## Mathematical Foundations

### Metric
- Function `$ d:K\times K \rightarrow \mathbb{R} $`
- Characteristics:
	- Non-negativity: `$ d(x,y)\geq 0 $`
	- Identity: `$ x=y \Rightarrow d(x,y) = 0 $`
	- Symmetry: `$ d(x,y) = d(y,x) $`
	- Triangular inequality: `$ d(x,z)\leq d(x,y)+d(y,z) $`

### Distance Functions
#### Minkowski Distance
- Generalization of Manhatten/Euclidean distance
- `$ x, y \in \mathbb{R}^n $`
- Parameter `$ r $`
- `$ d_r(x, y) = (\sum_{1\leq i\leq n}|x_i-y_i|^r)^\frac{1}{r} $`
- `$ r=1 $`: Manhattan distance
- `$ r=2 $`: Euclidean distance

#### Mahalanobis distance
- Distance between vectors
- Often used in statistics
- Covariance matrix `$ \Sigma $`
- `$ D(x,y)=\sqrt{(x-y)^T * \Sigma^{-1} * (x-y)} $`

### Transformation
- Idea:
	- Transform from one domain into another (keeping all information)
	- *See other things*
- Properties
	- Reversible
	- Information preserving (no loss)
- Using input signal
	- Discrete function

### Fourier Transformation
- High-level feature
- Transformation from original domain to *frequency domain*
- Original domain 
	- Images: Position space (image row)
	- Audio: Time domain
- Decompose periodic function into sum of simple oscillating functions (sin, cos)
- Lower frequency contain most information
- Discrete Fourier Transform
	- For discrete functions (i.e., sequence of real values)
	- Sequence of `$ n $` real numbers `$ x_0,...,x_{n-1} $`
	- Transform to sequence of coefficients
		- `$ a_0,...,a_{n/2} $` (for cos)
		- `$ b_0,...,b_{n/2} $` (for sin)
		- `$ a_k = \sum_{j=0}^{n/2} f(j)cos(kj) $`
	- Coefficients are amplitudes of sine and cosine waves (project signal into sine/cosive waves)
	- `$ x_k = \sum_{i=0}^{n/2}a_l cos(2\pi\frac{lk}{n}) + \sum_{i=0}^{n/2}b_l sin(2\pi\frac{lk}{n}) $`