$2.$ Calculate the shift $($i$.$e$.$ breathing motion$)$ between the first $1-$dimensional signal $($which we call the reference signal$)$ and all subsequent ones. To calculate the shift between any $1$dimensional signal and the reference signal, you should perform an exhaustive search of all possible shifts $($from $-10$ to $10$ in steps of $1)$ and return the shift that results in the best match between the reference signal and the shifted signal. The best match will be defined as the

$1$

Sianal $1$D profile

Figure $1$: $1-$dimensional signal acquired at two different phases of the breathing cycle $($endinspiration and end$-$expiration$).$

one with the smallest mean squared error $\backslash (($M S E$)\backslash ).$ The MSE between two signals $(\backslash ($ s $1\backslash )$ and $\backslash ($ s $2\backslash ))$ is defined as follows:

$\backslash [$

M S E$=\backslash$frac$\{1\}\{\backslash$max $-\backslash$min $+1\}\backslash$sum$\_\{$n$=\backslash$min $\}^\{\backslash$max $\}($s $1($n$)-$s $2($n$))^\{2\}$

$\backslash ]$

where $\backslash (\backslash$min $\backslash )$ and $\backslash (\backslash$max $\backslash )$ represent an area of interest in the $1-$dimensional signal which contains the liver$/$lung interface. In our case, min and max should be set to $20$ and $80.$ This enables the beginning and end of the signal $($which may contain signal from other organs$)$ to be ignored.

Hint: Read the documentation of the built$-$in MATLAB function circshift which can be used to apply a known shift to a $1-$dimensional signal.