A central issue in computer vision is the problem of signal to symbol transformation. In the case of texture, which is an important visual cue, this problem has hitherto received very little attention. This book presents a solution to the signal to symbol transformation problem for texture. The symbolic de- scription scheme consists of a novel taxonomy for textures, and is based on appropriate mathematical models for different kinds of texture. The taxonomy classifies textures into the broad classes of disordered, strongly ordered, weakly ordered and compositional. Disordered textures are described by statistical mea- sures, strongly ordered textures by the placement of primitives, and weakly ordered textures by an orientation field. Compositional textures are created from these three classes of texture by using certain rules of composition. The unifying theme of this book is to provide standardized symbolic descriptions that serve as a descriptive vocabulary for textures. The algorithms developed in the book have been applied to a wide variety of textured images arising in semiconductor wafer inspection, flow visualization and lumber processing. The taxonomy for texture can serve as a scheme for the identification and description of surface flaws and defects occurring in a wide range of practical applications.

1 Introduction.- 1.1 Scope of the book.- 1.1.1 Defining texture.- 1.1.2 Problem definition.- 1.2 Importance of texture.- 1.3 Potential applications of this research.- 1.4 Issues in automated process control involving computer vision.- 1.4.1 Background.- 1.4.2 Problems involving texture.- 1.4.3 The desired solution.- 1.5 A taxonomy for texture.- 1.6 Outline.- 2 Computing oriented texture fields.- 2.1 Introduction.- 2.2 Background.- 2.3 Oriented Texture Fields.- 2.4 Experimental Methods.- 2.4.1 A best estimate for dominant local orientation.- 2.4.2 Derivation using the moment method.- 2.4.3 Squaring the Gradient Vectors: Kass and Witkin's scheme.- 2.4.4 Inverse Arctangent.- 2.4.5 Flow Orientation Coherence.- 2.4.6 The effect of varying ?1 on the estimate of dominant orientation.- 2.5 Experimental Results.- 2.5.1 Comparing Calculations for Orientation.- 2.5.2 Comparing Measures of Coherence.- 2.6 Analyzing texture at different scales.- 2.7 Processing of the intrinsic images.- 2.7.1 The coherence intrinsic image.- 2.7.2 The angle intrinsic image.- 2.8 Conclusions.- 3 The analysis of oriented textures through phase portraits.- 3.1 Introduction.- 3.1.1 Overview of approach.- 3.2 Background.- 3.2.1 Related Research.- 3.2.2 Potential Applications.- 3.2.3 Obtaining the intrinsic images.- 3.2.4 Problems in post-processing.- 3.3 Geometric theory of differential equations.- 3.3.1 Preliminary definitions.- 3.3.1.1 One dimensional case.- 3.3.1.2 Two dimensional case.- 3.3.2 Linear systems.- 3.3.2.1 Case 1: Matrix A is non-singular.- 3.3.2.2 Case 2: Matrix A is singular.- 3.3.2.3 Affine transformations.- 3.3.3 Relevance of the theory.- 3.3.4 The perceptual significance of flow-like patterns.- 3.4 Experimental Methods.- 3.4.1 A Distance Measure.- 3.4.2 Non-linear least squares fitting.- 3.4.3 Implementation details.- 3.4.3.1 Generating the orientation field O2.- 3.4.3.2 Normalization of orientation fields.- 3.4.3.3 Uniqueness of the solution.- 3.4.3.4 Real Images.- 3.4.4 Segmentation.- 3.4.4.1 A measure for closeness of fit.- 3.4.4.2 Selecting the size of the window.- 3.4.5 Locating fixed points.- 3.4.6 Reconstructing the original texture.- 3.5 Experimental Results.- 3.5.1 Flow past an inclined plate.- 3.5.2 Image of secondary streaming.- 3.5.3 Image of wood knots.- 3.5.4 Image of complex flow.- 3.5.5 Image of vortex flow.- 3.5.6 Analysis of a resist gel defect.- 3.5.7 Analysis of textured paint brush strokes.- 3.6 Experiments with noise addition.- 3.7 A related model from fluid flow analysis.- 3.7.1 A comparison between the fluid motion viewpoint and the phase portrait viewpoint.- 3.7.2 Classification of velocity fields.- 3.7.3 Importance of divergence and curl.- 3.8 Discussion.- 3.8.1 Extensions to three dimensions.- 3.9 Conclusion.- 4 Analyzing strongly ordered textures.- 4.1 Introduction.- 4.2 Extraction of primitives.- 4.2.1 Edge based features.- 4.2.2 Region based features.- 4.2.2.1 Thresholding the response to ?2G.- 4.2.2.2 Solving for the response to a disk at multiple scales.- 4.3 Extracting structure from primitives.- 4.3.1 Syntactic approaches.- 4.3.2 Nearest Neighbor Histograms.- 4.3.3 Using co-occurrence matrices.- 4.3.4 Graph based approaches.- 4.4 Models for strongly ordered textures.- 4.4.1 Directions for further research.- 4.5 Symbolic descriptions: models from petrography.- 4.5.1 Description of primitives.- 4.5.1.1 Terminology from petrology.- 4.5.2 Description of placement of primitives.- 4.6 Frieze groups and wallpaper groups.- 4.6.1 Background.- 4.6.2 Preliminary definitions.- 4.6.3 Frieze groups.- 4.6.4 Wallpaper groups.- 4.7 Implications for computer vision.- 4.8 Summary.- 5 Disordered textures.- 5.1 Statistical measures for disordered textures.- 5.1.1 Computing the entropy as a measure for disorder.- 5.2 Describing disordered textures by means of the fractal dimension.- 5.2.1 Advantages.- 5.3 Computing the fractal dimension.- 5.3.1 Using the expected values of intensity differences.- 5.3.2 The reticular cell counting method for computing the fractal dimension.- 5.3.3 A comparison of the methods for measuring fractal dimension.- 5.4 Experimental Results.- 5.4.1 Implementation results for computing the entropy as a measure for disorder.- 5.4.2 Using the expected values of intensity differences.- 5.4.3 Reticular cell counting method for the fractal dimension.- 5.4.4 Discussion of results.- 5.5 Conclusion.- 6 Compositional textures.- 6.1 Introduction.- 6.2 Primitive textures.- 6.3 A Parametrized symbol set.- 6.4 Three types of composition.- 6.5 Linear combination (transparent overlap).- 6.6 Functional composition.- 6.6.1 W ? S.- 6.6.2 D ? S.- 6.6.3 D ? W.- 6.6.4 W ? D.- 6.6.5 S ? D.- 6.6.6 S ? W.- 6.6.7 S ? S, D ? D and W ? W.- 6.7 Opaque overlap.- 6.8 Definition of texture.- 6.9 A complete taxonomy for texture.- 6.10 Implementing the taxonomy.- 6.10.1 Operators for texture.- 6.10.2 Sequencing of the operators.- 6.11 Conclusion.- 7 Conclusion.- 7.1 Summary of results.- 7.1.1 Analysis of weakly ordered textures.- 7.1.2 Strongly Ordered Textures.- 7.1.3 Disordered Textures.- 7.1.4 Compositional Textures.- 7.2 Contributions.- 7.3 Future Work.- 7.3.1 Application to inspection problems.- 7.3.2 An automatic texture interpretation scheme.- 7.3.3 Taxonomies for other visual cues.- 7.3.4 Improving algorithms for Oriented Texture Analysis.- B Region Refinement.- C Preparation of the manuscript.- Permissions.