Название: Muography
Автор: Группа авторов
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9781119723066
isbn:
We now consider the meaning of the covariance matrices used in equations 2.3 and 2.4 in the field of probability theory. If
where V(x i ) is the deviation of x i and Cov(x i , x j ) = E[(x i − E[x i ])(x j − E[x j ])] (E[x i ] is the expected value of the probability variable x i ) and qualitatively reflects the correlation between x i and x j .
When solving for
where σ ρ is the density contrast deviation, l(i, j) is the distance between the i th and j th voxel, and L 0 is the correlation length. Equation 2.6 assumes that the internal density is continuous on a spatial scale L 0, and that the density contrast is typically within σ ρ . However, equation 2.6 is just one possible example, and it is possible to assume different constraints in the model based on the expected structure. In this case, ρ 0, σ ρ , and L 0 are a priori parameters.
We now consider the matrix elements of
(2.7)
where
In equations 2.1 and 2.2, we described d i and
The elements of matrix A ij are different between equations 2.1 and 2.8. In equation 2.1, the elements of Aij can be calculated from the topology, size, and shape of the voxels that are defined. To calculate the elements of matrix