Fractal Surfaces. John C. Russ. Springer Science+Business Media, LLC. North Carolina State University Raleigh, North Carolina

Transcription

1 Fractal Surfaces

2 Fractal Surfaces John C. Russ North Carolina State University Raleigh, North Carolina Springer Science+Business Media, LLC

3 Library of Congress Cataloging-in-Publication Data Russ. John C. Fractal surfaces ( John C. Russ. p. cm. Includes bibliographical references and index. 1. Surfaces (Physics)--Measurement. 2. Fractals. 1. Title. QCI73.4.S94R '.74--dc CIP ISBN ISBN (ebook) DOI / Springer Science+Business Media New York Originally published by Plenum Press, New York in Softcover reprint ofthe hardcover 1st edition 1994 All 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, rnicrofilrning, recording, or otherwise, without written perrnission from the Publisher.

4 Preface Fractals occupy a borderline between Euclidean geometry and complete ran~omness. In terms of the frequency distribution, one method that can be used to measure the fractal dimension, this is the space between a pure tone and white noise. This is similar, and perhaps related to, the fact that chaos theory (also called complexity theory) in mathematics lies in the critical boundary between pure Newtonian deterministic physics and complete random unpredictability. It appears that Nature tends toward this "self-organized criticality" as an attractor. Like all new paradigms, the notion of fractals has been as much abused as used. It remains to be seen how broadly useful it will become in many different fields of application, or whether it really does offer any explanation of surfaces, their history, and properties. But as a phenomenological description of surface geometry, fractal dimensions "work" in a surprising number of instances. Since this is the way Nature has chosen to behave, it will be important for researchers to apply the new methods for characterization discussed in this book, and for models that attempt to describe the generation and behavior of surfaces to incorporate this geometry. In writing this book, I have attempted to merge several different facets of the overall problem. Most of the early chapters are devoted to discussions of fractal geometry, and the methods available for the measurement of the dimension of surfaces. This can be done by a few direct methods and many indirect ones, eitherdealing with the surface in its entirety or by working in a lower dimension by interseetion of the surface with a horizontal or vertical plane. Each of these choices presents some difficulties of technique or interpretation, and in any case the various "dimensions" that are measured are not all equivalent, a point that has been seriously under-appreciated by many workers in this young field. A second section of the book discusses the modeling of fractal surfaces. This is not done with an eye to making pretty images for magazine covers or sets for movies, but to duplicate real surfaces produced by various processes, and to provide a basis for understanding the various measurement tools and the role of instrument noise and other limits to performance. The final chapter attempts to compile some of the applications of fractal geometry to real surfaces, produced by a variety of operations such as machining, fracture, deposition, and so forth. Some of these examples are also used to illustrate the earlier sections on methods, and some duplication of information between the various chapters has been unavoidable. The breadth of applications of fractal geometry is astounding, but the depth of the literature is still rather shallow in most places. Much more work remains to be done before conclusions about the relationship(s) between surface fractal dimension and the history or properties of the surfaces can be reached. The wide variety of applications has two consequences. Publication of preliminary results showing that some particular surfaces can be described by fractal geometry has taken place in v

5 vi Prefacc a great many different journals, reaching different audiences. This entails a certain duplication of effort and rediscovery or reinvention of principles. Secondly, the fohow-on work to find the quantitative relationships and model the underlying physics, in an attempt to "explain" the observations, has been quite spotty. In many fields, this has not yet taken place. One of the frustrations in writing the book has been keeping up with the ongoing publication of new results and methods in the field. I will apologize in advance for missing some papers. A second difficulty has been finding an organization that makes it possible to read about a particular application or technique without jumping all over the book. There is a certain amount of necessary duplication as a result. Some ideas appear and are at least briefly described in several places; the alternative was to force the reader to skip back and forth. It is hoped that this book will stimulate more work, provide some access to the basic tools for measurement and interpretation of data, and encourage more researchers to apply fractal geometry to their individual problems. As more quantitative data are accumulated, and more correlations between fractal parameters and surface and material properties are discovered, it may lead to an understanding of the reasons that Nature has adopted this geometry to shape so many parts of our world. It is a pleasure to acknowledge the contributions of coworkers to this book. Ron Scattergood, Tom Hare, and Mark Ray have ah endured head-banging sessions at the blackboard helping to find ways through strange territory. Several collaborators have provided data and feedback to questions, and exchanged papers before publication, particularly Miguel Aguilar, Johannes BueHer, Brian Kaye, Michael Hamblin, and Paul Scott. Carl Zanoni (Zygo Corporation) and George Collins (Topometrix) have kindly provided data on specimens from state-of-the-art high-resolution surface characterization tools. A number of graduate students, particularly Yusef Fahmy, Sreeram Srinivasan, Michael Tidwell, and John Tyner, have obtained data as part of their thesis pro grams which are incorporated here. Mark Schaffer has been a willing and able "gofer" to track down elusive and often incomplete references. Chris Russ wrote the original Macintosh pro gram shell and contributed portions of the code. Marty Esterrnan translated the finished program to run under Windows. Helen Adams has been a willing ear, asounding board to see whether my explanations make sense, and has made possible a program to teach these concepts to 4th and 5th grade ehildren (Adams and Russ 1992). She has also provided the moral support and enthusiasm necessary to write the book. John C. Russ Oetober, 1993

6 Contents 1. Introduction A Little History Classes of Fractal Surfaces Monster Curves Random Fractals Dusts Zerosets Korcak Islands Deposited Surfaces Pore Structures L-Systems Measuring the Fractal Dimension of Boundary Lines Richardson Plots U sing a Computer Digitized Images Mosaic Amalgamation and the Kolmogorov Dimension The Minkowski Sausage Implementation with the Euclidean Distance Transfonn Fitting Lines to Data Other Methods Comparison of Dimensions The Relationship between Boundary Lines and Surfaces Analogies Direct Methods Zerosets Dimensional Analysis vii

7 viii Contents The Minkowski Comforter Data Formats Effects ofnoise Simulated Noise Hurst and Fourier Analysis Time-Based Data The Hurst Plot Fractal Brownian Motion Elevation Profiles Hurst Analysis of Range Images The Hurst Orientation Transform Fourier Analysis Fourier Analysis ofboundary Lines White and lifnoise Fourier Analysis in Two Dimensions Anisotropy Characterizing the Magnitude ofthe Roughness The Topothesy Light Reflection and Scattering Visual Appearance Fractal Brownian Profiles Electrons. Radar. etc Local Texture Measurement Brightness Patterns from Rough Surfaces Relating the Surface and Brightness Dimensions Light Scattering Scattering of Diffuse Light from Rough Surfaces Range Measurement Methods Range Images and Surface Parameters Modeling Fractal Profiles and Surfaces Fractal Profiles Particle Aggregation Deposited Surfaces Modeling a Fractal Surface Fractal Brownian Surfaces Mandelbrot-Weierstrass Functions

8 Contents ix Comparing Models and Measurements...,, Takagi Functions Weak and Strong Anisotropy.., Modeling an Anisotropic Surface Mixed Fractals Limited Self-Similarity... " 191 Kaye's Definition Simulating the Projection... " 195 Vicsek's Fat Fractal.... ' Mixed Fractals The Mandelbrot Conjectures Measurement of Fractal Dimensions Addition of Fractals Variation with Scale Splicing Fractals Together Directionality Tentative Conclusions Examples of Fractal Surfaces Brittle Fracture Machining and Wear Deposited Surfaces Pore Networks Other Surface Applications Very Flat Surfaces I: Interferometry Very Flat Surfaces II: Atomic Force Microscopy References Appendix Index