## Introduction to Inverse Problems in ImagingThis 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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### Contents

1 | |

5 | |

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 | |

### Other editions - View all

Introduction to Inverse Problems in Imaging, Mario Bertero,Patrizia Boccacci No preview available - 1998 |

### Common terms and phrases

appendix approximate solutions approximation error bandlimited basic behaviour blur chapter components compute condition number consider convergence convolution operator corresponding deconvolution defined in equation denoted density function direct problem discrepancy functional discrete discussed in section domain eigenvalues element estimate Euclidean space example exists figure Fourier transform frequency Gaussian given by equation holds true ill-posed problem image deconvolution image g image restoration imaging system implies instance integral operator introduced inverse problem least-squares solution likelihood function linear operator mathematical minimal minimum noise noise-free image noisy image norm null space number of iterations obtained optical orthogonal orthonormal Parseval equality Poisson projected Landweber method propagation Radon transform regularization methods regularization parameter regularized solutions representation restoration error result satisfied scalar product sequence singular value decomposition solution of equation square square-integrable functions subspace tends to zero Tikhonov regularization tomography unique unknown object variables window functions

### Popular passages

Page vi - Most people, if you describe a train of events to them, will tell you what the result would be. They can put those events together in their minds, and argue from them that...

Page vi - In the everyday affairs of life it is more useful to reason forward, and so the other comes to be neglected. There are fifty who can reason synthetically for one who can reason analytically." "I confess," said I, "that I do not quite follow you.

Page 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.

Page 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.

Page 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...