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Mathematics
Morphology
1. Introduction to Mathematical Morphology
2. Foundational Mathematical Concepts
3. Structuring Elements
4. Primary Morphological Operations
5. Compound Morphological Operations
6. Morphological Filtering and Enhancement
7. Pattern Detection and Shape Analysis
8. Grayscale Morphology
9. Advanced Morphological Concepts
10. Theoretical Frameworks and Extensions
11. Implementation and Computational Aspects
12. Practical Applications
4.
Primary Morphological Operations
4.1.
Dilation
4.1.1.
Mathematical Definition
4.1.1.1.
Minkowski Addition Formulation
4.1.1.2.
Set-Theoretic Definition
4.1.2.
Geometric Interpretation
4.1.2.1.
Expansion of Objects
4.1.2.2.
Growth Process
4.1.3.
Properties
4.1.3.1.
Increasingness
4.1.3.2.
Translation Invariance
4.1.3.3.
Associativity
4.1.3.4.
Commutativity with Respect to Structuring Element
4.1.4.
Effects on Image Features
4.1.4.1.
Bridging Small Gaps
4.1.4.2.
Expanding Object Boundaries
4.1.4.3.
Filling Small Holes
4.2.
Erosion
4.2.1.
Mathematical Definition
4.2.1.1.
Minkowski Subtraction Formulation
4.2.1.2.
Set-Theoretic Definition
4.2.2.
Geometric Interpretation
4.2.2.1.
Shrinking of Objects
4.2.2.2.
Removal Process
4.2.3.
Properties
4.2.3.1.
Anti-Extensivity
4.2.3.2.
Translation Invariance
4.2.3.3.
Associativity
4.2.4.
Effects on Image Features
4.2.4.1.
Removing Small Objects
4.2.4.2.
Shrinking Object Boundaries
4.2.4.3.
Separating Connected Objects
4.3.
Duality Relationship
4.3.1.
Mathematical Duality
4.3.2.
Complementarity Properties
4.3.3.
De Morgan's Laws in Morphology
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5. Compound Morphological Operations