Image noise types: salt and pepper noise, random noise, Gaussian noise, Poisson noise, heteroscedastic Gaussian noise model and parameter estimation

noise type

1. Gaussian noise and Poisson noise

The following formula represents the Poisson distribution approximated by a heteroscedastic Gaussian:

https://people.csail.mit.edu/hasinoff/pubs/hasinoff-photon-2012-preprint.pdf

However, there is still an error in this under dark light (very dark, pixel value <2), which is explained in Section 2 in detail

Probability distribution and Poisson distribution, normal distribution

http://hongyitong.github.io/2016/11/13/%E4%BA%8C%E9%A1%B9%E5%88%86%E5%B8%83%E3%80%81%E6%B3%8A%E6%9D%BE%E5%88%86%E5%B8%83%E3%80%81%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83/

2. The difference between Gaussian distribution and Poisson distribution:

Key ReferencesGaussian Noise is Added, Poisson Noise is Applied

As shown below:

1. The first line is the original image pixel values ​​are 0, 1, 2, 3, 4, 5
2. The second line apply posion noise
3. The third line uses the heteroscedastic Gaussian approximation to represent the possion noise
4. Gaussian noise with fixed variance in the fourth row.

In general, heteroscedastic Gaussian can express Poisson noise more accurately, but there is a slight difference when it is particularly small.

3. Heteroscedastic Gaussian noise model (mixture of Poisson and Gaussian noise)

论文 Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data. 和 sidd:A High-Quality Denoising Dataset for Smartphone Cameras

It is mentioned that __Poisson and Gaussian noise__ are modeled together as a heteroscedastic Gaussian model: the

left side of the equation is the variance, and the right side y is the noise-free image value.
Beta1 is a noise parameter related to the signal. For example, Poisson noise
beta2 is related to Signal-independent noise parameters, such as Gaussian noise variance

How to understand the above formula?
Since the poisson distribution can be approximated as:

Then

$$\begin{gathered} P\left(lamda\right)+G(sigma) \\ =Gaussian(lamda,lamda)+Gaussian(0,sigm{a}^2) \\ =Gaussian(image,image)+Gaussian(0,sigm{a}^2) \\ =image+gaussian(0,image)+Gaussian(0,sigm{a}^2) \\ =image+Gaussian(0,beta1\ast image+beta2) \end{gathered}$$

So:
beta1 = 1, beta2 = sigma^2, image is the actual noise-free pixel value.

4. Several noise models

As shown in the figure below, they are
the original image, salt and pepper noise (50% of the pixels are polluted), random noise (50% of the pixels are polluted), Gaussian noise and Poisson noise

5. The calculation method of SNR is

mean/standard deviation

https://www.23bei.com/tool/920.html

6. Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data

This article first establishes a poisson-gaussian model:
refer to the second section of this article.
This approximation is usually very accurate. For small photon counts, photon noise is usually dominated by other signal-independent noise sources, and for larger photon counts, the central limit theorem ensures that the Poisson distribution is close to a Gaussian distribution.

Then estimate the beta1 and beta2 parameters:
the estimation method must be to fit a straight line through the mean and variance.

Several patches (h=100pixel, w100pixel) with value = np.arange(0, 256, 5), and then add Poisson noise and Gaussian noise.
Calculate the mean value of each patch as the abscissa, and
the variance of each patch as the ordinate, and
draw the following figure.
Due to the clip, the variance close to 0 and close to 255 is inaccurate (it will drop), so the points in the [50, 200] part are used to fit the straight line to obtain the straight line coefficients, namely beta1, beta2.

https://blog.csdn.net/qq_34901691/article/details/119304917