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Tables of Contents for Shape Analysis and Classification
Chapter/Section Title
Page #
Page Count
Introduction
1
26
Introduction to Shape Analysis
1
4
Case Studies
5
4
Case Study: Morphology of Plant Leaves
5
2
Case Study: Morphometric Classification of Ganglion Cells
7
2
Computational Shape Analysis
9
15
Shape Pre-Processing
9
5
Shape Transformations
14
7
Shape Classification
21
3
Organization of the Book
24
3
Basic Mathematical Concepts
27
170
Basic Concepts
27
24
Propositional Logic
28
1
Functions
29
2
Free Variable Transformations
31
2
Some Special Real Functions
33
11
Complex Functions
44
7
Linear Algebra
51
39
Scalars, Vectors and Matrices
52
4
Vector Spaces
56
6
Linear Transformations
62
3
Metric Spaces, Inner Products and Orthogonality
65
5
More about Vectors and Matrices
70
20
Differential Geometry
90
11
2D Parametric Curves
90
4
Arc Length, Speed and Tangent Fields
94
3
Normal Fields and Curvature
97
4
Multivariate Calculus
101
9
Multivariate Functions
101
6
Directional, Partial and Total Derivatives
107
2
Differential Operators
109
1
Convolution and Correlation
110
12
Continuous Convolution and Correlation
111
6
Discrete Convolution and Correlation
117
3
Nonlinear Correlation as a Coincidence Operator
120
2
Probability and Statistics
122
27
Events and Probability
122
3
Random Variables and Probability Distributions
125
6
Random Vectors and Joint Distributions
131
4
Estimation
135
9
Stochastic Processes and Autocorrelation
144
2
The Karhunen-Loeve Transform
146
3
Fourier Analysis
149
48
Brief Historical Remarks
150
1
The Fourier Series
151
6
The Continuous One-Dimensional Fourier Transform
157
13
Frequency Filtering
170
6
The Discrete One-Dimensional Fourier Transform
176
4
Matrix Formulation of the DFT
180
4
Applying the DFT
184
10
The Fast Fourier Transform
194
1
Discrete Convolution Performed in the Frequency Domain
195
2
Shape Acquisition and Processing
197
68
Image Representation
198
18
Image Formation and Gray Level Images
198
3
Case Study: Image Sampling
201
2
Binary Images
203
3
Shape Sampling
206
2
Some Useful Concepts from Discrete Geometry
208
2
Color Digital Images
210
3
Video Sequences
213
2
Multispectral Images
215
1
Voxels
216
1
Image Processing and Filtering
216
19
Histograms and Pixel Manipulation
218
5
Local or Neighborhood Processing
223
1
Average Filtering
224
2
Gaussian Smoothing
226
2
Fourier-Based Filtering
228
6
Median and Other Nonlinear Filters
234
1
Image Segmentation: Edge Detection
235
13
Edge Detection in Binary Images
237
1
Gray-Level Edge Detection
237
2
Gradient-Based Edge Detection
239
1
Roberts Operator
240
2
Sobel, Prewitt and Kirsch Operators
242
1
Fourier-Based Edge Detection
243
1
Second-Order Operators: Laplacian
244
1
Multiscale Edge Detection: The Marr-Hildreth Transform
245
3
Image Segmentation: Additional Algorithms
248
7
Image Thresholding
248
3
Region-Growing
251
4
Binary Mathematical Morphology
255
7
Image Dilation
255
4
Image Erosion
259
3
Further Image Processing References
262
3
Shape Concepts
265
66
Introduction to Two-Dimensional Shapes
265
2
Continuous Two-Dimensional Shapes
267
6
Continuous Shapes and their Types
268
5
Planar Shape Transformations
273
2
Characterizing 2D Shapes in Terms of Features
275
5
Classifying 2D Shapes
280
1
Representing 2D Shapes
281
8
General Shape Representations
283
3
Landmark Representations
286
3
Shape Operations
289
1
Shape Metrics
290
11
The 2n Euclidean Norm
291
4
The Mean Size
295
1
Alternative Shape Sizes
295
1
Which Size?
296
2
Distances between Shapes
298
3
Morphic Transformations
301
30
Affine Transformations
308
6
Euclidean Motions
314
1
Rigid Body Transformations
315
1
Similarity Transformations
315
1
Other Transformations and Some Important Remarks
316
1
Thin-Plate Splines
317
14
Two-Dimensional Shape Representation
331
90
Introduction
331
4
Parametric Contour
335
16
Contour Extraction
335
6
A Contour Following Algorithm
341
7
Contour Representation by Vectors and Complex Signals
348
2
Contour Representation Based on the Chain Code
350
1
Sets of Contour Points
351
1
Curve Approximations
352
13
Polygonal Approximation
352
2
Ramer Algorithm for Polygonal Approximation
354
6
Split-and-Merge Algorithm for Polygonal Approximation
360
5
Digital Straight Lines
365
11
Straight Lines and Segments
366
1
Generating Digital Straight Lines and Segments
367
7
Recognizing an Isolated Digital Straight Segment
374
2
Hough Transforms
376
24
Continuous Hough Transforms
376
2
Discrete Image and Continuous Parameter Space
378
4
Discrete Image and Parameter Space
382
7
Backmapping
389
2
Problems with the Hough Transform
391
2
Improving the Hough Transform
393
7
General Remarks on the Hough Transform
400
1
Exact Dilations
400
5
Distance Transforms
405
2
Exact Distance Transform Through Exact Dilations
407
1
Voronoi Tessellations
408
4
Scale-Space Skeletonization
412
7
Bounding Regions
419
2
Shape Characterization
421
46
Statistics for Shape Descriptors
421
1
Some General Descriptors
422
20
Perimeter
423
1
Area
424
1
Centroid (Center of Mass)
425
1
Maximum and Minimum Distance to Centroid
426
1
Mean Distance to the Boundary
427
1
Diameter
427
2
Norm Features
429
1
Maximum Arc Length
429
1
Major and Minor Axes
429
3
Thickness
432
1
Hole-Based Shape Features
432
1
Statistical Moments
433
1
Symmetry
434
1
Shape Signatures
435
3
Topological Descriptors
438
1
Polygonal Approximation-Based Shape Descriptors
438
1
Shape Descriptors based on Regions and Graphs
439
1
Simple Complexity Descriptors
439
3
Fractal Geometry and Complexity Descriptors
442
7
Preliminary Considerations and Definitions
442
1
The Box-Counting Approach
443
1
Case Study: The Classical Koch Curve
443
2
Implementing the Box-Counting Method
445
2
The Minkowsky Sausage or Dilation Method
447
2
Curvature
449
10
Biological Motivation
449
2
Simple Approaches to Curvature
451
5
c-Measure
456
1
Curvature-Based Shape Descriptors
457
2
Fourier Descriptors
459
8
Some Useful Properties
461
3
Alternative Fourier Descriptors
464
3
Multiscale Shape Characterization
467
66
Multiscale Transforms
467
17
Scale-Space
468
3
Time-Frequency Transforms
471
1
Gabor Filters
472
1
Time-Scale Transforms or Wavelets
473
3
Interpreting the Transforms
476
4
Analyzing Multiscale Transforms
480
4
Fourier-Based Multiscale Curvature
484
18
Fourier-Based Curvature Estimation
484
3
Numerical Differentiation Using the Fourier Property
487
3
Gaussian Filtering and the Multiscale Approach
490
1
Some Simple Solutions for the Shrinking Problem
491
3
The Curvegram
494
6
Curvature-Scale Space
500
2
Wavelet-Based Multiscale Contour Analysis
502
22
Preliminary Considerations
502
2
The w-Representation
504
2
Choosing the Analyzing Wavelet
506
2
Shape Analysis from the w-Representation
508
1
Dominant Point Detection Using the w-Representation
509
6
Local Frequencies and Natural Scales
515
3
Contour Analysis using the Gabor Transform
518
2
Comparing and Integrating Multiscale Representations
520
4
Multiscale Energies
524
9
The Multiscale Bending Energy
524
3
Bending Energy-Based Neuromorphometry
527
3
The Multiscale Wavelet Energy
530
3
Shape Recognition and Classification
533
84
Introduction to Shape Classification
533
32
The Importance of Classification
534
2
Some Basic Concepts in Classification
536
2
A Simple Case Study in Classification
538
7
Some Additional Concepts in Classification
545
5
Feature Extraction
550
6
Feature Normalization
556
9
Supervised Pattern Classification
565
12
Bayes Decision Theory Principles
565
6
Bayesian Classification: Multiple Classes and Dimensions
571
3
Bayesian Classification of Leaves
574
1
Nearest Neighbors
574
3
Unsupervised Classification and Clustering
577
23
Basic Concepts and Issues
577
3
Scatter Matrices and Dispersion Measures
580
3
Partitional Clustering
583
6
Hierarchical Clustering
589
11
A Case Study: Leaves Classification
600
8
Choice of Method
602
1
Choice of Metrics
603
1
Choice of Features
604
3
Validation Considering the Cophenetic Correlation Coefficient
607
1
Evaluating Classification Methods
608
9
Case Study: Classification of Ganglion Cells
609
1
The Feature Space
610
2
Feature Selection and Dimensionality Reduction
612
5
Epilogue-Future Trends in Shape Analysis and Classification
617
4
Bibliography
621
28
Index
649