Fuzzy Sets Methods in Image Processing and Understanding: Medical Imaging Applications

Acknowledgments
Contents
1 Introduction
	1.1 Fuzzy Sets and Image Understanding Under Imprecision
		1.1.1 Sources of Imprecision
		1.1.2 Advantages and Usefulness of Fuzzy Sets
		1.1.3 Semantic Gap
		1.1.4 A Short Review of Existing Books
	1.2 Representations
	1.3 Low Level—Clustering, Enhancement, Filtering, Edge Detection
	1.4 Intermediate Level
	1.5 Higher Level
		1.5.1 Representations of Structural Information
		1.5.2 Fusion
		1.5.3 Scene Understanding
	1.6 Emerging Topics
		1.6.1 Mining and Retrieval
		1.6.2 Towards Bipolarity
		1.6.3 Towards More Interactions Between Knowledge and Image Information
		1.6.4 Deep Neuro-Fuzzy Systems
	References
2 Preliminaries
	2.1 Imprecision in Images and Related Knowledge
	2.2 Basic Definitions of Fuzzy Sets Theory
		2.2.1 Fuzzy Sets
		2.2.2 Set Theoretical Operations: Original Definitions of L. Zadeh
		2.2.3 Structure and Types of Fuzzy Sets
		2.2.4 α-Cuts
		2.2.5 Cardinality
		2.2.6 Convexity
		2.2.7 Fuzzy Number
	2.3 Main Operators on Fuzzy Sets
		2.3.1 Fuzzy Complementation
		2.3.2 Triangular Norms and Conorms
		2.3.3 Mean Operators
		2.3.4 Symmetric Sums
		2.3.5 Adaptive Operators
		2.3.6 Logical Connectives
	2.4 Linguistic Variable
		2.4.1 Definition
		2.4.2 Example of Linguistic Variable
		2.4.3 Modifiers
	2.5 Translating a Crisp Operation into a Fuzzy Operation
		2.5.1 Extension Principle
			Definition
			Application to the Compatibility of Two Fuzzy Sets
			Application to Fuzzy Numbers
		2.5.2 Combination of Results on α-Cuts
			Reconstruction from α-Cuts
			Extension Principle Based on α-Cuts
		2.5.3 Translating Binary Terms into Functional Ones
		2.5.4 Comparison
	2.6 Summary of the Main Notations
	References
3 Fuzzy Spatial Objects
	3.1 Fuzzy Sets in the Spatial Domain
	3.2 Set Theoretical Operations
		3.2.1 Degree of Intersection
			Crisp Case
			Direct Extension
			Introducing the Volume of the Overlapping Domain
			Properties
			Application to the Non-contradiction Principle
		3.2.2 Degree of Union and Covering
		3.2.3 Degree of Inclusion
			Inclusion from Other Set Operations
			Inclusion from Fuzzy Implication
			Other Axiomatic Definitions for the Fuzzy Inclusion
			Inclusion and Fuzzy Entropy
		3.2.4 Degree of Equality
	3.3 Topology: Neighborhood, Boundary, and Connectedness of a Fuzzy Set
		3.3.1 Fuzzy Neighborhood
		3.3.2 Boundary of a Fuzzy Set
		3.3.3 Connectedness
	3.4 Fuzzy Geometry
		3.4.1 Fuzzy Points and Lines
		3.4.2 Fuzzy Rectangles and Fuzzy Convex Polygons
		3.4.3 Fuzzy Disks
		3.4.4 Fuzzy Geometrical Measures
			Area of a Fuzzy Set
			Perimeter of a Fuzzy Set
			Compactness of a Fuzzy Set
			Height, Width, and Diameter of a Fuzzy Set
			Intersection and Parallelism Between Fuzzy Lines
			Geometrical Measures as Fuzzy Numbers
	3.5 Fuzzy Geometric Transformations
		3.5.1 Transformation of a Fuzzy Set by a Crisp Operation
		3.5.2 Transformation of a Fuzzy Set by a Fuzzy Operation
	References
4 Fuzzy Mathematical Morphology
	4.1 Lattice Structure of ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper F) /StPNE pdfmark [/StBMC pdfmarkFps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	4.2 Algebraic Operators
	4.3 Structuring Elements and Basic Morphological Operators
	4.4 An Example in Medical Imaging
	4.5 Towards a Fuzzy Mathematical Morphology Toolbox
		4.5.1 Neighborhood and Boundary from Fuzzy Dilation and Erosion
		4.5.2 Fuzzy Morphological Filters
		4.5.3 Conditioning and Fuzzy Geodesic Operators
		4.5.4 Fuzzy Skeleton and Skeleton by Influence Zones
			Distance-Based Approaches
			Morphological Approaches to Compute the Centers of Maximal Balls
			Morphological Thinning
			Fuzzy Skeleton of Influence Zones
			Discussion
		4.5.5 Fuzzy Median, Application to Interpolation Between Fuzzy Sets
		4.5.6 Extensions
	References
5 Fusion
	5.1 Definitions
	5.2 Fusion Systems and Architectures Types
	5.3 Fuzzy Modeling in Fusion
	5.4 Defining and Estimating Membership Functions
	5.5 Fuzzy Combination
	5.6 Decision in Fuzzy Fusion
	5.7 Exploiting Spatial Information
	5.8 Illustrative Examples
	References
6 Spatial Relations
	6.1 Set Theoretical and Topological Relations
		6.1.1 Adjacency
		6.1.2 Fuzzy Region Connection Calculus
	6.2 Distances Between Image Regions or Objects
		6.2.1 Representations
		6.2.2 Comparison of Membership Functions
		6.2.3 Combination of Spatial and Membership Comparisons
		6.2.4 Discussion and Examples
	6.3 Fuzzy Hamming Distance
	6.4 Directional Relations
		6.4.1 Fuzzy Relations Describing Relative Position
		6.4.2 Centroid Method
		6.4.3 Histogram of Angles: Compatibility Method
		6.4.4 Aggregation Method
		6.4.5 Histogram of Forces
		6.4.6 Projection Based Approach
		6.4.7 Morphological Approach
		6.4.8 Discussion and Examples
	6.5 Complex Relations: Surround, Between, Along, Across, Parallel, Aligned
		6.5.1 Surround
		6.5.2 Between
		6.5.3 Across
		6.5.4 Along
		6.5.5 Aligned
		6.5.6 Parallel
	6.6 Fuzzy Perceptual Organization for Image Understanding
		6.6.1 Fuzzy Grouping Operator to Produce Straight LineSegments
		6.6.2 Discrimination: Overlap of Two Segments
		6.6.3 Obtaining Junctions
		6.6.4 Obtaining Symmetric Line Structures
			Symmetry of Non-parallel Line Segments
			Symmetry of Parallel Line Structures
		6.6.5 Obtaining Curves and Closed Regions
	6.7 Comparison of Spatial Relations
		6.7.1 Relations Represented as Numbers or Intervals
		6.7.2 Relations Represented as Distributions
		6.7.3 Relations Represented as Spatial Fuzzy Sets
	References
7 Fuzzy Sets and Machine Learning
	7.1 Fuzzy IF-THEN Rules
	7.2 Unsupervised Learning
		7.2.1 Fuzzy Clustering
		7.2.2 Spatial Information and Bias
	7.3 Fuzzy Sets and Connectionist Approaches
		7.3.1 Conventional 2D Hopfield Neural Network
		7.3.2 Fuzzy Sets and Deep Learning
	References
8 Structural and Linguistic Representations
	8.1 Fuzzy Representation of Image Information and of Related Knowledge
		8.1.1 Image Features
		8.1.2 Knowledge and Semantics
		8.1.3 Semantic Gap
	8.2 Linguistic Representations
		8.2.1 Description of Some Properties or Characteristics
		8.2.2 Quantifiers
		8.2.3 Associating Linguistic Representations and the Spatial Domain
	8.3 Knowledge-Based Systems
	8.4 Fuzzy Graphs and Hypergraphs
	8.5 Fuzzy Logics and Fuzzy Rules
	8.6 Ontologies
	8.7 Fuzzy Decision Trees
	8.8 Fuzzy Association Rules
	8.9 Fuzzy Formal Concept Analysis
	References
9 Structural and Linguistic Reasoning for Image Understanding
	9.1 From Linguistic Descriptions to Image Understanding
		9.1.1 Representations of Structural Information
		9.1.2 Fusion
		9.1.3 Scene Understanding
	9.2 From Image Analysis to Image Content Descriptions
	9.3 A Few Examples in Medical Image Understanding
		9.3.1 Interpretation as Graph Reasoning
		9.3.2 Interpretation as Constraint Satisfaction Problem
		9.3.3 Recognition Based on Ontological Reasoning
		9.3.4 Interpretation as Abductive Reasoning
		9.3.5 Deriving Linguistic Descriptions
	9.4 Interpretability and Explainability
	References
Index