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I am reading a paper on measuring respiratory patterns from video data. In defining the model, the authors formulate the problem mathematically as:

$x_i(t)=h_i(t) \ast g(t) + n_i(t) $

Where $n_i(t)$ is a zero mean noise. My question is why do we model the pixel data as having zero mean noise? What prevents it from having a mean?

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The brutally honest answer here is: The noise is considered zero-mean because that's what the author decided to do. Without looking deeper into the signal model employed, it's impossible to answer.

However, for many systems this makes a lot of sense physically, since the processes leading to a noise realization are very often zero-mean in nature. For example, the thermal noise over any resistor should be zero-mean, simply because, well, that's how these random fluctuations work.

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  • $\begingroup$ Ah so looking for this answer I should just look at the physical nature of whatever signal is there. In the case of pixel data I suppose the writer models any noise as zero mean because it just makes sense. $\endgroup$ – yodama Feb 28 '17 at 15:47
  • $\begingroup$ An interesting note here, the noise model for an imaging sensor can also be defined in terms of Poisson distribution, where the noise level increases as $\sqrt{n}$, with $n$ being the signal intensity (in counts). $\endgroup$ – PhilMacKay Feb 28 '17 at 16:43
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To add on @MarcusMüller: in image processing, a constant pixel value shift is often not perceived (as a global scaling), leaving aside saturation issues. In video sequences, illumination may change from one frame to the other. This may induce the noise to have a mean (as a scaling). Also, quantization on integers can change the mean. So often the presence of this shift in noise in undecidable, and related images are sometimes corrected to have the same central tendency (128, mean, median) before further processing. Common image analysis tools and measures are shift insensitive (edge detection, textural analysis, SSIM).

if the subsequent analysis should reveal patterns, fluctuations, assuming zero mean (no DC component) is thus common, unless it is important to use a mean as a reference.

More important in image are background disturbances, where the average illumination (slowly ) varies non-uniformly across the image (like shadow effects), as in the following (darker on the bottom)

non-uniform background

Here, oner may ask what is the true mean: the dark gray on the bottom, or the light gray on the top (or something else)?

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