Dear Huashuai,
On 17/10/2019 18:21, Huashuai wrote:
> Recently, I am a little confused about the calculation of Euler
> Characteristic(EC).
I hope I'll get this right and clear the confusion. Have a look at e.g.:
http://www.math.mcgill.ca/keith/noniso/noniso.pdf
> Most of papers think that the expected value of the Euler
> characteristic of this excursion set is given by the Gaussian
> Kinematic Formula, the sum of intrinsic volume * EC density.
True. That's equation 7: E[φ{s∈S:T(s)≥t}]
> But sometimes they also mention that the 0-order of the intrinsic
> volume is the Euler Characteristic of the image.
True. That's equation 9: μ0(S) = φ(S)
See also equation 33 and the mention of the Gauss-Bonnet theorem.
> So my question here is what the difference between them is. I have
> simulated white Gaussian noise, and compare the results. The results
> show they are not the same. Which one should we use to calculate EC?
So the difference is that the former is about the Euler characteristic
of the excursion set (the thresholded map {s∈S:T(s)≥t}) while the latter
is the Euler characteristic of the entire search volume (S) and is used
in the computation of the former via the Gaussian Kinematic formula.
Best regards,
Guillaume.
--
Guillaume Flandin, PhD
Wellcome Centre for Human Neuroimaging
UCL Queen Square Institute of Neurology
London WC1N 3BG
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