Introduction to Inverse Problems in ImagingCRC Press, 30 Ağu 2020 - 364 sayfa This is a graduate textbook on the principles of linear inverse problems, methods of their approximate solution, and practical application in imaging. The level of mathematical treatment is kept as low as possible to make the book suitable for a wide range of readers from different backgrounds in science and engineering. Mathematical prerequisites are first courses in analysis, geometry, linear algebra, probability theory, and Fourier analysis. The authors concentrate on presenting easily implementable and fast solution algorithms. With examples and exercises throughout, the book will provide the reader with the appropriate background for a clear understanding of the essence of inverse problems (ill-posedness and its cure) and, consequently, for an intelligent assessment of the rapidly growing literature on these problems. |
İçindekiler
Introduction | 1 |
12 What is an illposed problem? | 5 |
13 How to cure illposedness | 9 |
14 An outline of the book | 11 |
IMAGE DECONVOLUTION | 17 |
Some mathematical tools | 19 |
22 Bandlimited functions and sampling theorems | 22 |
23 Convolution operators | 27 |
73 The ML method in the case of Poisson noise | 175 |
74 Bayesian methods | 183 |
75 The Wiener filter | 184 |
LINEAR INVERSE IMAGING PROBLEMS | 189 |
Examples of linear inverse problems | 191 |
82 Xray tomography | 194 |
83 Emission tomography | 200 |
84 Inverse diffraction and inverse source problems | 206 |
24 The Discrete Fourier Transform DFT | 30 |
25 Cyclic matrices | 36 |
26 Relationship between FT and DFT | 39 |
27 Discretization of the convolution product | 42 |
Examples of image blurring | 50 |
32 Linear motion blur | 54 |
33 Outoffocus blur | 58 |
34 Diffractionlimited imaging systems | 60 |
35 Atmospheric turbulence blur | 67 |
36 Nearfield acoustic holography | 69 |
The illposedness of image deconvolution | 75 |
42 Wellposed and illposed problems | 77 |
43 Existence of the solution and inverse filtering | 79 |
from illposedness to illconditioning | 81 |
leastsquares solutions and generalized solution | 86 |
46 Approximate solutions and the use of a priori information | 90 |
47 Constrained leastsquares solutions | 94 |
Regularization methods | 98 |
52 Approximate solutions with minimal energy | 104 |
53 Regularization algorithms in the sense of Tikhonov | 107 |
54 Regularization and filtering | 113 |
resolution and Gibbs oscillations | 119 |
56 Choice of the regularization parameter | 127 |
Iterative regularization methods | 137 |
62 The projected Landweber method | 147 |
63 The projected Landweber method for the computation of constrained regularized solutions | 154 |
64 The steepest descent and the conjugate gradient method | 157 |
65 Filtering properties of the conjugate gradient method | 165 |
Statistical methods | 168 |
72 The ML method in the case of Gaussian noise | 172 |
85 Linearized inverse scattering problems | 214 |
Singular value decomposition SVD | 220 |
92 SVD of a matrix | 225 |
93 SVD of a semidiscrete mapping | 231 |
94 SVD of an integral operator with squareintegrable kernel | 234 |
95 SVD of the Radon transform | 240 |
Inversion methods revisited | 247 |
102 The Tikhonov regularization method | 253 |
103 Truncated SVD | 256 |
104 Iterative reguiarization methods | 259 |
105 Statistical methods | 263 |
Fourierbased methods for specific problems | 268 |
112 The filtered backprojection FBP method in tomography | 272 |
113 Implementation of the discrete FBP | 277 |
114 Resolution and superresolution in image restoration | 280 |
115 Outofband extrapolation | 284 |
116 The Gerchberg method and its generalization | 289 |
Comments and concluding remarks | 295 |
122 In praise of simulation | 302 |
MATHEMATICAL APPENDICES | 309 |
Euclidean and Hilbert spaces of functions | 311 |
Linear operators in function spaces | 317 |
Euclidean vector spaces and matrices | 322 |
Properties of the DFT and the FFT algorithm | 328 |
Minimization of quadratic functionals | 335 |
Contraction and nonexpansive mappings | 339 |
The EM method | 343 |
References | 346 |
| 347 | |
Diğer baskılar - Tümünü görüntüle
Sık kullanılan terimler ve kelime öbekleri
algorithm appendix applied approximation associated assume bandlimited basic behaviour blur bounded called chapter coincides components compute condition consider consists contains convergence convolution corresponding deconvolution defined definition denoted depends described direct discrete discussed distance distribution domain effect eigenvalues element energy equality equation estimate example exists figure filter Fourier transform frequency function given ill-posed imaging system implies important instance integral introduced inverse problem iterations Landweber method least-squares linear mathematical matrix means minimal Moreover noise noisy image norm object observe obtained operator orthogonal particular physical positive possible precisely projection properties prove regularization parameter remark representation respect restoration restoration error result sampling satisfied scalar product sequence shown solution space square square-integrable step Tikhonov true unique variables vector window zero