Additive noise is quite common. But what is the difference between additive and multiplicative noise? What are the factors that make them multiplicative noise? Can anyone suggest literature or present mathematical representation and explanation on multiplicative noise?
Let's consider a variable x(t) which follows a stochastic differential equation. If the corresponding random term in the equation does not depend on the state of the system (i.e. on x(t)), we call it additive noise. On the other hand, if the random term depends on the state of the system x(t), then the noise term is called multiplicative.
To have a better understanding of multiplicative nois, you may have a look at speckle noise in image processing - especially for Synthetic Aperture (SAR) images.
You may find some information in the following papers
- Swami, A., "Multiplicative noise models: parameter estimation using cumulants," Spectrum Estimation and Modeling, 1990., Fifth ASSP Workshop on , vol., no., pp.212,216, 10-12 Oct. 1990
- Sancho, José M.; San Miguel Ruibal, Maximino; Katz, S. L.; Gunton, J. D. "Analytical and numerical studies of multiplicative noises" The American Physical Society 1982
- Gilles Aubert, Jean-François Aujol "A Nonconvex Model to Remove Multiplicative Noise" Scale Space and Variational Methods in Computer Vision Lecture Notes in Computer Science Volume 4485, 2007, pp 68-79
Bo Chen, Jin-Lin Cai, Wen-Sheng Chen, and Yan Li "A Multiplicative Noise Removal Approach Based on Partial Differential Equation Model" Mathematical Problems in Engineering 2012
R.L. Stratonovich "Topics In the Theory of Random Noise", Volume 2
Dear iwan Ninglekhu, The following article may provide some help.
"Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization"
The article is accessible via link given below:
http://arxiv.org/pdf/0912.1845.pdf
"What is the difference between additive noise and multiplicative noise? "
The LOG function. It transforms multiplicative noise in additive noise.
Let's consider a variable x(t) which follows a stochastic differential equation. If the corresponding random term in the equation does not depend on the state of the system (i.e. on x(t)), we call it additive noise. On the other hand, if the random term depends on the state of the system x(t), then the noise term is called multiplicative.
To have a better understanding of multiplicative noise, let's consider motion of a Brownian particle. In absence of temperature gradient, thermal noise and subsequently diffusion coefficient of particle don't depend on position of particle, so noise is called additive noise. Now let's add temperature gradient in the space. Particle will sweep more space in time in positions with higher temperature than positions with lower temperature.So thermal noise and diffusion coefficient of particle depend on position of particle and in this case noise is called multiplicative noise.
To have a better understanding of speckle noise,
Here a good book "Processing of SAR images"
I think there is no difference. If you look closer at a 2D SDE it's easy to realize this
dX = f(X,Y)*dt + dR
dY = g(X,Y)*dt + dS
R and S are two stochastic processes, so result both X and Y , and f and g are two non-linear functions. Indeed let's suppose you have the following SDE with a multiplicative noise
dX = S*h(X)*dt + dR
where S is a stochastic process, one can recast this equation to the 2D case
dX = m(X,Y)*dt + dR
dY = dS
where m(x,y)=y*h(x), so the trick is to increase the dimensionality of the SDE system and you're back to an additive noise.
Best,
F
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