# Advances in Imaging and Electron Physics, Volume 175

## 1st Edition

**Serial Editors:**Peter Hawkes

**Hardcover ISBN:**9780124076709

**eBook ISBN:**9780124077973

**Imprint:**Academic Press

**Published Date:**7th February 2013

**Page Count:**360

**View all volumes in this series:**Advances in Imaging and Electron Physics

## Table of Contents

Editor-In-Chief

Preface

Future Contributions

Contributors

Chapter One. Small Angle Scatter with Correlation, Scatter and Intermediate Functions

1 Overview of Scatter: Neutron and X-ray Small-angle Scatter in Perspective

2 Neutron and X-ray Scatter Amplitude from Non–bravais Lattice Crystal

3 Neutron and X-ray Scatter Intensity from Non–Bravais Lattice Crystal

4 Small-angle Scatter: Scatter Length Density and Particle Structure Factor

5 Small-angle Scatter: Scatter Amplitudes and Intensity

6 The Rayleigh–Gans Equation, Babinet’s Principle, and Differential Cross Sections

7 Random Variables: Correlation and Independence

8 The Scattering Vector Resolution of SANS Instruments for Neutrons and X-rays

9 Macroscopic Differential Cross Section: Scatter Length Density Convolution and Correlation

10 The Pair Correlation Function

11 The Macroscopic Differential Cross Section for Elastic Scatter Expressed by Intermediate Function, and Fourier Transform of Patterson Function

12 Scatter Function for Elastic and Inelastic Scatter from Sample Solute Particles Obtained from Neutron (X-ray), Sample-Averaged Scatter Intensity

13 Sample-averaged Scatter Intensity Versus Scattering Vector q for the Guinier, Intermediate, and Porod Regimes

14 Small-angle Scatter: Measurement of Solute Particle Size and Shape—Guinier Regime

15 Small-angle Scatter from Spherical Particles—Guinier Regime

16 Small-angle Scatter from Particles of Various Shapes, Log-log Plots, Method of Contrast Variation—Guinier Regime

17 Small-angle Scatter—Intermediate Regime and Porod Regime

18 Small-angle Scatter—Porod Regime and Porod’s Law

19 Neutron Incoherent Scatter: Solution of the Diffusion Equation with Self-Correlation and Incoherent Scatter Functions, Measurable by Small-angle Scatter

20 Neutron Coherent Partial Differential Cross Section: The Scatter Function and the Principle of Detailed Balance

21 The Coherent and Incoherent Pair Correlation Function, Intermediate Function, and Scatter Function: The Static Approximation

22 The Particle Number Density Operator and the Coherent Intermediate and Pair Correlation Functions

23 Neutron Elastic Scatter Occurs Only in Forward Direction for Liquids and Gases Shown by Coherent Intermediate and Scatter Functions

24 Neutron Coherent Differential Cross Sections in Crystals Derived from Coherent Scatter and Intermediate Functions

25 Pair Correlation Function Expressed by Particle Density Operator and the Patterson Function

26 Neutron Coherent Differential Cross Section in Crystals Expressed by Particle Density Operators

27 Neutron Incoherent Elastic Differential Cross Section in Crystals Derived from Incoherent Intermediate Function

28 Neutron Incoherent Differential Cross Section in Crystals Derived from Incoherent Pair Correlation Function

29 No Elastic Scatter Except Forward Direction in Liquids and Gases is Shown by Intermediate and Pair Correlation Functions

30 Moments of the Neutron Scatter Function

References

Chapter Two. Nuclear Scatter of Neutron Spin States

1 Angular Momentum Vectors, Spin Vectors, and Vector Operators

2 Heisenberg Uncertainty Principle and Commutation of Operators

3 The Neutron Spin Operator

4 The Neutron Spin–Lowering and –Raising Operators

5 Nuclear Scatter of Neutron Spin States: Partial Differential Cross Section

6 Combined Neutron and Nuclear spin Operators for Polarized Neutron Scatter

7 Neutron Nuclear Scatter Lengths for Neutron Spin States

8 Partial Differential Cross Section for Single Transition Neutron Spin-State Scatter

9 Thermal Averaging: Total Partial Differential Cross Section for Neutron Spin-State Scatter

10 Neutron Spin-State Scatter Lengths for Ensemble of Nuclear Spins and Isotopes

11 Coherent Partial Differential Cross Section for Neutron Spin-State Scatter

12 Incoherent Partial Differential Cross Section for Neutron Spin-State Scatter

References

Chapter Three. Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope

1 Introduction

2 Practical Aspects

3 Theoretical Aspects

4 Selected Applications

5 Concluding Remarks

References

Chapter Four. Point Spread Function Engineering for Super-Resolution Single-Photon and Multiphoton Fluorescence Microscopy

1 Introduction

2 Theory

3 Results and Discussions

4 Conclusions

Acknowledgments

References

Chapter Five. A Review of Recent Advances in the Hit-or-Miss Transform

1 Introduction

2 Preliminaries and Properties of Mathematical Morphology

3 Fundamental Morphological Operations

4 Extensions of the Hit-or-Miss Transform

5 Conclusions

References

Chapter Six. Perspectives on Color Image Processing by Linear Vector Methods Using Projective Geometric Transformations

1 History and Background

2 Linear Filtering

3 Linear Quaternion Systems and Homogeneous Coordinates

4 Quaternion Derivations of Geometric Operations

5 The Difficulty Of Designing Linear Filters

6 Conclusion

References

Subject Index

Contents of Volumes 151-174

Color Plates

## Description

Editor-In-Chief

Preface

Future Contributions

Contributors

Chapter One. Small Angle Scatter with Correlation, Scatter and Intermediate Functions

1 Overview of Scatter: Neutron and X-ray Small-angle Scatter in Perspective

2 Neutron and X-ray Scatter Amplitude from Non–bravais Lattice Crystal

3 Neutron and X-ray Scatter Intensity from Non–Bravais Lattice Crystal

4 Small-angle Scatter: Scatter Length Density and Particle Structure Factor

5 Small-angle Scatter: Scatter Amplitudes and Intensity

6 The Rayleigh–Gans Equation, Babinet’s Principle, and Differential Cross Sections

7 Random Variables: Correlation and Independence

8 The Scattering Vector Resolution of SANS Instruments for Neutrons and X-rays

9 Macroscopic Differential Cross Section: Scatter Length Density Convolution and Correlation

10 The Pair Correlation Function

11 The Macroscopic Differential Cross Section for Elastic Scatter Expressed by Intermediate Function, and Fourier Transform of Patterson Function

12 Scatter Function for Elastic and Inelastic Scatter from Sample Solute Particles Obtained from Neutron (X-ray), Sample-Averaged Scatter Intensity

13 Sample-averaged Scatter Intensity Versus Scattering Vector q for the Guinier, Intermediate, and Porod Regimes

14 Small-angle Scatter: Measurement of Solute Particle Size and Shape—Guinier Regime

15 Small-angle Scatter from Spherical Particles—Guinier Regime

16 Small-angle Scatter from Particles of Various Shapes, Log-log Plots, Method of Contrast Variation—Guinier Regime

17 Small-angle Scatter—Intermediate Regime and Porod Regime

18 Small-angle Scatter—Porod Regime and Porod’s Law

19 Neutron Incoherent Scatter: Solution of the Diffusion Equation with Self-Correlation and Incoherent Scatter Functions, Measurable by Small-angle Scatter

20 Neutron Coherent Partial Differential Cross Section: The Scatter Function and the Principle of Detailed Balance

21 The Coherent and Incoherent Pair Correlation Function, Intermediate Function, and Scatter Function: The Static Approximation

22 The Particle Number Density Operator and the Coherent Intermediate and Pair Correlation Functions

23 Neutron Elastic Scatter Occurs Only in Forward Direction for Liquids and Gases Shown by Coherent Intermediate and Scatter Functions

24 Neutron Coherent Differential Cross Sections in Crystals Derived from Coherent Scatter and Intermediate Functions

25 Pair Correlation Function Expressed by Particle Density Operator and the Patterson Function

26 Neutron Coherent Differential Cross Section in Crystals Expressed by Particle Density Operators

27 Neutron Incoherent Elastic Differential Cross Section in Crystals Derived from Incoherent Intermediate Function

28 Neutron Incoherent Differential Cross Section in Crystals Derived from Incoherent Pair Correlation Function

29 No Elastic Scatter Except Forward Direction in Liquids and Gases is Shown by Intermediate and Pair Correlation Functions

30 Moments of the Neutron Scatter Function

References

Chapter Two. Nuclear Scatter of Neutron Spin States

1 Angular Momentum Vectors, Spin Vectors, and Vector Operators

2 Heisenberg Uncertainty Principle and Commutation of Operators

3 The Neutron Spin Operator

4 The Neutron Spin–Lowering and –Raising Operators

5 Nuclear Scatter of Neutron Spin States: Partial Differential Cross Section

6 Combined Neutron and Nuclear spin Operators for Polarized Neutron Scatter

7 Neutron Nuclear Scatter Lengths for Neutron Spin States

8 Partial Differential Cross Section for Single Transition Neutron Spin-State Scatter

9 Thermal Averaging: Total Partial Differential Cross Section for Neutron Spin-State Scatter

10 Neutron Spin-State Scatter Lengths for Ensemble of Nuclear Spins and Isotopes

11 Coherent Partial Differential Cross Section for Neutron Spin-State Scatter

12 Incoherent Partial Differential Cross Section for Neutron Spin-State Scatter

References

Chapter Three. Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope

1 Introduction

2 Practical Aspects

3 Theoretical Aspects

4 Selected Applications

5 Concluding Remarks

References

Chapter Four. Point Spread Function Engineering for Super-Resolution Single-Photon and Multiphoton Fluorescence Microscopy

1 Introduction

2 Theory

3 Results and Discussions

4 Conclusions

Acknowledgments

References

Chapter Five. A Review of Recent Advances in the Hit-or-Miss Transform

1 Introduction

2 Preliminaries and Properties of Mathematical Morphology

3 Fundamental Morphological Operations

4 Extensions of the Hit-or-Miss Transform

5 Conclusions

References

Chapter Six. Perspectives on Color Image Processing by Linear Vector Methods Using Projective Geometric Transformations

1 History and Background

2 Linear Filtering

3 Linear Quaternion Systems and Homogeneous Coordinates

4 Quaternion Derivations of Geometric Operations

5 The Difficulty Of Designing Linear Filters

6 Conclusion

References

Subject Index

Contents of Volumes 151-174

Color Plates

## Key Features

- Contributions from leading authorities
- Informs and updates on all the latest developments in the field

## Readership

Physicists, electrical engineers and applied mathematicians in all branches of image processing and microscopy as well as electron physics in general

## Details

- No. of pages:
- 360

- Language:
- English

- Copyright:
- © Academic Press 2013

- Published:
- 7th February 2013

- Imprint:
- Academic Press

- eBook ISBN:
- 9780124077973

- Hardcover ISBN:
- 9780124076709

## About the Serial Editors

### Peter Hawkes Serial Editor

Peter Hawkes graduated from the University of Cambridge and subsequently obtained his PhD in the Electron Microscopy Section of the Cavendish Laboratory. He remained there for several years, working on electron optics and digital image processing before taking up a research position in the CNRS Laboratory of Electron Optics (now CEMES-CNRS) in Toulouse, of which he was Director in 1987. During the Cambridge years, he was a Research Fellow of Peterhouse and a Senior Research fellow of Churchill College. He has published extensively, both books and scientific journal articles, and is a member of the editorial boards of Ultramicroscopy and the Journal of Microscopy. He was the founder-president of the European Microscopy Society, CNRS Silver Medallist in 1983 and is a Fellow of the Optical Society of America and of the Microscopy Society of America (Distinguished Scientist, Physics, 2015), Fellow of the Royal Microscopical Society and Honorary Member of the French Microscopy Society. In 1982, he was awarded the ScD degree by the University of Cambridge.

In 1982, he took over editorship of the Advances in Electronics & Electron Physics (now Advances in Imaging & Electron Physics) from Claire Marton (widow of the first editor, Bill Marton) and followed Marton's example in maintaining a wide range of subject matter. He added mathematical morphology to the topics regularly covered; Jean Serra and Gerhard Ritter are among those who have contributed.

In 1980, he joined Professor Wollnik (Giessen University) and Karl Brown (SLAC) in organising the first international conference on charged-particle optics, designed to bring together opticians from the worlds of electron optics, accelerator optics and spectrometer optics. This was so successful that similar meetings have been held at four-year intervals from 1986 to the present day. Peter Hawkes organised the 1990 meeting in Toulouse and has been a member of the organising committee of all the meetings. He has also participated in the organization of other microscopy-related congresses, notably EMAG in the UK and some of the

He is very interested in the history of optics and microscopy, and recently wrote long historical articles on the correction of electron lens aberrations, the first based on a lecture delivered at a meeting of the Royal Society. He likewise sponsored biographical articles for the Advances on such major figures as Ernst Ruska (Nobel Prize 1986), Helmut Ruska, Bodo von Borries, Jan Le Poole and Dennis Gabor (Nobel Prize, 1971). Two substantial volumes of the series were devoted to 'The Beginnings of Electron Microscopy' and 'The Growth of Electron Microscopy'. and others have covered 'Cold Field Emission Scanning Transmission Electron Microscopy' and 'Aberration-corrected Electron Microscopy', with contributions by all the main personalities of the subject.

### Affiliations and Expertise

Laboratoire d'Optique Electronique du Centre National de la Recherche Scientifique (CEMES), Toulouse, France