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Combining the Errors

To get meaningful plots, where error bars faithfully reflect on truth, we ought to better understand our error sources. They appear to be independent, but this observation does not simplify matters.

Let us look at the plots vertically, taking one value of mean pixel displacement at the time. Each value of displacement, donated by $d$, is produced by warping a set of aligned images, i.e. images where there is no inherent displacement.

Let us define $\sigma_{mi}$ to be the predicted error in the estimate of $m$ for a warp instance $i:(1..N)$. We can then obtain the mean

$$\overline{\sigma_{m}}=\begin{array}{c} \\ \sum\\ i\end{array}\frac{\sigma_{mi}}{N}$$ (1)

and the standard error is thus

$$SE_{\sigma_{m}}=\frac{SD(\sigma_{mi})}{\sqrt{N-1}}.$$ (2)

As for the mean of the measurement $m$ for the given displacement value $m$,

$$\overline{m_{d}}=\begin{array}{c} \\ \sum\\ i\end{array}m_{i}/N$$ (3)

and the corresponding standard error

$$SE_{\overline{m}}=\frac{SD(m_{i})}{\sqrt{N-1}}.$$ (4)