Introduction to Inverse Problems in Imaging

Ön Kapak
CRC Press, 1 Oca 1998 - 352 sayfa
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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 exercised 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.
 

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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
Index
347
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