This book is one in a series dedicated to fine particle science and technology. Topics covered in the book include the role of definitions, concepts, hypothesis, and laws; morphological analysis of fine particles and surfaces; analytical three-dimensional representations of particle and surface morphologies; the problem of invariance with respect to rotational transformations, as well as transformations characterized by reflection and inversion; matrix mechanics of particle characterization; and general applications of morphological analysis in other areas of science.
INTRODUCTION: WHAT IS A SCIENTIFIC THEORY? The Theory of Morphology. The Development of a Theory. Elements of a Theory. Morphology as a Theory. A CRITIQUE OF CLASSICAL SIZE AND SHAPE ANALYSIS. Classical Size. Classical Shape. CLASSICAL FOURIER ANALYSIS OF A SINGLE PARTICLE. Fourier Series. The R(q) Method. The f*(L) Method. The (R,S) Method. Classical Feature Extraction. THE THEORY OF MORPHOLOGY. The Need. Particle Characterization. Morphological Analysis. Characterization of a Physical Process. Characterization of a Physical Process-An Example in Fluid Flow. THE BOUNDARY FUNCTIONS OF PARTICLE REPRESENTATION. A Unified Approach to Particle Representation. Boundary Functions and the Variational Principle. The Particle Profile, R(q). The Extended Surface, G(x,y). The Extended Surface, G(r,q). The Three-Dimensional "Bulky" Particle. The Finite Fiber. The Repeating Fiber. Infinite Fibers and Threads. Flakes. The Future. DEVELOPMENT OF THE MORPHOLOGICAL VARIATIONAL PRINCIPLE AND DERIVATION OF THE BOUNDARY FUNCTIONS. The Concept of Shape. The Development of the Morphological Variational Principle. Derivation of the Boundary Function, R(q). The Boundary Function, G(r,q), for Extended Surfaces. Extending the Morphological Variational Principle to Higher Dimensions. The Theory of Continuous Functions. The Morphological Variational Principle. The Surface Area for an Irregular Surface. Image Analysis. The Derivation of the Gray Level Function for Extended Surfaces. Applications of the Gray Level Function in Microscopy. The Bessel-Fourier Coefficients for Extended Surfaces. Limitations of the Gray Level Function for Extended Surfaces. The Three Dimensional Particle, R(q,Æ). The Finite Fiber, R(q,Z). The Repeating Fiber, R(q,Z). The Infinite Fiber, R*(q,Z). Flakes, R(q), Independent of Z. Commentary on the Derivations of the Boundary Functions as Particle Representations. Closure-The Boundary Functions of the Theory of Morphology. FEATURE EXTRACTION FROM PARTICLE REPRESENTATIONS. Guiding Principles of Feature Extraction in the Theory of Morphology. The Concept of Particle Shape. The Concept of Particle Size. Extracting the Size Feature. The Morphological Variational Principle as a Fundamental Law or Hypothesis in the Theory of Morphology. The Equivalent Radius as a Derived Law in the Theory of Morphology. Statistical Features of the Particle Profile. Rotational Invariance of the Moments of the Radial Distribution. Invariant Fourier Shape Features. Symmetry Analysis of a Particle Profile. Partial Symmetry Operations. The Rotational Partial Symmetry Element, Cm. The Reflectional Partial Symmetry Element, m. Group Properties of the Classical Symmetry Elements. A Symmetry Classification Scheme When a Plane of Symmetry Exists in the Particle Profile. A Symmetry Classification Scheme Utilizing Group Properties When a Plane of Symmetry Does Not Exist in the Profile. Physical Features Associated With Boundary Functions. Feature Extraction From Sets of Particles. The Future of the Theory of Morphology. THE GENERALIZED R(q) METHOD-THE REENTRANT PARTICLE. The Reentrant Particle Profile. Reparameterization of the Reentrant Particle. Derivation of the Boundary Function, R(t), for the Reentrant Particle. Closure. AN INTRODUCTION TO THE MORPHOLOGICAL ANALYSIS OF THE REGULAR FIGURES. The Circle. The Cardioid. The Lemniscate. The Triangle. The Square. The Pentagon. The Hexagon. The Ellipse. The Rectangle. APPLICATIONS OF THE THEORY OF MORPHOLOGY. The Effect of Particle Morphology of the Flow of a Dextrose Powder. Quality Assessment of Industrial Sieve Mesh. Effects of Powder Production and Material Processing on the Morphology of Adipic Acid. Differentiation Between Three Races of Giraffe Based on the Morphic Features of Trunk Spots. The Morphological Features of the Lower 48. Closure. GENERAL REMARKS ON THE THEORY OF MORPHOLOGY. The Theory of Morphology as a Scientific Theory. Future Technical Developments for the Theory of Morphology. The Future for the Theory of Morphology. Opportunity. APPENDIX I: A COMPUTER PROGRAM TO CALCULATE FOURIER COEFFICIENTS FOR NON-REENTRANT PARTICLE PROFILES. APPENDIX II: THE GENERAL PROPERTIES OF THE FOURIER MOMENT FUNCTION. The Fourier Moment Function. The Derivation of the General Moment Function of the Fourier Series. The Second Moment Function.