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Horwood Publishing, 2005. — 825 p.Digital Image Processing complements Digital Signal Processing (Horwood Publishing 2003) which was based on teaching material developed for the MSc programme in Digital Systems Engineering at Loughborough University. Digital Image Processing extends this material further by exploring the characteristics of imaging systems, the computational techniques used to process digital images and the interpretation of the information which an image conveys through an understanding of the physical processes that occur.

Many excellent image processing systems, software libraries and packages are currently available for low-level general applications whereas others have been designed for specific applications. Users can process images using either a command line language (e.g. the MATLAB image processing toolbox) or a graphical user interface (e.g. Adobe Photoshop) to improve the general quality and fidelity of a digital image and/or to achieve results conveying specific aspects of its information content (feature extraction). This can be accomplished without the user having a thorough understanding of the computational methods involved or how and why such methods have evolved, e.g. the application of a particular filter. For those who are only interested in using a particular processing system to 'get the job done' working in a commercial environment for example, application of a specific commercial package or packages with an appropriate selection of image processing options is all that is required. However, for those who wish to contribute to the future development of such systems and/or develop their own 'home-spun' versions for research purposes, a deeper understanding of the mathematical and computational techniques is, by necessity, required.

This work provides a study of the computational methods that are used to process images, but in such a way that there is a direct link (where possible) between the process that is used, the data to which it is applied and, most of all, the 'physics' that underpins the generation of the data. In order to do this, it is necessary to spend some time discussing the principles of how waves and wavefields propagate and interact with objects whose images are required. Depending on the wavelength of the field, the interactions that occur are usually in the form of some scattered wavefield. Hence, after a review of the mathematical and computational background to the subject given in Part I (which includes material on vector fields, the 2D Fourier transform and the 2D FIR filter), we provide an introduction to the field equations and wave equations used to model different types of wavefields and the scattering theory needed to develop appropriate models for the images that these wavefields produce in terms of the information on the imaged object that they convey. We formulate some of the analytical methods and results that are required to compute a scattered wavefield and provide details on the equations that are used in later chapters. Some of this material is based on a previous work published by the author, namely, Quantitative Coherent Imaging (Academic Press, 1989), which was concerned with the principles of interpreting the structure and material properties of objects by the way in which they scatter electromagnetic and acoustic radiation with the aim of exploring tht theory, methods and some of the applications of incoherent and coherent imaging systems.

Having established the principal theoretical background to modelling an imaging system, we look at a range of imaging techniques which are classified into two main types, namely, incoherent and coherent imaging. In Part II, incoherent optical systems are studied and an introduction given to the method of projection tomography where it is assumed that the probe (i.e. the radiation field) used to interrogate an object can be described in terms of a sequence of rays traceable through the object and 'back-projected'. Part II includes a study of coherent imaging methods and investigates the principles of coherent optics, the imaging of layered media, diffraction tomography and synthetic aperture imaging. Both electromagnetic and acoustic imaging systems are discussed. In the case of diffraction tomography for example, the aim is to interpret the internal structure and composition of an object by the way in which it diffracts electromagnetic or acoustic radiation. Two types of diffraction tomography are discussed where the object is illuminated/insonified with a wavefield oscillating at a fixed frequency (Continuous Wave or CW case) or with a short pulse of radiation. In the material on synthetic aperture imaging, attention is focused on the use of Radar for imaging the surface of the Earth and a model presented to describe the scattering of a pulse of frequenc - modulated microwave radiation by the ground. This material also includes a case study which develops a solution to the so called 'sea spikes' problem.

In the 'light' of the preceding material, Part III introduces the basis of digital image processing including the problem of image restoration, image reconstruction and image enhancement. The methods discussed are all related in one form or another to the physical principles presented in Parts I and II and forms the basis for Part IV of this work which studies the principles of pattern recognition and computer vision. This includes an introduction to statistical modelling and analysis, an extended chapter on fractal images and fractal image processing, and a chapter on data coding and image compression, including fractal image compression.

The author has attempted to provide the reader with the mathematical methods required for image analysis which are then used to develop models and algorithms for processing digital images and, finally, to encourage the reader to design some example software solutions for Digital Image Processing (DIP). In this way, the reader is invited to develop a small DIP library that can then be developed further and tailored to his/her learning and/or research interests. This is accomplished by the inclusion of a series of tutorial problems which are given at the end of each Part with model ansv prs provided in Appendix A. These problems include theoretical, computational and programming exercises in the C programming language.

The emphasis throughout is on the mathematical foundations of the subject which are common to a variety of imaging systems and methods. In some cases, examples have been provided to illustrate the conversion of a computational algorithm into a computer program. Either pseudo code, C or MATLAB code is used for this purpose. The book has been designed to serve the reader with enough formal detail for him/her to acquire a firm foundation on which to build. References to other important texts and/or key scientific papers are included at the end of each chapter or within the text for this purpose.

The material presented in this book is based on the lecture notes and supplementary material developed by the author as part of an advanced taught MSc programme in 'Digital Signal Processing'. This programme was originally established at Cranfield University in 1990 and modified at De Montfort University in 1994. The programmes are still operating at these universities and the material has been used by more than 500 graduates since its creation and development in the early 1990s. The material was enhanced and developed further when the author moved to the Department of Electronic and Electrical Engineering at Loughborough University in 2003, and now forms part of the department's post-graduate teaching and learning activities. The original MSc programme was based on taught components covering a period of eight months and consisting of two semesters, each semester, being composed of four modules; the third semester focused on a minor research project. The material in this work covers the second semester and is 'index-linked' through this teaching programme to the publication Digital Signal Processing (Horwood, 2003) which covers the first semester. The classification of this work into four parts reflects the four modules given in the second semester. It has been necessary to include some of the material published previously with the view of revising some of the principal themes such as those concerned with the properties and computational methods associated with the Fourier transform. This has been done for reasons of completeness and to provide the reader with an account of the field that does not necessarily require significant reference to previous publications (by the author or otherwise).

An attempt has been made to cut through much of the jargon characterizing different fields of research in imaging science presenting an account of the fundamental physical principles common to nearly all imaging systems. This is done by illustrating the similarity of the underlying mathematical models used to process data on a wavefield in a variety of applications. The approach has been to unify the principles of different imaging systems and to provide a course text covering the theoretical foundations of imaging science in an integrated and complete form.Introduction

**I Mathematical and Computational Background**

Vector Fields

2D Fourier Theory

The 2D DFT, FFT and FIR Filter

Field and Wave Equations

Green Functions

**II Imaging Systems Modelling**

Scattering Theory

Imaging of Layered Media

Projection Tomography

Diffraction Tomography

Synthetic Aperture Imaging

Optical Image Formation

**III Digital Image Processing Methods**

Image Restoration and Reconstruction

Reconstruction of Band-limited Images

Bayesian Estimation Methods

Image Enhancement

**IV Pattern Recognition and Computer Vision**

Segmentation and Edge Detection

Statistical Modelling and Analysis

Fractal Images and Image Processing

Coding and Compression

**Appendices**

A: Solutions to Problems

B: Supplementary Problems

C: Fourier Transform of a Fractal

D: I/O and Graphics Utilities

Many excellent image processing systems, software libraries and packages are currently available for low-level general applications whereas others have been designed for specific applications. Users can process images using either a command line language (e.g. the MATLAB image processing toolbox) or a graphical user interface (e.g. Adobe Photoshop) to improve the general quality and fidelity of a digital image and/or to achieve results conveying specific aspects of its information content (feature extraction). This can be accomplished without the user having a thorough understanding of the computational methods involved or how and why such methods have evolved, e.g. the application of a particular filter. For those who are only interested in using a particular processing system to 'get the job done' working in a commercial environment for example, application of a specific commercial package or packages with an appropriate selection of image processing options is all that is required. However, for those who wish to contribute to the future development of such systems and/or develop their own 'home-spun' versions for research purposes, a deeper understanding of the mathematical and computational techniques is, by necessity, required.

This work provides a study of the computational methods that are used to process images, but in such a way that there is a direct link (where possible) between the process that is used, the data to which it is applied and, most of all, the 'physics' that underpins the generation of the data. In order to do this, it is necessary to spend some time discussing the principles of how waves and wavefields propagate and interact with objects whose images are required. Depending on the wavelength of the field, the interactions that occur are usually in the form of some scattered wavefield. Hence, after a review of the mathematical and computational background to the subject given in Part I (which includes material on vector fields, the 2D Fourier transform and the 2D FIR filter), we provide an introduction to the field equations and wave equations used to model different types of wavefields and the scattering theory needed to develop appropriate models for the images that these wavefields produce in terms of the information on the imaged object that they convey. We formulate some of the analytical methods and results that are required to compute a scattered wavefield and provide details on the equations that are used in later chapters. Some of this material is based on a previous work published by the author, namely, Quantitative Coherent Imaging (Academic Press, 1989), which was concerned with the principles of interpreting the structure and material properties of objects by the way in which they scatter electromagnetic and acoustic radiation with the aim of exploring tht theory, methods and some of the applications of incoherent and coherent imaging systems.

Having established the principal theoretical background to modelling an imaging system, we look at a range of imaging techniques which are classified into two main types, namely, incoherent and coherent imaging. In Part II, incoherent optical systems are studied and an introduction given to the method of projection tomography where it is assumed that the probe (i.e. the radiation field) used to interrogate an object can be described in terms of a sequence of rays traceable through the object and 'back-projected'. Part II includes a study of coherent imaging methods and investigates the principles of coherent optics, the imaging of layered media, diffraction tomography and synthetic aperture imaging. Both electromagnetic and acoustic imaging systems are discussed. In the case of diffraction tomography for example, the aim is to interpret the internal structure and composition of an object by the way in which it diffracts electromagnetic or acoustic radiation. Two types of diffraction tomography are discussed where the object is illuminated/insonified with a wavefield oscillating at a fixed frequency (Continuous Wave or CW case) or with a short pulse of radiation. In the material on synthetic aperture imaging, attention is focused on the use of Radar for imaging the surface of the Earth and a model presented to describe the scattering of a pulse of frequenc - modulated microwave radiation by the ground. This material also includes a case study which develops a solution to the so called 'sea spikes' problem.

In the 'light' of the preceding material, Part III introduces the basis of digital image processing including the problem of image restoration, image reconstruction and image enhancement. The methods discussed are all related in one form or another to the physical principles presented in Parts I and II and forms the basis for Part IV of this work which studies the principles of pattern recognition and computer vision. This includes an introduction to statistical modelling and analysis, an extended chapter on fractal images and fractal image processing, and a chapter on data coding and image compression, including fractal image compression.

The author has attempted to provide the reader with the mathematical methods required for image analysis which are then used to develop models and algorithms for processing digital images and, finally, to encourage the reader to design some example software solutions for Digital Image Processing (DIP). In this way, the reader is invited to develop a small DIP library that can then be developed further and tailored to his/her learning and/or research interests. This is accomplished by the inclusion of a series of tutorial problems which are given at the end of each Part with model ansv prs provided in Appendix A. These problems include theoretical, computational and programming exercises in the C programming language.

The emphasis throughout is on the mathematical foundations of the subject which are common to a variety of imaging systems and methods. In some cases, examples have been provided to illustrate the conversion of a computational algorithm into a computer program. Either pseudo code, C or MATLAB code is used for this purpose. The book has been designed to serve the reader with enough formal detail for him/her to acquire a firm foundation on which to build. References to other important texts and/or key scientific papers are included at the end of each chapter or within the text for this purpose.

The material presented in this book is based on the lecture notes and supplementary material developed by the author as part of an advanced taught MSc programme in 'Digital Signal Processing'. This programme was originally established at Cranfield University in 1990 and modified at De Montfort University in 1994. The programmes are still operating at these universities and the material has been used by more than 500 graduates since its creation and development in the early 1990s. The material was enhanced and developed further when the author moved to the Department of Electronic and Electrical Engineering at Loughborough University in 2003, and now forms part of the department's post-graduate teaching and learning activities. The original MSc programme was based on taught components covering a period of eight months and consisting of two semesters, each semester, being composed of four modules; the third semester focused on a minor research project. The material in this work covers the second semester and is 'index-linked' through this teaching programme to the publication Digital Signal Processing (Horwood, 2003) which covers the first semester. The classification of this work into four parts reflects the four modules given in the second semester. It has been necessary to include some of the material published previously with the view of revising some of the principal themes such as those concerned with the properties and computational methods associated with the Fourier transform. This has been done for reasons of completeness and to provide the reader with an account of the field that does not necessarily require significant reference to previous publications (by the author or otherwise).

An attempt has been made to cut through much of the jargon characterizing different fields of research in imaging science presenting an account of the fundamental physical principles common to nearly all imaging systems. This is done by illustrating the similarity of the underlying mathematical models used to process data on a wavefield in a variety of applications. The approach has been to unify the principles of different imaging systems and to provide a course text covering the theoretical foundations of imaging science in an integrated and complete form.Introduction

Vector Fields

2D Fourier Theory

The 2D DFT, FFT and FIR Filter

Field and Wave Equations

Green Functions

Scattering Theory

Imaging of Layered Media

Projection Tomography

Diffraction Tomography

Synthetic Aperture Imaging

Optical Image Formation

Image Restoration and Reconstruction

Reconstruction of Band-limited Images

Bayesian Estimation Methods

Image Enhancement

Segmentation and Edge Detection

Statistical Modelling and Analysis

Fractal Images and Image Processing

Coding and Compression

A: Solutions to Problems

B: Supplementary Problems

C: Fourier Transform of a Fractal

D: I/O and Graphics Utilities

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