Introduction to Inverse Problems in Imaging
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.
Kullanıcılar ne diyor? - Eleştiri yazın
Her zamanki yerlerde hiçbir eleştiri bulamadık.
73 The ML method in the case of Poisson noise
74 Bayesian methods
75 The Wiener filter
LINEAR INVERSE IMAGING PROBLEMS
Examples of linear inverse problems
82 Xray tomography
83 Emission tomography
84 Inverse diffraction and inverse source problems
24 The Discrete Fourier Transform DFT
25 Cyclic matrices
26 Relationship between FT and DFT
27 Discretization of the convolution product
Examples of image blurring
32 Linear motion blur
33 Outoffocus blur
34 Diffractionlimited imaging systems
35 Atmospheric turbulence blur
36 Nearfield acoustic holography
The illposedness of image deconvolution
42 Wellposed and illposed problems
43 Existence of the solution and inverse filtering
from illposedness to illconditioning
leastsquares solutions and generalized solution
46 Approximate solutions and the use of a priori information
47 Constrained leastsquares solutions
52 Approximate solutions with minimal energy
53 Regularization algorithms in the sense of Tikhonov
54 Regularization and filtering
resolution and Gibbs oscillations
56 Choice of the regularization parameter
Iterative regularization methods
62 The projected Landweber method
63 The projected Landweber method for the computation of constrained regularized solutions
64 The steepest descent and the conjugate gradient method
65 Filtering properties of the conjugate gradient method
72 The ML method in the case of Gaussian noise
85 Linearized inverse scattering problems
Singular value decomposition SVD
92 SVD of a matrix
93 SVD of a semidiscrete mapping
94 SVD of an integral operator with squareintegrable kernel
95 SVD of the Radon transform
Inversion methods revisited
102 The Tikhonov regularization method
103 Truncated SVD
104 Iterative reguiarization methods
105 Statistical methods
Fourierbased methods for specific problems
112 The filtered backprojection FBP method in tomography
113 Implementation of the discrete FBP
114 Resolution and superresolution in image restoration
115 Outofband extrapolation
116 The Gerchberg method and its generalization
Comments and concluding remarks
122 In praise of simulation
Euclidean and Hilbert spaces of functions
Linear operators in function spaces
Euclidean vector spaces and matrices
Properties of the DFT and the FFT algorithm
Minimization of quadratic functionals
Contraction and nonexpansive mappings
The EM method
Diğer baskılar - Tümünü görüntüle
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
Sayfa 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...
Sayfa 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.
Sayfa 78 - Data in nature cannot possibly be conceived as rigidly fixed; the mere process of measuring them involves small errors. Therefore a mathematical problem cannot be considered as realistically corresponding to physical phenomena unless a variation of the given data in a sufficiently small range leads to an arbitrary small change in the solution. This requirement of stability is not only essential for meaningful problems in mathematical physics, but also for approximation methods'.
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 340 - X is said to be a contraction mapping if there exists a constant Л, 0 < A < 1 such that d(Tx, Ту) < \d(x, y) for all x, у in X.
Sayfa 78 - The problem is said to be well-posed in the sense of Hadamard if the following conditions hold: (i) For each g 6 Q there exists / € / such that A(f) = g.
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...