Introduction to Inverse Problems in Imaging: 1st Edition (Paperback) book cover

Introduction to Inverse Problems in Imaging

1st Edition

By M. Bertero, P. Boccacci

CRC Press

352 pages

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Paperback: 9780750304351
pub: 1998-01-01
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Description

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.

Reviews

"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

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

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

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 REMARKS

Does there exist a general-purpose method?

In praise of simulation

References

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

Subject Categories

BISAC Subject Codes/Headings:
TEC015000
TECHNOLOGY & ENGINEERING / Imaging Systems