Recently, I am a little confused about the calculation of Euler Characteristic(EC). 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. But sometimes they also mention that the 0-order of the intrinsic volume is the Euler Characteristic of the image.

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?