* Textbook written for graduate courses.
* Appropriate level of mathematics used.
* Based on many years experience of teaching.
* Subject relevant to a wide range of disciplines.
* Examples and exercises throughout.
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.
Table of Contents
What is an inverse problem?
What is an ll-posed problem?
How to cure ill-posedness
An outline of the book
SOME MATHEMATICAL TOOLS
The Fourier transform (FT)
Bandlimited functions and sampling theorems
The discrete Fourier transform (DFT)
Relationship between FT and DFT
Discretization of the convolution product
EXAMPLES OF IMAGE BLURRING
Blurring and noise
Linear motion blur
Diffraction-limited imaging systems
Atmospheric turbulence blur
Near-field acoustic holography
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
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
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
Maximum likelihood (ML) methods
The ML method in the case of Gaussian noise
The ML method in the case of Poisson noise
The Wiener filter
LINEAR INVERSE IMAGING PROBLEMS
EXAMPLES OF LINEAR INVERSE PROBLEMS
Space-variant imaging systems
Inverse diffraction and inverse source problems
Linearized inverse scattering problems
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
INVERSION METHODS REVISITED
The generalized solution
The Tikhonov regularization method
Iterative regularization methods
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
The Gerchberg method and its generalization
COMMENTS AND CONCLUDING REMARKS
Does there exist a general-purpose method?
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 non-expansive mappings
The EM method
"This is an essential book for all those interested imaging systems and the image derived from them … I would like to take this opportunity to congratulate the authors on an excellent publication which should find a wide audience in an area of application mathematics that is becoming increasingly important."
-Jonathan Blackledge, Mathematics Today
"Although this valuable book is both an introduction into a modern field and instruction for the solution of difficult identification problems in imaging and metrology. It can be recommended to all students, scientists, and engineers who are interested in the state of the art of solving inverse problems or who are practically dealing with modern approaches of image processing such as active vision."
-Optics and Laser Technology