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I'm building a particle filter and have two measurements, say one from a camera (C) and one from a GPS (G). Each measurement has a PDF associated with it defined by a mean and sigma. It turns out that the sigma for C is very small while the one for G is rather large.

When I run a measurement update step of the particle filter, I evaluate PDF_G(measurement_g) and PDF_C(measurement_c) for each particle in order to determine which particle's pose most closely matches the measurements. Since sigma_C << sigma_G, the PDF_C(measurement_c) term always dominates the weighting since the result is always really large or small (as opposed to the much wider Gaussian that describes PDF_G). This is because I am multiplying the two PDF results together but this seems incorrect since the units do not match and I haven't done any normalization.

Is this the correct way to combine the PDF results from two independent measurements when trying to determine which hypothesis (particle) most closely matches the measurements? Thanks!

CPayne
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1 Answers1

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If the two functions are probability density functions, and the measurements are independent, then by definition:

  • PDF_GC(measurement_g, measurement_c) = PDF_G(measurement_g)*PDF_C(measurement_c)

You state that you have not performed any normalisation, so it is plausible that your functions are actually not probability density functions.   Normalise!

$$f_G(g) = \dfrac{\exp(-(g-\mu_{\small G})^2/2\sigma^2_{\small G})}{\sigma_{\small G}\sqrt{2\pi}}$$

PDF_G(measure_g) = exp(-(measure_g - mean_g)^2/(2*sigma_g^2)) / (sigma_g*sqrt(2*pi))

The other issue is the assumption that the GPS and camera measurements are independent.   Depending on what are being measured this may not be the case.

Graham Kemp
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