Types of Noise
Additive noise
Additive noise is independent from image signal. The image g with nosie can be considered as the sum of ideal image f and noise n.^{[1]}
$$ g = f + n $$
Multiplicative noise
Multifplicative noise is often dependent on image signal. The relation of image and noise is^{[1]}:
$$ g = f + fn $$
Gaussian noise
Gaussian noise, named after Carl Friedrich Gauss, is statistical noise having a probability density function (PDF) equal to that of the normal distribution, aka. the Gaussian distribution. i.e. the values that the noise can take on are Gaussiandistributed.
The PDF \( p \) of a Gaussian random variable \( z \) is given by^{[2]}:
$$ p_G(z) = \frac{1}{ \sigma \sqrt{2\pi} } e^{  \frac{ (z\mu)^2 }{ 2 \sigma^2 } } $$
Saltandpepper noise
Fattail distributed or "impulsive" noise is sometimes called saltandpepper nosie or spike noise. An image containing saltandpepper noise will have dark pixels in bright regions and bright pixels in dark regions.^{[2]}
The PDF of (Bipolar) Impulse noise is given by:
$$ p(z) = \left\{ \begin{array}{ll} p_a \qquad & for \, z = a \\ p_b \qquad & for \, z = b \\ 0 \qquad & otherwise \\ \end{array} \right. $$
if b > a, graylevel b appears as a light dot in the image. Conversely, level a appears like a dark dot. If either \( p_a \) or \( p_b \) is zero, the impulse noise is called unipolar.^{[3]}
Types of Filters^{[4]}

Spatial domain

Frequency domain (Transform domain)

Integrated Spatial and Frequency Domain
Spatial domain filtering
Lowpass filter
Typical lowpass filters can be:
$$ \frac{1}{9} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $$
and:
$$ \frac{1}{8} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} $$
and a typical Gaussian filter:
$$ \frac{1}{16} \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \\ \end{bmatrix} $$
More generally, a 2 dimensional Gaussian filter in spatial domain is:
$$ G(x,y) = \frac{1}{2 \pi \sigma^2} e^{ \frac{x^2+y^2}{2 \sigma^2}} $$
Highpass filter
Basically, we can obtain a highpass filtering kernel corresponding to each of the lowpass filter kernels by subtracting the lowpass kernel from the allpass kernel.^{[5]}
A typical highpass filter can be:
$$ \frac{1}{9} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 8 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $$
and a typical Laplacian filter:
$$ \begin{bmatrix} 0.17 & 0.67 & 0.17 \\ 0.67 & 3.33 & 0.67 \\ 0.17 & 0.67 & 0.17 \\ \end{bmatrix} $$
Median filter
Replace a value with the median value of its surroundings.
$$ \begin{bmatrix} 2 & 4 & 1 \\ 3 & 8 & 1 \\ 2 & 5 & 1 \end{bmatrix} \Rightarrow \begin{bmatrix} 2 & 4 & 1 \\ 3 & 2 & 1 \\ 2 & 5 & 1 \end{bmatrix} $$
Frequency domain filtering
Lowpass filter
The Gaussian filter in frequency domain:
The derivation:
https://www.youtube.com/watch?v=iLQE0FA85Q
References
Shahriar Kaisar, et al. Salt and Pepper Noise Detection and removal by Tolerance based Selective Arithmetic Mean Filtering Technique for image restoration. (June 2008) International Journal of Computer Science and Network Security. Vol 8 No.6.
Tram Tran Nguyen Quynh, Hung Do Phi. Comparative Study of Image Denoise Algorithms