PRACTICAL STEREO LOGY Second Edition

PRACTICAL STEREO LOGY Second Edition

John C. Russ North Carolina State University Raleigh, North Carolina

and

Robert T. Dehoff University of Florida Gainesville, Florida

Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Russ, John C. Practical stereology/John C. RlIss and Robert T. Dehoff.-2nd ed.

p. cm. Includes bibliographical references and index. ISBN 978-1-4613-5453-6 ISBN 978-1-4615-1233-2 (eBook)

DOI 10.1007/978-1-4615-1233-2 1. Stereology. 1. DeHoff, Robert T. II. Title.

Q175 .R855 2000 502' .87 -dc21

Additiona1 material to this book can be downloaded from http://extras.springer.com. ISBN 978-1-4613-5453-6

©20oo Springer Science+Business Media New York

00-046612

Origina11y published by Kluwer Academic / Plenum Publishers, New York in 2000 Softcover reprint of the hardcover 1 st edition http://www.wkap.nl/

10 9 8 7 6 5 4 3 2 1

A C.I.P. record for this book is available from the Library of Congress

AH rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, withollt writ!en permission from the Publisher

Preface

Since the first edition of this book was published more than a decade ago, a revolution in stereological techniques has occurred. The new emphasis is on design-based techniques in which new types of probes are used to extract specific kinds of information from structures, without the traditional requirements that they be isotropic, uniform or random. The literature on new methods for sampling struc­tures using oriented plane and line probes is extensive and continuously expanding, and there is some danger of overlooking the utility of the traditional and classical methods which retain important areas of applicability.

In rewriting the second edition of this book, we have tried to treat each of the three major classes of stereological tools. Most general and easily applied are those that measure global microstructural parameters such as volume fraction, mean intercept length, surface area and curvature, and line length. The classical model-based methods that make geometric assumptions about the structures of interest are covered along with the simulation methods that are used to generate the required models. And finally the new design-based techniques are introduced, using vertical sections and cycloids, structured sampling strategies, and oriented surface probes. Some chapters emphasize the procedures for applying stereological mea­surement techniques, while others emphasize the derivation of the underlying rela­tionships. The chapters are intended to be largely self-contained and so some duplication of topics occurs.

All of these methods can be applied relatively simply by understanding what is being measured and practicing the step-by-step methods of implementation. It is not required that the user be able to derive the techniques from basic geometric probability (which is fortunate). The most important requirement for the user is to become comfortable with what aspects of structure are important and how they can be measured. In most cases, the resulting measurements are used to compare one structure to another or to correlate the geometric properties of the structure with its history of development or its properties and performance. This must be done statistically, and so there is also a brief but useful chapter on the statistical tools of primary importance. Again, these are introduced as practical tools for usage and not derived.

The basic stereological tools are equally useful in a broad variety of appli­cations. They are used in biological and medical research, materials science, geology, food science, and other fields. There are some minor differences of emphasis based on how sampling and sectioning can be accomplished with these different types of materials, and some differences in terminology used to describe the results, but the underlying methods are the same. Metallurgists may discuss grains where biologists see cells, but the idea of a space-filling tesselation of subunits is the same. Although

v

vi Preface

our own field is materials science, we have tried to use a mixture of terminologies throughout in hopes of communicating the ideas to a broad spectrum of potential users of these methods.

We are indebted to Dr. Jeanette Norden, from the Department of Cell Biology at Vanderbilt University, who has collaborated with us in teaching this material at a series of annual workshops at North Carolina State University for more than fifteen years. These workshops consider in depth both computer-based image processing and measurement, and the stereological interpretation of the mea­surements. It has been our experience that these two topics are generally taught quite separately and individual researchers are often not familiar with both areas, which is unfortunate. Our collaboration in teaching the workshops has led to the writing of this book, which we believe combines our two somewhat different points of emphasis to provide a balanced treatment of the topics.

The text was written making extensive use of color images and figures, as used in the workshops. It has not proved practical to print these in the text, so, as an alternative, the entire book is also supplied on CD-ROM in the form of "pdf' (Acrobat) files containing the original color figures. The reader is encouraged to refer to those figures when color is important for discriminating details.

John C. Russ Raleigh, NC

Robert T. Dehoff Gainesville, FL

Contents

Chapter 1: Introduction Elements of microstucture Geometric properties of features 4 Typical stereo logical procedures 6 Fundamental relationships 8 Intercept length and grain size 10 Curvature 13 Second order stereo logy 14 Stereo logy of single objects 16

Chapter 2: Basic Stereo logical Procedures 19 What stereology is 19 How stereo logy works 20 Why stereology works 24 Ground rules for applying stereology 27

Chapter 3: Geometry of Microstructures 29 The qualitative microstructural state 31 The quantitative microstructural state 35

Metric properties 35 Topological properties 37 Ratios of global properties 39

The topographic microstructural state 39 Gradients: variation with position 39 Anisoptropies: variation with orientation 41 Associations 42

Chapter 4: Classical Stereo logical Measures 45 Two-dimensional structures; area fraction

from the point count 45 Volume fraction from the point count 51 Two-dimensional structures; feature perimeter

from the line intercept count 56 Three-dimensional structures: surface area

and the line intercept count 63 Three-dimensional microstructures; line length

and the area point count 69

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viii Contents

Chapter 5: Less Common Stereo logical Measures 79 Three-dimensional features: topological

properties and the volume tangent count 79 Three-dimensional features: the mean

caliper diameter 88 Mean surface curvature and its integral 90 The sweeping line probe in two dimensions 99 Edges in three-dimensional Microstructures 104

Chapter 6: Sample Design in Stereo logy 111 Population of point probes 111 Population of lines in two dimensions 115 Line probes in three dimensions 115 Planes in three-dimensional space 118 Disectors in three-dimensional space 118 Sampling strategies in 3D 119

Chapter 7: Procedures for IUR Sampling 127 Volume fraction 128 Sampling planes 131 Isotropic planes 132 Isotropic line probes 136 Volume probes-the Disector 142 Networks 145

Chapter 8: Statistical Interpretation of Data 149 Sources of variability in measurement 149 Distributions of values 152 The mean, median and mode 155 The central limit theorem and the Gaussian

distribution 156 Variance and standard deviation 157 Testing distributions for normality-skew and

kurtosis 160 Some other common distributions 161 Comparing sets of measurements-the T-test 165 Nonparametric comparisons 168 Linear regression 175 Nonlinear regression 179

Chapter 9: Computer-Assisted Methods 183 Getting the image to the computer 183 Display and storage 187 Image processing 191

Contrast manipulation 193 Correcting nonuniform brightness 194

Contents ix

Reducing image noise 198 Rectifying image distortion 200 Enhancement 202

Overlaying grids onto images 204 Basic stereo logical calculations 210 Appendix 211

Chapter 10: Computer Measurement of Images 223 Measurement using grids 223 Measuring area with pixels 229 Measurement parameters-size 237 Other feature measurements: shape and position 246 Image processing to enable thresholding and

measurement 249 Image processing to extract measurable

information 255 Combining multiple images 261

Chapter 11: Geometric Modeling 271 Methods: analytic and sampling 272 Sphere intercepts 275 Intercept lengths in other bodies 275 Intercept lengths in three dimensions 280 Intersections of planes with objects 286 Bertrand's paradox 287 The Buffon needle problem 290 Appendix 293

Chapter 12: Unfolding Size Distributions 297 Linear intercepts in spheres 298 Plane intersections 300 Other shapes 303 Simpler methods 309 Lamellae 310

Chapter 13: Anisotropy and Gradients 313 Grain structures in rolled metals 313 Boundary orientation 320 Gradients and neighbor relationships 323 Distances and irregular gradients 327 Alignment 330

Chapter 14: Finite Section Thickness 331 Projected images 331 Bias in stereological measurements 337 Measurements within sections 340

x Contents

Chapter 15: Three-Dimensional Imaging 345 Limitations of stereology 345 Serial methods for acquiring 3D image data 345 Inversion to obtain 3D data 347 Stereoscopy as a 3D technique 349 Visualization 352 Processing 358 Measurement 361

References 367 Index 373