next up previous
Next: 8.4 Projection through Group Up: 8 Uncertain Transformations Previous: 8.2 Transforming

8.3 Distribution of Inverse

Similarly, the distribution of the inverse transformation is easily computed:

$\displaystyle \tilde{z}$ $\displaystyle \equiv$ $\displaystyle \tilde{x}^{-1}$ (225)
  $\displaystyle =$ $\displaystyle x^{-1}\cdot\exp\left(-\boldsymbol{\delta}\right)$ (226)
  $\displaystyle =$ $\displaystyle \exp\left(-\mathrm{Adj}_{x^{-1}}\boldsymbol{\delta}\right)\cdot x^{-1}$ (227)
$\displaystyle \tilde{z}$ $\displaystyle \in$ $\displaystyle \mathcal{N}\left(x^{-1};\:\mathrm{Adj}_{x^{-1}}\cdot\boldsymbol{\Sigma}\cdot\mathbf{\mathrm{Adj}}_{x^{-1}}^{T}\right)$ (228)


Ethan Eade 2012-02-16