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Let's say we have an image/signal with some noise in it. When would it be beneficial to model the signal as an outcome of a stochastic process? More specifically:

How significant would noise have to be in for example an image for it to be beneficial to estimate the power spectrum density instead of taking the fourier transform directly?

And would it matter what kind of noise that was present in the image/signal, if it is additive or multiplicative? I don't think I'm seeing how a stochastic process could be a beneficial model for a signal yet.

I hope these weren't stupid questions. I'd be really grateful for any answers to them!

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When would it be beneficial to model the signal as an outcome of a stochastic process?

When the process that generates the image/signal has a strong element of chance.

This is not related to noise necessarily. Sometimes, a deterministic model for the behaviour of a quantity simply does not exist.

For example, modern supermarkets have barcode readers that go "beep" when customers scan an item. Suppose now that we want to create a model of that "beep" sequence from which we can derive the rate at which customers are bying products from the supermarket.

What is a sequence of "beep"s? It is Beep-Random Pause-Beep-Random Pause-Beep and so on.

There is no deterministic equation for the inter-beep pauses. Deterministic models of the form $y = \alpha t + \beta x \dots$ return the same output $y$ for the same parameters $\alpha, \beta, \dots$.

But in the case of the supermarket, we have no idea about how customers arrive at the barcode scanner to scan their baskets. They just arrive...at random.

So, in this case, the inter-beep random interval is governed by a Poisson distribution which only requires a rate parameter. This rate denotes average rate of events per unit of time. In our case, the rate at which customers scan their products.

As this model is stochastic, it generates a different series of interbeep intervals everytime that you "run it" because there are infinite ways by which you can generate a sequence of events at some average rate $\lambda$...at random.

The exact same mathematics govern radioactive decay, the output of a suitable detector (that detects byproducts of the decay) and after a few more parameters are taken into consideration, the progressive formation of a gamma camera image.

Now, a stochastic process does not have to produce discrete events at some rate $\lambda$ only. It might represent the random fluctuation of some other quantity. What is the birth rate of a society at a given year?.....I don't know....But let's say that it is a random variable with characteristics governed by some probability distribution. Random variables represent quantities that either are inherently uncertain (e.g. radioactive decay) or, our understanding during the construction of the model is uncertain.

With this in mind, you might find exploring the rest of the examples provided in Stanley Pawlukiewicz's post really entertaining too. Drunken sailor walks are everywhere around us and they are the starting point of a lot of wonderful generative art and graphics effects today.

How significant would noise have to be in for example an image for it to be beneficial to estimate the power spectrum density instead of taking the fourier transform directly?

The choice between obtaining the Power Spectral Density (PSD) or Discrete Fourier Transform (DFT) is not exactly based on the amount of noise present in the image but rather, what are you trying to model.

And would it matter what kind of noise that was present in the image/signal, if it is additive or multiplicative?

Yes it would. It is the difference between an additive noise component, like background crowd murmur in a shopping centre when you are trying to hear a person talking to you and a fault at the cable of the microphone recording the person. In the first case, you record two sources simultaneously and both are present. In the second case, when one is present the other is not. If your microphone cable is faulty, it kind of makes and does not make contact so your rBZZZord is fuZZZBZ f artifacBZZZSPAK!ZZZ ike thiZZZZ. You either have the person's voice or not. Well, when do I have the person's voice? I don't know...what is the rate at which the microphone connection fails at? Another manifestation of multiplicative noise is wow and flutter in record players. That is, the variation that is inserted in the recording because of the variable rate of rotation of the disc platter. There, the spectrum of the playback is the modulation (or product) of the song's spectrum with the variation of the rate of turn of the disc platter. What could that be? I don't know, but let's say that it's a random variable that is $0$ (that is, bang on the right speed), plus or minus $\sigma$ turns per minute.

In fact, if we were talking about disc players that use an elastic middle wheel to couple the motor to the disc platter, this "model" would directly represent the variation in the radius of the elastic wheel that progressively gets worn out and inserts this "modulation" of the player's speed.

I am mentioning this last part more to show that there is a direct link between reality and how you choose to represent it in a model.

So, the real question here is "What are you trying to model"? And one of the best places to look for experience in creating models of the real world is Automatic Control.

Hope this helps.

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  • $\begingroup$ Thank you, beautiful answer. I think I'm a bit confused as to what the PSD actually models. Would it be the average frequency spectrum over many outcomes from the stochastic process? Let's say we have a signal with noise. How would taking the PSD differ from the DFT, they would both describe the signal with noise, right? $\endgroup$ – epe2308 Mar 6 at 9:41
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    $\begingroup$ @epe2308 You are welcome, I am glad if this was useful and a bit confused that Stanley Pawlukiewicz removed his answer, I hope it comes back with a few more words for each example he was providing. Regarding the question in your comment, what if the DFT cannot be defined?. In general, I take PSD to be the DFT of the autocorrelation rather than the "Power Spectrum" to be able to differentiate the two (see also link provided about PSD). $\endgroup$ – A_A Mar 6 at 13:48

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