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I'm facing the problem of visual tracking in computer vision. I have some observation (image blobs by background subtraction) produced by some moving object, and the task is to infer the state (position, velocity) of the object given the observation. I assume that each state depends only on the immediate predecessor (Markov assumption), ruled by a temporal model (the dynamics) that describes how a state $X_t$ evolve in $X_{t+1}$, and that the measurement model do not depends on time, so that $y_t$ do not depends on $X_{t-1}$ but only on $X_t$.

Everywhere in the web I have found that this problem can be described by this graphical model:

enter image description here

For inferring the posterior distribution, after receiving new measurements, a Kalman filter or Particle filter is used.

But anywhere I explicitly read that this is an Hidden Markov Model.

Is this a HMM? Why? Why not?

nkint
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2 Answers2

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In short: you don't know yet. The plate diagram for a State Space Model and a Hidden Markov Model are the same (and they are what you have presented). This is deliberate.

The distinction between the two models is in the nature of X. If X is multinomial with K states then this is an K-state HMM of order 1. If X is (possibly multivariate and) continuous, then it is a SSM. If X also happens to be Normal then it is a linear Normal SSM, perhaps the most common variety, for which the Kalman filter and RT smoother was designed.

For typological purposes it doesn't apparently matter what Y is or the form of the conditional distributions, although clearly these strongly affect how easy it is to filter, smooth and estimate parameters for.

conjugateprior
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  • An SSM is an HMM: nothing in HMM says the states (or outputs) have to be discrete (although I admit they typically are). – daknowles Feb 06 '14 at 03:42
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    Nope. See line 1 of the original paper http://projecteuclid.org/download/pdf_1/euclid.aoms/1177699147 or section two of http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.ps.gz an old Zoubin back issue. – conjugateprior Feb 08 '14 at 21:02
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If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.

It looks like an HMM to me.

daknowles
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