Introduction to Inverse Problems in Imaging

CRC 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.

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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 Telif Hakkı

Popüler pasajlar

Sayfa 3 - We call two problems inverses of one another if the formulation of each involves all or part of the solution of the other. Often for historical reasons, one of the two problems has been studied extensively for some time, while the other has never been studied and is not so well understood. In such cases, the former is called direct problem, while the latter is the inverse problem.
Sayfa 4 - The direct problem is also a problem directed towards a loss of information: its solution defines a transition from a physical quantity with a certain information content to another quantity with a smaller information content. This implies that the solution is much smoother than the corresponding object.
Sayfa 2 - I(i,j) at the pixel location (ij)) to its cause (in that case the displacement d(u,v,w) of the related object point P(x,y,z)). In other words, an inverse problem has to be solved. But what is an inverse problem? From the point of view of a mathematician the concept of an inverse problem has a certain degree of ambiguity which is well illustrated by a frequently quoted statement of JB Keller...

Yazar hakkında (2020)

M. Bertero and P. Boccacci