# Noise modeling and Probability Theory

Noise modeling refers to modeling and describing noises using a mathematical way. In physics, telecommunication, imaging processing and other fields, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. In imaging process, the sources of noise arise from the disturbance during image acquisition, digitization, transimission and storage.

From physics point of view, the term noise has the following meanings:

1. An undesired disturbance within the frequency band of interest; the summation of unwanted or disturbing energy introduced into a communications system from man-made and natural sources.

2. A disturbance that affects a signal and that may distort the information carried by the signal.

3. Random variations of one or more characteristics of any entity such as voltage, current, or data.

4. A random signal of known statistical properties of amplitude, distribution, and spectral density.

Noise has long been studied. People analyze its property, type, influence and what can be done about it. Most of the research is done in mathematics and close related to Probability Theory.

## Some noise models

Although in many cases noise is dependent upon physical properties of device,media,etc, people often consider noise as random variables, characterized by a probability density function (PDF). The following are some noise models and their PDFs which are commonly used in signal/imaging processing.

Uniform noise

The Uniform PDF is given by:

The plot of uniform PDF

The mean of this density function is given by

and its variance by

Uniform noise is not often encountered in real-world imaging systems, but provides a useful comparison with Gaussian noise. The linear average is a comparatively poor estimator for the mean of a uniform distribution. This implies that nonlinear filters should be better at removing uniform noise than Gaussian noise.

Gaussian noise

The PDF of a Gaussian random variable x, is given by

x is the gray level, µ is the mean, σ is the standard deviation and σ2 is the variance.

The plot of Gaussian PDF is like:

Gaussian noise is defined as noise with a Gaussian amplitude distribution[1].

The Gaussian distribution has an important property: to estimate the mean of a stationary Gaussian random variable, one can't do any better than the linear average. This makes Gaussian noise a worst-case scenario for nonlinear image restoration filters, in the sense that the improvement over linear filters is least for Gaussian noise. To improve on linear filtering results, nonlinear filters can exploit only the non-Gaussianity of the signal distribution.

Rayleigh noise

The PDF of Rayleigh noise is given by

The plot of Rayleigh PDF is like:

The mean of this density function is given by

and variance is given by

Exponential noise

The PDF of exponential noise is given by

The plot of exponential PDF is like:

The mean of this density function is given by

and variance is given by

Impulse noise

The PDF of impulse noise (also known as salt-and-pepper noise) is given by

The plot of its PDF is like:

Impulse noise consists of random occurrences of energy spikes having random amplitude and and spectral content.

Erlang(Gamma) noise

The PDF of Erlang noise is given by

The plot of its PDF is like:

The mean of this density function is given by

and variance is given by

## Noise sources in image processing

A large portion of images are formed from light using modern electro-optics. In particular the use of modern, charge-coupled device (CCD) cameras where photons produce electrons that are commonly referred to as photoelectrons. Nevertheless, most of the observations about noise and its various sources hold equally well for other imaging modalities. The major sources of noise during image acquisition from electro-optics devices are photon noise, thermal noise, on-chip electronic noise, amplifier noise, and quantization noise.