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. 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.

Table of Contents:
Introduction: What is an inverse problem? What is an ll-posed problem? How to cure ill-posedness. An outline of the book. Reference. Part 1 Image deconvolution - Some mathematical tools: The Fourier Transform (FT). Bandlimited functions and sampling theorems. Convolution operators. The Discrete Fourier Transform (DFT). Cyclic matrices. Relationship between FT and DFT. Discretization of the convolution product. References. Examples of image blurring: Blurring and noise. Linear motion blur. Out-of-focus blur. Diffraction-limited imaging systems. Atmospheric turbulence blur. Near-field acoustic holography. References. The ill-posedness of image deconvolution: Formulation of the problem. Well-posed and ill-posed problems. Existence of the solution and inverse filtering. Discretization: from ill-posedness to ill-conditioning. Bandlimited systems: least-squares solutions and generalized solution. Approximate solutions and the use of "a priori" information. Constrained least-squares. References. Regularization methods: Least squares solutions with prescribed energy. Approximate solutions with minimal energy. Regularization algorithms in the sense of Tikhonov. Regularization and filtering. The global point spread function. Choice of the regularization parameter. References. Iterative regularization methods: The Landweber method. The projected Landweber method for the computation of constrained regularized solutions. The steepest descent and the conjugate gradient method. References. Statistical methods: Maximum Likelihood (ML) methods. The ML method in the case of gaussian noise. The ML method in the case of Poisson noise. Bayesian methods. The Wiener filter. References. Part 2 Linear inverse imaging problems - Examples of linear inverse problems: Space-variant imaging systems. X-ray tomography. Emission tomography. Inverse diffraction and inverse source problems. Linearized inverse scattering problems. References. Singular value decomposition (SVD): Mathematical description of linear imaging systems. SVD of a matrix. SVD of a semi-discrete mapping. SVD of an integral operator with square-integrable kernel. SVD of the Radon transform. References. Inversion methods revisited: The generalized solution. The Tikhonov regularization method. Truncated SVD. Iterative regularization methods. Statistical methods. References. Fourier based methods for specific problems: The Fourier slice theorem in tomography. The filtered backprojection (FBP) method in tomography. Implementation of the discrete FBP. Resolution and super-resolution in image restoration. Out-of-band extrapolation. The Gerchberg method and its generalization. References. Comments and concluding remakrs: Does there exist a general-purpose method? In praise of simulation. References. Part 3 Mathematical appendices: 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 non-expansive mappings. The EM method. References.