Table 1.

Iterative Gauss-Newton algorithm used to estimate motion for the one-dimensional Butterfly navigator in the x direction.

	Weighted Gauss-Newton Motion Estimate
Goal:	${\text{min}}_{(d_x,c_{\varphi })}{\sum }_l{w}^2\left[l\right]{\mid s_0\left[l\right]-s\left[l\right]e^{-i2\pi (k_x\left[l\right]d_x+c_{\varphi })}\mid }^2$
Inputs:	kx[l] – x navigator trajectory of length L
	s0[l] – reference signal evaluated at kx[l]
	s[l] – navigator signal evaluated at kx[l]
	w[l] – weights
	∊1 – stopping criteria 1
	∊2 – stopping criteria 2
Outputs:	cφ – bulk phase difference
	dx – motion estimate as linear translation
Algorithm:	cφ = 0, dx = 0, sj [l] = s[l]	Initialization
	do{
	Construct r: r[l] = w[l] × (s0[l] − sj[l])	Weighted residual
	Construct a1: a1[l] = w[l] × i2πkx[l]sj[l]	$\approx \frac{\delta }{\delta d_x}r$
	Construct a2: a2[l] = w[l] × i2πsj [l]	$\approx \frac{\delta }{\delta c_{\varphi }}r$
	$\text{Solve}:\left[\begin{array}{c}\hfill \mid \hfill \\ \hfill r\hfill \\ \hfill \mid \hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \mid \hfill & \hfill \mid \hfill \\ \hfill a_1\hfill & \hfill a_2\hfill \\ \hfill \mid \hfill & \hfill \mid \hfill \end{array}\right]\left[\begin{array}{c}\hfill d_j\hfill \\ \hfill c_j\hfill \end{array}\right]$	Using weighted least-squares
	dx = dx + dj, cφ = cφ + cj	Update
	sj [l] = s[l]e−i2π(kx[l]dx+cφ)
	}while |dj| > ∊1 and |cj| > ∊2