates of the uniform and L1- convergence -- 1.9. The change-point problem -- 2. Minimax Lower Bounds -- 2.1. General statistical model and minimax rates of convergence -- 2.2. The basic idea -- 2.3. Distances between distributions -- 2.4. Examples -- 2.5. The main theorem on lower bounds -- 2.6. Assouadโs lemma -- 2.7. Examples: uniform and integral metrics -- 2.8. Arbitrary design -- 3. The Problem of Edge and Image Estimation -- 3.1. Introduction -- 3.2. Assumptions and notation -- 3.3. Lower bounds on the accuracy of estimates -- 4. Optimal Image and Edge Estimation for Boundary Fragments -- 4.1. Optimal edge estimation -- 4.2. Preliminary lemmas -- 4.3. Proof of Theorem 4.1.1 -- 4.4. Optimal image estimation -- 4.5. Proof of Theorem 4.4.5 -- 5. Generalizations and Extensions -- 5.1. High-dimensional boundary fragments. Non-Gaussian noise -- 5.2. General domains in high dimensions: a simple and rough estimator -- 5.3. Optimal estimators for general domains in two dimensions -- 5.4. Dudleyโs classes of domains -- 5.5. Maximum likelihood estimation on ?-net -- 5.6. Optimal edge estimators for Dudleyโs classes -- 5.7. On calculation of optimal edge estimators for general domains -- 6. Image Reconstruction Under Restrictions on Estimates -- 6.1. Naรฏve linewise processing -- 6.2. Modified linewise procedure -- 6.3. Proofs -- 6.4. Linear image estimators -- 7. Estimation of Support of a Density -- 7.1. Problem statement -- 7.2. A simple and rough support estimator -- 7.3. Minimax lower bounds for support estimation -- 7.4. Optimal support estimation for boundary fragments -- 7.5. Optimal support estimation for convex domains and for Dudleyโs classes -- 8. Estimation of The Domainโs Area -- 8.1. Preliminary discussion -- 8.2. Domainโs area estimation in continuous parametric models -- 8.3. Theorem on the lower bound -- 8.4. Optimal estimator for the domainโs area -- 8.5. Generalizations and extensions -- 8.6. Functionals of support of a density -- 9. Image Estimation from Indirect Observations -- 9.1. The blurred image model -- 9.2. High-dimensional blurred image models -- 9.3. Upper bounds in non-regular case -- 9.4. The stochastic problem of tomography -- 9.5. Minimax rates of convergence -- References -- Author Index