SHAPE ANALYSIS AND CLASSIFICATION: THEORY AND PRACTICE

CRC Press Book Series on Image Processing - CRC Press

1. INTRODUCTION

• 1.1 INTRODUCTION TO SHAPE ANALYSIS
• 1.2 CASE STUDIES
  • 1.2.1 Case Study: Morphology of Plant Leaves
  • 1.2.2 Case Study: Morphometric Classification of Ganglion Cells
• 1.3 COMPUTATIONAL SHAPE ANALYSIS
  • 1.3.1 SHAPE PRE-PROCESSING
  • 1.3.2 SHAPE TRANSFORMATIONS
  • 1.3.3 SHAPE CLASSIFICATION
• 1.4 ORGANIZATION OF THE BOOK

2. MATHEMATICAL CONCEPTS

• 2.1 BASIC CONCEPTS
  • 2.1.1 Propositional Logic
  • 2.1.2 Functions
  • 2.1.3 Free Variable Transformations
  • 2.1.4 Some Special Real Functions
  • 2.1.5 Complex Functions

• 2.2 LINEAR ALGEBRA
  • 2.2.1 Scalars, Vectors, and Matrices
  • 2.2.2 Vector Spaces
  • 2.2.3 Linear Transformations
  • 2.2.4 Metric Spaces, Inner Products, and Orthogonality
  • 2.2.5 More about Vectors and Matrices

• 2.3 DIFFERENTIAL GEOMETRY
  • 2.3.1 2D Parametric Curves
  • 2.3.2 Arc Length, Speed and Tangent Fields
  • 2.3.3 Normal Fields and Curvature

• 2.4 MULTIVARIATE CALCULUS
  • 2.4.1 Multivariate Functions
  • 2.4.2 Directional, Partial, and Total Derivatives
  • 2.4.3 Differential Operators

• 2.5 CONVOLUTION AND CORRELATION
  • 2.5.1 Continuous Convolution and Correlation
  • 2.5.2 Discrete Convolution and Correlation
  • 2.5.3 Non-Linear Correlation as a Coincidence Operator

• 2.6 PROBABILITY AND STATISTICS
  • 2.6.1 Events and Probability
  • 2.6.2 Random Variables and Probability Distributions
  • 2.6.3 Random Vectors and Joint Distributions
  • 2.6.4 Estimation
  • 2.6.5 Autocorrelation 1242.6.6 The Karhunen-Loève Transform

• 2.7 FOURIER ANALYSIS
  • 2.7.1 Brief Historical Remarks
  • 2.7.2 The Fourier Series
  • 2.7.3 Cookie Recipe
  • 2.7.4 Fourier Series
  • 2.7.5 The Continuous One-Dimensional Fourier Transform
  • 2.7.6 Frequency Filtering
  • 2.7.7 The Discrete One-Dimensional Fourier Transform
  • 2.7.8 Matrix Formulation of the DFT
  • 2.7.9 Applying the DFT
  • 2.7.10 The Fast Fourier Transform
  • 2.7.11 Discrete Convolution Performed in the Frequency Domain

3. IMAGE ACQUISITION AND PROCESSING

• 3.1 IMAGE REPRESENTATION
  • 3.1.1 Image Formation and Gray Level Images
  • 3.1.2 Sampling and Quantization
  • 3.1.3 Shape Sampling
  • 3.1.4 Binary Images
  • 3.1.5 Color Digital Images
  • 3.1.6 Video Sequences
  • 3.1.7 Multispectral Images
  • 3.1.8 Voxels
• 3.2 IMAGE PROCESSING AND FILTERING
  • 3.2.1 Histograms and Pixel Manipulation
  • 3.2.2 Local or neighborhood processing
  • 3.2.3 Average Filtering
  • 3.2.4 Linear Filtering
  • 3.2.5 Linear Filtering
  • 3.2.6 Gaussian Smoothing
  • 3.2.7 Fourier-based filtering
  • 3.2.8 Median and other Non-Linear Filters
• 3.3 IMAGE SEGMENTATION: EDGE DETECTION
  • 3.3.1 Edge Detection in Binary Images
  • 3.3.2 Gray-Level Edge Detection
  • 3.3.3 Gradient-Based Edge Detection
  • 3.3.4 Roberts Operator
  • 3.3.5 Sobel, Prewitt and Kirsch Operator
  • 3.3.6 Second Order Operators: Laplacian
  • 3.3.7 Fourier-Based Edge Detection
  • 3.3.8 Multiscale Edge Detection: The Marr-Hildreth Transform
• 3.4 IMAGE SEGMENTATION: FURTHER ALGORITHMS
  • 3.4.1 Image Thresholding
  • 3.4.2 Region-Growing
• 3.5 BINARY MATHEMATICAL MORPHOLOGY
  • 3.5.1 Image Dilation
  • 3.5.2 Image Erosion
• 3.6 CONCLUDING REMARKS

4. SHAPE CONCEPTS

• 4.1 INTRODUCTION TO TWO-DIMENSIONAL SHAPES

• 4.2 CONTINUOUS TWO-DIMENSIONAL SHAPES
  • 4.2.1 Continuous Shapes and their Types

• 4.3 PLANAR SHAPE TRANSFORMATIONS

• 4.4 CHARACTERIZING 2D SHAPES IN TERMS OF FEATURES

• 4.5 CLASSIFYING 2D SHAPES

• 4.6 REPRESENTING 2D SHAPES
  • 4.6.1 General Shape Representations
  • 4.6.2 Landmark Representations

• 4.7 SHAPE OPERATIONS

• 4.8 SHAPE METRICS
  • 4.8.1 The 2n Euclidean Norm
  • 4.8.2 The Mean Size
  • 4.8.3 Alternative Shape Sizes
  • 4.8.4 Which Size?
  • 4.8.5 Distances Between Shapes

• 4.9 MORPHIC TRANSFORMATIONS
  • 4.9.1 Affine Transformations
  • 4.9.2 Euclidean Motions
  • 4.9.3 Rigid Body Transformations
  • 4.9.4 Similarity Transformations
  • 4.9.5 Other Transformations and Some Important Remarks
  • 4.9.6 Thin-Plate Splines
  • 4.9.6.1 Single Thin-Plate Splines
  • 4.9.6.2 Pairs of Thin-Plate Splines

5. TWO-DIMENSIONAL SHAPE REPRESENTATION

• 5.1 INTRODUCTION
• 5.2 PARAMETRIC CONTOUR
  • 5.2.1 Contour Extraction
  • 5.2.2 A Contour Following Algorithm
  • 5.2.3 Contour Representation by Vectors and Complex Signals
  • 5.2.4 Contour Representation based on the Chain Code
• 5.3 SET OF CONTOUR POINTS
• 5.4 CURVE APPROXIMATION
  • 5.4.1 Poligonal Approximation
  • 5.4.2 Ramer Algorithm for Polygonal Approximation
  • 5.4.3 Split-and-Merge Algorithm for Polygonal Approximation
• 5.5 DIGITAL STRAIGHT LINES
  • 5.5.1 Straight Lines and Segments
  • 5.5.2 Generating Digital Straight Lines and Segments
  • 5.5.3 Recognizing an Isolated Digital Straight Segment
• 5.6 HOUGH TRANSFORMS
  • 5.6.1 Continuous Hough Transforms
  • 5.6.2 Discrete Image and Continuous Parameter Space
  • 5.6.3 Discrete Image and Parameter Space
  • 5.6.4 Backmapping
  • 5.6.5 Problems with the Hough Transform
  • 5.6.6 Improving the Hough Transform
  • 5.6.7 General Remarks on the Hough Transform
• 5.7 EXACT DILATIONS
• 5.8 DISTANCE TRANSFORMS
• 5.9 EXACT DISTANCE TRANSFORM THROUGH EXACT DILATIONS
• 5.10 VORONOI DIAGRAMS
• 5.11 SCALE SPACE SKELETONIZATION

6. SHAPE CHARACTERIZATION

• 6.1 STATISTICS FOR SHAPE DESCRIPTORS
• 6.2 SOME GENERAL DESCRIPTORS
  • 6.2.1 Perimeter
  • 6.2.2 Area
  • 6.2.3 Centroid (center of mass)
  • 6.2.4 Maximum and Minimum Distance to Centroid
  • 6.2.5 Distance to the Boundary
  • 6.2.6 Diameter
  • 6.2.7 Maximum Chord
  • 6.2.8 Norm Sizes
  • 6.2.9 Maximum Arc Length
  • 6.2.10 Major and Minor Axis
  • 6.2.11 Thickness
  • 6.2.12 Holes-based Shape Features
  • 6.2.13 Topological Descriptors
  • 6.2.14 Polygonal Approximation-Based Shape Descriptors
  • 6.2.15 Shape Descriptors based on Regions and on Graphs
  • 6.2.16 Complexity Descriptors 203
• 6.3 FRACTAL GEOMETRY FOR COMPLEXITY DESCRIPTORS
  • 6.3.1 Preliminary Considerations and Definitions
  • 6.3.2 The Box-Counting Approach
  • 6.3.3 Case Example: The Classical Koch Curve
  • 6.3.4 Implementing the Box-Counting Method
  • 6.3.5 The Minkowski Sausage or Dilation Method
• 6.4 CURVATURE
  • 6.4.1 Biological Motivation
  • 6.4.2 Simple Approaches to Curvature
  • 6.4.3 Curvature-Based Shape Descriptors
  • 6.4.4 c-Curvature
• 6.5 SHAPE SIGNATURES
• 6.6 FOURIER DESCRIPTORS
  • 6.6.1 Alternative Fourier Descriptors

7. MULTISCALE SHAPE CHARACTERIZATION

• 7.1 MULTISCALE TRANSFORMS
  • 7.1.1 The Scale-Space
  • 7.1.2 Time-Frequency Transforms
  • 7.1.3 Gabor Filters
  • 7.1.4 Time-Scale Transforms or Wavelets
  • 7.1.5 A Unified Approach to Linear Multiscale Transforms
  • 7.1.6 Case Study: Interpreting the Transforms
  • 7.1.7 Analyzing the Multiscale Transforms
• 7.2 A FOURIER APPROACH TO MULTISCALE CURVATURE
  • 7.2.1 Curvature Estimation using a Fourier Property
  • 7.2.2 Numerical Differentiation using the Fourier Property
  • 7.2.3 Gaussian Filtering and the Multiscale Approach
  • 7.2.4 Some Simple Solutions for the Shrinking Problem
  • 7.2.5 The Curvegram
  • 7.2.6 Some Experimental Results
  • 7.2.7 Curvature-Scale Space
• 7.3 MULTISCALE CONTOUR ANALYSIS USING WAVELETS
  • 7.3.1 Preliminary Considerations
  • 7.3.2 The W-Representation
  • 7.3.3 Choosing the Analyzing Wavelet
  • 7.3.4 Shape Analysis From the W-Representation
  • 7.3.5 Dominant Point Detection using the W-Representation
  • 7.3.6 Local Frequencies and Natural Scales
  • 7.3.7 Contour Analysis using the Gabor Transform
  • 7.3.8 Comparing and Integrating the Multiscale Representations
• 7.4 MULTISCALE ENERGIES
  • 7.4.1 The Multiscale Bending Energy
  • 7.4.2 The Multiscale Bending Energy
  • 7.4.3 Neuromorphometry with Bending Energy
  • 7.4.4 The Multiscale Wavelet Energy
  • 7.4.5 Case of Study: Classification of Ganglion Cells
  • 7.4.6 The Feature Space
  • 7.4.7 Feature Selection and Dimensionality Reduction

8. SHAPE RECOGNITION AND CLASSIFICATION

• 8.1 INTRODUCTION TO SHAPE CLASSIFICATION
  • 8.1.1 The Importance of Classification
  • 8.1.2 Some Basic Concepts in Classification
  • 8.1.3 A Simple Case Study in Classification
  • 8.1.4 Some Additional Concepts in Classification
  • 8.1.5 Feature Extraction
  • 8.1.6 Feature Normalization
• 8.2 SUPERVISED PATTERN CLASSIFICATION
  • 8.2.1 Bayes Decision Theory Principles
  • 8.2.2 Bayesian Classification Involving Multiple Classes and Dimensions
  • 8.2.3 Bayesian Classification of Leaves
  • 8.2.4 Nearest Neighbours
• 8.3 UNSUPERVISED CLASSIFICATION AND CLUSTERING
  • 8.3.1 Basic Concepts and Issues
  • 8.3.2 Scatter Matrices and Dispersion Measures
  • 8.3.3 Partitional Clustering
  • 8.3.4 Hierarchical Clustering
• 8.4 A CASE STUDY: LEAVES CLASSIFICATION
  • 8.4.1 Choice of Method
  • 8.4.2 Choice of Metric
  • 8.4.3 Choice of Features
  • 8.4.4 Effects of Unit Variance Normalization and Principal Component Analysis
  • 8.4.5 Validation Considering the Cophenetic Correlation Coefficient
• 8.5 EVALUATING CLASSIFICATION METHODS

9. FUTURE TRENDS IN SHAPE ANALYSIS AND CLASSIFICATION

References