Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
CHAPTER 1: Elements of Algebra
	1.1. SETS, FUNCTIONS, AND NOTATION
		1.1.1. Special Sets and Families of Sets
		1.1.2. Functions
		1.1.3. Finite, Countable, and Uncountable Sets
	1.2. ALGEBRAIC STRUCTURES
		1.2.1. Operations on Sets
		1.2.2. Semigroups and Groups
		1.2.3. Rings and Fields
		1.2.4. Vector Spaces
		1.2.5. Homomorphisms and Linear Transforms
CHAPTER 2: Pertinent Properties of Euclidean Space
	2.1. ELEMENTARY PROPERTIES OF R
		2.1.1. Foundations
		2.1.2. Topological Properties of R
	2.2. ELEMENTARY PROPERTIES OF EUCLIDEAN SPACES
		2.2.1. Metrics on Rn
		2.2.2. Topological Spaces
		2.2.3. Topological Properties of Rn
		2.2.4. Aspects of Rn, Artificial Intelligence, Pattern Recognition, and Artificial Neural Networks
CHAPTER 3: Lattice Theory
	3.1. HISTORICAL BACKGROUND
	3.2. PARTIAL ORDERS AND LATTICES
		3.2.1. Order Relations on Sets
		3.2.2. Lattices
	3.3. RELATIONS WITH OTHER BRANCHES OF MATHEMATICS
		3.3.1. Topology and Lattice Theory
		3.3.2. Elements of Measure Theory
		3.3.3. Lattices and Probability
		3.3.4. Fuzzy Lattices and Similarity Measures
CHAPTER 4: Lattice Algebra
	4.1. LATTICE SEMIGROUPS AND LATTICE GROUPS
	4.2. MINIMAX ALGEBRA
		4.2.1. Valuations, Metrics, and Measures
	4.3. MINIMAX MATRIX THEORY
		4.3.1. Lattice Vector Spaces
		4.3.2. Lattice Independence
		4.3.3. Bases and Dual Bases of l-Vector Spaces
	4.4. THE GEOMETRY OF S (X)
		4.4.1. Affine Structures in Rn
		4.4.2. The Shape of S(X)
CHAPTER 5: Matrix-Based Lattice Associative Memories
	5.1. HISTORICAL BACKGROUND
		5.1.1. The Classical ANN Model
	5.2. LATTICE ASSOCIATIVE MEMORIES
		5.2.1. Basic Properties of Matrix-Based LAMs
		5.2.2. Lattice Auto-Associative Memories
		5.2.3. Pattern Recall in the Presence of Noise
		5.2.4. Kernels and Random Noise
		5.2.5. Bidirectional Associative Memories
		5.2.6. Computation of Kernels
		5.2.7. Addendum
CHAPTER 6: Extreme Points of Data Sets
	6.1. RELEVANT CONCEPTS OF CONVEX SET THEORY
		6.1.1. Convex Hulls and Extremal Points
		6.1.2. Lattice Polytopes
	6.2. AFFINE SUBSETS OF EXT(P(X))
		6.2.1. Simplexes and Affine Subspaces of Rn
		6.2.2. Analysis of ext(P(X)) ˆ Rn
			6.2.2.1. The case n = 2
			6.2.2.2. The case n = 3
			6.2.2.3. The case n >= 4
CHAPTER 7: Image Unmixing and Segmentation
	7.1. SPECTRAL ENDMEMBERS AND LINEAR UNMIXING
		7.1.1. The Mathematical Basis of the WM-Method
		7.1.2. A Validation Test of the WM-Method
		7.1.3. Candidate and Final Endmembers
	7.2. AVIRIS HYPERSPECTRAL IMAGE EXAMPLES
	7.3. ENDMEMBERS AND CLUSTERING VALIDATION INDEXES
	7.4. COLOR IMAGE SEGMENTATION
		7.4.1. About Segmentation and Clustering
		7.4.2. Segmentation Results and Comparisons
CHAPTER 8: Lattice-Based Biomimetic Neural Networks
	8.1. BIOMIMETIC ARTIFICIAL NEURAL NETWORKS
		8.1.1. Biological Neurons and Their Processes
		8.1.2. Biomimetic Neurons and Dendrites
	8.2. LATTICE BIOMIMETIC NEURAL NETWORKS
		8.2.1. Simple Examples of Lattice Biomimetic Neural Networks
CHAPTER 9: Learning in Biomimetic Neural Networks
	9.1. LEARNING IN SINGLE-LAYER LBNNS
		9.1.1. Training Based on Elimination
		9.1.2. Training Based on Merging
		9.1.3. Training for Multi-Class Recognition
		9.1.4. Training Based on Dual Lattice Metrics
	9.2. MULTI-LAYER LATTICE BIOMIMETIC NEURAL NETWORKS
		9.2.1. Constructing a Multi-Layer DLAM
		9.2.2. Learning for Pattern Recognition
		9.2.3. Learning Based on Similarity Measures
Epilogue
Bibliography
Index