Fundamentals of Image Processing and Transformations

📋 Course Outline

1. Mathematical Notation
2. Image Foundations
3. Discrete Images
4. Coordinate System
5. Image Visualization
6. Discretization Methods
7. Image Interpolation
8. Image Transformations
9. Linear Transformations
10. Geometric Transformations
11. Affine Transformations
12. Image Characteristics

📖 1. Mathematical Notation

🔑 Key Concepts & Definitions

• Vectors and Matrices: Small boldface letters (e.g., v) denote vectors; capital boldface letters (e.g., A) denote matrices. Vectors can be transposed (vᵗ) to switch between row and column forms.

• Complex Conjugation: The symbol * indicates complex conjugation, performed component-wise for vectors and matrices.

• Hadamard Product and Division: Element-wise multiplication ($\circ$) and division ($/$) between vectors or matrices, applied element-by-element.

• Image Function: Images are modeled as functions f(x) with domain Ω (spatial coordinates) and range in color space, possibly continuous or discrete, multi-dimensional (e.g., 2D, 3D+t).

• Coordinate Transformations: Mapping between physical and normalized coordinates using scaling and offset parameters, enabling switching between image and real-world sizes.

• Discrete vs. Continuous Images: Discrete images are represented on index sets (pixels/voxels), while continuous images are functions over real-valued domains.